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User:Ilya Gutkovskiy
Location: Tula, Russia.
Born in 1982.
Graduated from Tula State University (Faculty of Technology) and Tula State Pedagogical University (Faculty of Psychology).
I work in the field of metal processing industry.
I am interested in elementary and analytic number theory. In addition to mathematics I am fond of poetry. My books
OEIS sequences which I submitted
OEIS sequences which I submitted and/or edited
Contents
- 1 Generalization of generating functions
- 1.1 The ordinary generating function for the alternating sum of k-gonal numbers
- 1.2 The ordinary generating function for the alternating sum of centered k-gonal numbers
- 1.3 The ordinary generating function for the alternating sum of k-gonal pyramidal numbers
- 1.4 The ordinary generating function for the alternating sum of centered k-gonal pyramidal numbers
- 1.5 The ordinary generating function for the first bisection of k-gonal numbers
- 1.6 The ordinary generating function for the first trisection of k-gonal numbers
- 1.7 The ordinary generating function for the squares of k-gonal numbers
- 1.8 The ordinary generating function for the convolution of nonzero k-gonal numbers and nonzero h-gonal numbers
- 1.9 The ordinary generating function for the convolution of nonzero k-gonal numbers with themselves
- 1.10 The ordinary generating function for the convolution of nonzero k-gonal numbers and nonzero triangular numbers
- 1.11 The ordinary generating function for the generalized k-gonal numbers
- 1.12 The ordinary generating function for the Sum_{k = 0..n} m^k
- 1.13 The ordinary generating function for the Sum_{k = 0..n} (-1)^k*m^k
- 1.14 The ordinary generating function for the Sum_{k=0..n} floor(k/m)
- 1.15 The ordinary generating function for the sums of m consecutive squares of nonnegative integers
- 1.16 The ordinary generating function for the number of ways of writing n as a sum of k squares
- 1.17 The ordinary generating function for the values of quadratic polynomial p*n^2 + q*n + k
- 1.18 The ordinary generating function for the values of cubic polynomial p*n^3 + q*n^2 + k*n + m
- 1.19 The ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r
- 1.20 The ordinary generating function for the values of quintic polynomial b*n^5 + p*n^4 + q*n^3 + k*n^2 + m*n + r
- 1.21 The ordinary generating function for the characteristic function of the multiples of k
- 1.22 The ordinary generating function for the continued fraction expansion of phi^(2*k + 1), where phi = (1 + sqrt(5))/2), k = 1, 2, 3,...
- 1.23 The ordinary generating function for the continued fraction expansion of phi^(2*k), where phi = (1 + sqrt(5))/2), k = 1, 2, 3,...
- 1.24 The ordinary generating function for the continued fraction expansion of exp(1/k), with k = 1, 2, 3....
- 1.25 The ordinary generating function for the Fibonacci(k*n)
- 1.26 The ordinary generating function for the Sum_{k = 0..n} (k mod m)
- 1.27 The ordinary generating function for the recurrence relation b(n) = k^n - b(n-1), with n>0 and b(0)=0
- 1.28 The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n>0 and b(0)=1
- 1.29 The ordinary generating function for the recurrence relation b(n) = r*b(n - 1) + s*b(n - 2), with n>1 and b(0)=k, b(1)=m
- 1.30 The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*b(n - 2), with n>1 and b(0)=0, b(1)=1
- 1.31 The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - b(n - 2), with n>1 and b(0)=1, b(1)=1
- 1.32 The ordinary generating function for the recurrence relation b(n) = 2*b(n - 2) - b(n - 1), with n>1 and b(0)=k, b(1)=m
- 1.33 The ordinary generating function for the recurrence relation b(n) = b(n - 1) + 2*b(n - 2) + 3*b(n - 3) + 4*b(n - 4) + ... + k*b(n - k), with n > k - 1 and initial values b(i-1) = i for i = 1..k
- 1.34 The ordinary generating function for the recurrence relation b(n) = b(n - 1) + b(n - 2) + b(n - 3), with n>2 and b(0)=k, b(1)=m, b(2)=q
- 1.35 The ordinary generating function for the recurrence relation b(n) = floor(phi^(2*k)*b(n - 1)), with n>0, b(0)=1 and k = 0,1,2,3, ...
- 1.36 The ordinary generating function for the recurrence relation b(n) = floor(phi^(2*k+1)*b(n - 1)), with n>0, b(0)=1 and k = 0,1,2,3, ...
- 1.37 The ordinary generating function for the recurrence relation b(n) = floor((1 + sqrt(2))^(2*k)*b(n - 1)), with n>0, b(0)=1 and k = 1,2,3,4, ...
- 1.38 The ordinary generating function for the recurrence relation b(n) = floor((1 + sqrt(2))^(2*k+1)*b(n - 1)), with n>0, b(0)=1 and k = 0,1,2,3, ...
- 1.39 The ordinary generating function for the integers repeated k times
- 1.40 The ordinary generating function for the partial sums of numbers that are repdigits in base k (for k > 1)
- 1.41 The ordinary generating function for the binomial coefficients C(n,k)
- 1.42 The ordinary generation function for the Gaussian binomial coefficients [n,k]_q
- 1.43 The ordinary generating function for the transformation of the Wonderful Demlo numbers
- 1.44 The ordinary generating function for the sequences of the form k^n + m
- 1.45 The ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2
- 1.46 The ordinary generating function for the number of partitions of n into parts congruent to 1 mod m (for m>0)
- 1.47 The ordinary generating function for the number of partitions of n into parts larger than 1 and congruent to 1 mod m (for m>0)
- 1.48 The ordinary generating function for the surface area of the n-dimensional sphere of radius r
- 2 The sum of reciprocals of Catalan numbers (with even indices, with odd indices)
- 3 Double hyperfactorial
- 4 Polynomials
- 5 Conjectures
Generalization of generating functions
The ordinary generating function for the alternating sum of k-gonal numbers
The ordinary generating function for the alternating sum of centered k-gonal numbers
The ordinary generating function for the alternating sum of k-gonal pyramidal numbers
The ordinary generating function for the alternating sum of centered k-gonal pyramidal numbers
The ordinary generating function for the first bisection of k-gonal numbers
The ordinary generating function for the first trisection of k-gonal numbers
The ordinary generating function for the squares of k-gonal numbers
The ordinary generating function for the convolution of nonzero k-gonal numbers and nonzero h-gonal numbers
The ordinary generating function for the convolution of nonzero k-gonal numbers with themselves
The ordinary generating function for the convolution of nonzero k-gonal numbers and nonzero triangular numbers
The ordinary generating function for the generalized k-gonal numbers
The ordinary generating function for the Sum_{k = 0..n} m^k
The ordinary generating function for the Sum_{k = 0..n} (-1)^k*m^k
The ordinary generating function for the Sum_{k=0..n} floor(k/m)
The ordinary generating function for the sums of m consecutive squares of nonnegative integers
The ordinary generating function for the number of ways of writing n as a sum of k squares
The ordinary generating function for the values of quadratic polynomial p*n^2 + q*n + k
The ordinary generating function for the values of cubic polynomial p*n^3 + q*n^2 + k*n + m
The ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r
The ordinary generating function for the values of quintic polynomial b*n^5 + p*n^4 + q*n^3 + k*n^2 + m*n + r
The ordinary generating function for the characteristic function of the multiples of k
The ordinary generating function for the continued fraction expansion of phi^(2*k + 1), where phi = (1 + sqrt(5))/2), k = 1, 2, 3,...
The ordinary generating function for the continued fraction expansion of phi^(2*k), where phi = (1 + sqrt(5))/2), k = 1, 2, 3,...
The ordinary generating function for the continued fraction expansion of exp(1/k), with k = 1, 2, 3....
The ordinary generating function for the Fibonacci(k*n)
The ordinary generating function for the Sum_{k = 0..n} (k mod m)
The ordinary generating function for the recurrence relation b(n) = k^n - b(n-1), with n>0 and b(0)=0
The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n>0 and b(0)=1
The ordinary generating function for the recurrence relation b(n) = r*b(n - 1) + s*b(n - 2), with n>1 and b(0)=k, b(1)=m
The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*b(n - 2), with n>1 and b(0)=0, b(1)=1
The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - b(n - 2), with n>1 and b(0)=1, b(1)=1
The ordinary generating function for the recurrence relation b(n) = 2*b(n - 2) - b(n - 1), with n>1 and b(0)=k, b(1)=m
The ordinary generating function for the recurrence relation b(n) = b(n - 1) + 2*b(n - 2) + 3*b(n - 3) + 4*b(n - 4) + ... + k*b(n - k), with n > k - 1 and initial values b(i-1) = i for i = 1..k
The ordinary generating function for the recurrence relation b(n) = b(n - 1) + b(n - 2) + b(n - 3), with n>2 and b(0)=k, b(1)=m, b(2)=q
The ordinary generating function for the recurrence relation b(n) = floor(phi^(2*k)*b(n - 1)), with n>0, b(0)=1 and k = 0,1,2,3, ...
The ordinary generating function for the recurrence relation b(n) = floor(phi^(2*k+1)*b(n - 1)), with n>0, b(0)=1 and k = 0,1,2,3, ...
The ordinary generating function for the recurrence relation b(n) = floor((1 + sqrt(2))^(2*k)*b(n - 1)), with n>0, b(0)=1 and k = 1,2,3,4, ...
The ordinary generating function for the recurrence relation b(n) = floor((1 + sqrt(2))^(2*k+1)*b(n - 1)), with n>0, b(0)=1 and k = 0,1,2,3, ...
The ordinary generating function for the integers repeated k times
The ordinary generating function for the partial sums of numbers that are repdigits in base k (for k > 1)
The ordinary generating function for the binomial coefficients C(n,k)
The ordinary generation function for the Gaussian binomial coefficients [n,k]_q
The ordinary generating function for the transformation of the Wonderful Demlo numbers
The ordinary generating function for the sequences of the form k^n + m
The ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2
The ordinary generating function for the number of partitions of n into parts congruent to 1 mod m (for m>0)
The ordinary generating function for the number of partitions of n into parts larger than 1 and congruent to 1 mod m (for m>0)
The ordinary generating function for the surface area of the n-dimensional sphere of radius r
The sum of reciprocals of Catalan numbers (with even indices, with odd indices)
Double hyperfactorial
Polynomials
Polynomials T_n(x) = -((-1)^n*2^(-n-1)*cos(Pi*sqrt(8*x+1)/2)*Gamma(n-sqrt(8*x+1)/2+3/2)*Gamma(n+sqrt(8*x+1)/2+3/2))/Pi
Polynomials Q_n(x) = 2^(-n)*((x+sqrt(x*(x+6)-3)+1)^n-(x-sqrt(x*(x+6)-3)+1)^n)/sqrt(x*(x+6)-3)
Polynomials C_n(x) = Sum_(k=0..n} (2*k)!*(x - 1)^(n-k)/((k + 1)!*k!)
Conjectures
Every number > 15 can be represented as a sum of 3 semiprimes.
Every number is the sum of at most 6 square pyramidal numbers.
Every number is the sum of at most k+2 k-gonal pyramidal numbers (except k = 5).
Every number is the sum of at most 12 squares of triangular numbers (or partial sums of cubes).
Every number > 27 can be represented as a sum of 4 proper prime powers.
Every number > 8 can be represented as a sum of a proper prime power and a squarefree number > 1.
Every number > 108 can be represented as a sum of a proper prime power and a nonprime squarefree number.
Every number > 10 can be represented as a sum of a prime and a nonprime squarefree number.
Every number > 30 can be represented as a sum of a prime and a squarefree semiprime.
Every number > 30 can be represented as a sum of a twin prime and a squarefree semiprime.
Every number > 108 can be represented as a sum of a perfect power and a squarefree semiprime.
Every number > 527 can be represented as a sum of a prime with prime subscript and a semiprime (only 18 positive integers cannot be represented as a sum of a prime with prime subscript and a semiprime).
Every number > 51 can be represented as a sum of 2 multiplicatively perfect numbers.
Any sufficiently large number can be represented as a sum of 3 squarefree palindromes.
Every number > 3 can be represented as a sum of 4 squarefree palindromes.
Every number > 82 can be represented as a sum of 2 numbers that are the product of an even number of distinct primes (including 1).
Every number > 57 can be represented as a sum of 2 numbers that are the product of an odd number of distinct primes.
Every number > 10 can be represented as a sum of 2 numbers, one of which is the product of an even number of distinct primes (including 1) and another is the product of an odd number of distinct primes.
Every number > 1 is the sum of at most 5 сentered triangular numbers.
Every number > 1 is the sum of at most 6 centered square numbers.
Every number > 1 is the sum of at most k+2 centered k-gonal numbers.
Every number is the sum of at most k-4 generalized k-gonal numbers (for k >= 8).
Every number is the sum of at most 15 icosahedral numbers.
Every number > 23 is the sum of at most 8 squares of primes.
Every number > 131 can be represented as a sum of 13 squares of primes.
Every number > 16 is the sum of at most 4 primes of form x^2 + y^2.
Every number > 7 is the sum of at most 4 twin primes.
Every number > 3 is the sum of at most 5 partial sums of primes.
Let a_p(n) be the length of the period of the sequence k^p mod n where p is a prime, then a_p(n) = n/p if n == 0 (mod p^2) else a_p(n) = n.
Let a(n) be the sum of largest prime power factors of numbers <= n, then a(n) = O(n^2/log(n)).
Let a(n) = n - a(floor(a(n-1)/2)) with a(0) = 0, then a(n) ~ c*n, where c = sqrt(3) - 1.
Recurrences (Pisot and related sequences)
a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 3, a(1) = 16.
a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 4, a(1) = 14.
a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 5, a(1) = 13.
a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 5, a(1) = 12.
a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 6, a(1) = 15.