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# User:Ilya Gutkovskiy

### From OeisWiki

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

# 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) = Sum_{k=1..n} sigma(k)/k, where sigma(k) is the sum of the divisors of k, it is assumed that the value of a(n)/n approaches Pi^2/6.**

**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) = 4, a(1) = 14.**

**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.**