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User:Ilya Gutkovskiy

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I gut.jpg




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

Generalization of generating functions

The ordinary generating function for the alternating sum of k-gonal numbers

A266088

The ordinary generating function for the alternating sum of centered k-gonal numbers

A270693

The ordinary generating function for the alternating sum of k-gonal pyramidal numbers

A266677

The ordinary generating function for the alternating sum of centered k-gonal pyramidal numbers

A270694

The ordinary generating function for the first bisection of k-gonal numbers

A270704

The ordinary generating function for the first trisection of k-gonal numbers

A268351

The ordinary generating function for the squares of k-gonal numbers

A100255

The ordinary generating function for the convolution of nonzero k-gonal numbers and nonzero h-gonal numbers

A271663

The ordinary generating function for the convolution of nonzero k-gonal numbers with themselves

A271662

The ordinary generating function for the convolution of nonzero k-gonal numbers and nonzero triangular numbers

A271567

The ordinary generating function for the generalized k-gonal numbers

A277082

The ordinary generating function for the Sum_{k = 0..n} m^k

A269025

The ordinary generating function for the Sum_{k = 0..n} (-1)^k*m^k

A268413

The ordinary generating function for the Sum_{k=0..n} floor(k/m)

A269445

The ordinary generating function for the sums of m consecutive squares of nonnegative integers

A276026

The ordinary generating function for the number of ways of writing n as a sum of k squares

A276285

The ordinary generating function for the values of quadratic polynomial p*n^2 + q*n + k

A270710

The ordinary generating function for the values of cubic polynomial p*n^3 + q*n^2 + k*n + m

A268644

The ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r

A269792

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

A125083

The ordinary generating function for the characteristic function of the multiples of k

A267142

The ordinary generating function for the continued fraction expansion of phi^(2*k + 1), where phi = (1 + sqrt(5))/2), k = 1, 2, 3,...

A267319

The ordinary generating function for the continued fraction expansion of phi^(2*k), where phi = (1 + sqrt(5))/2), k = 1, 2, 3,...

A267319

The ordinary generating function for the continued fraction expansion of exp(1/k), with k = 1, 2, 3....

A267318

The ordinary generating function for the Fibonacci(k*n)

A269500

The ordinary generating function for the Sum_{k = 0..n} (k mod m)

A268291

The ordinary generating function for the recurrence relation b(n) = k^n - b(n-1), with n>0 and b(0)=0

A271427

The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n>0 and b(0)=1

A268414

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

A268409

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

A268344

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

A269028

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

A268741

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

A268349

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

A268410

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, ...

A278475

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, ...

A278475

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, ...

A278476

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, ...

A278476

The ordinary generating function for the integers repeated k times

A004526

The ordinary generating function for the partial sums of numbers that are repdigits in base k (for k > 1)

A277209

The ordinary generating function for the binomial coefficients C(n,k)

A017764

The ordinary generation function for the Gaussian binomial coefficients [n,k]_q

A275944

The ordinary generating function for the transformation of the Wonderful Demlo numbers

A271528

The ordinary generating function for the sequences of the form k^n + m

A271527

The ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2

A269457

The ordinary generating function for the number of partitions of n into parts congruent to 1 mod m (for m>0)

A277090

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)

A277210

The ordinary generating function for the surface area of the n-dimensional sphere of radius r

A072478

The sum of reciprocals of Catalan numbers (with even indices, with odd indices)

A276483


A276484

Double hyperfactorial

A271385

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

T n(x).gif

A271386

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)

Q n(x).gif

A271451

Polynomials C_n(x) = Sum_(k=0..n} (2*k)!*(x - 1)^(n-k)/((k + 1)!*k!)

C n(x).gif

A271453

Conjectures

Every number > 15 can be represented as a sum of 3 semiprimes.

A282135

Number of ways to write n as an ordered sum of three semiprimes.jpg



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).

A282173

Number of ways to write n as an ordered sum of 6 square pyramidal numbers.jpg



Every number is the sum of at most 12 squares of triangular numbers (or partial sums of cubes).

A284641

Number of ways to write n as an ordered sum of 12 squares of triangular numbers.jpg



Every number > 27 can be represented as a sum of 4 proper prime powers.

A282289

Number of ways to write n as an ordered sum of 4 proper prime powers.jpg



Every number > 8 can be represented as a sum of a proper prime power and a squarefree number > 1.

A282290

Number of ways of writing n as a sum of a proper prime power and a squarefree number.jpg



Every number > 108 can be represented as a sum of a proper prime power and a nonprime squarefree number.

A287299

Number of ways of writing n as a sum of a proper prime power and a nonprime squarefree number.jpg



Every number > 10 can be represented as a sum of a prime and a nonprime squarefree number.

A282318

Number of ways of writing n as a sum of a prime and a nonprime squarefree number.jpg



Every number > 30 can be represented as a sum of a prime and a squarefree semiprime.

A282192

Number of ways of writing n as a sum of a prime and a squarefree semiprime.jpg



Every number > 30 can be represented as a sum of a twin prime and a squarefree semiprime.

A283929

Number of ways of writing n as a sum of a twin prime and a squarefree semiprime 2.jpg



Every number > 108 can be represented as a sum of a perfect power and a squarefree semiprime.

A282947

Number of ways of writing n as a sum of a perfect power and a squarefree semiprime.jpg



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).

A282355

Number of ways of writing n as a sum of a prime with prime subscript and a semiprime.jpg



Every number > 51 can be represented as a sum of 2 multiplicatively perfect numbers.

A282570

Number of ways to write n as an ordered sum of two multiplicatively perfect numbers.jpg



Any sufficiently large number can be represented as a sum of 3 squarefree palindromes.

A282585

Number of ways to write n as an ordered sum of 3 squarefree palindromes in base 10.jpg

Every number > 3 can be represented as a sum of 4 squarefree palindromes.

Number of ways to write n as an ordered sum of 4 squarefree palindromes in base 10.jpg



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).

A285796



Every number > 57 can be represented as a sum of 2 numbers that are the product of an odd number of distinct primes.

A285797



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.

A286971



Every number > 1 is the sum of at most 5 сentered triangular numbers.

A282502



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.

A282504



Every number is the sum of at most k-4 generalized k-gonal numbers (for k >= 8).

A290943



Every number is the sum of at most 15 icosahedral numbers.

A282350



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.

A275001



Every number > 16 is the sum of at most 4 primes of form x^2 + y^2.

A282971



Every number > 7 is the sum of at most 4 twin primes.

A283875



Every number > 3 is the sum of at most 5 partial sums of primes.

A282906



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.

A282779



Let a(n) be the sum of largest prime power factors of numbers <= n, then a(n) = O(n^2/log(n)).

A284521



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.

A284648



Let a(n) = n - a(floor(a(n-1)/2)) with a(0) = 0, then a(n) ~ c*n, where c = sqrt(3) - 1.

A286389



A279586



Recurrences (Pisot and related sequences)



a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 3, a(1) = 16.


A278681


a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 4, a(1) = 14.


A278692


a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 5, a(1) = 13.


A278764


a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 4, a(1) = 14.


A277084


a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 5, a(1) = 12.


A277088


a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 6, a(1) = 15.


A277089