A165718 Integers of the form k*(k+7)/6.

3, 5, 10, 13, 20, 24, 33, 38, 49, 55, 68, 75, 90, 98, 115, 124, 143, 153, 174, 185, 208, 220, 245, 258, 285, 299, 328, 343, 374, 390, 423, 440, 475, 493, 530, 549, 588, 608, 649, 670, 713, 735, 780, 803, 850, 874, 923, 948, 999, 1025, 1078, 1105, 1160, 1188

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 24 2009

Keywords

• nonn
• easy

Comments

Integers of the form k + k*(k+1)/6 = k + A000217(k)/3; for k see A007494, for A000217(k)/3 see A001318. - R. J. Mathar, Sep 25 2009

Only 3 terms are prime numbers (3,5,13). Are all the rest composite?

The only prime terms in this sequence are 3, 5, and 13. If k=6j+1 or k=6j+4, k*(k+7) is congruent to 2 mod 6 and will never be an integer. If k=6j, k*(k+7)/6 = j*(6j+7) which is prime only for j=1 (i.e., 13 is in the sequence). If k=6j+2, k*(k+7)/6 = (3j+1)*(2j+3) which is prime only for j=0 (i.e., 3 is in the sequence). If k=6j+3, k*(k+7)/6 = (2j+1)*(3j+5) which is prime only for j=0 (i.e., 5 is in the sequence). If k=6j+5, k*(k+7)/6 = (6j+5)*(j+2) which is never prime. Thus {3,5,13} are the only primes in this sequence. - Derek Orr, Feb 26 2017

Conjecturally, the sequence terms are the exponents in the expansion of x/(1 + x) + Sum_{n >= 1} (-1)^n * x^(2*n-1) / Product_{k = 1..n+1} (1 + x^(2*k-1)) = x^3 - x^5 + x^10 - x^13 + x^20 - x^24 + - .... - Peter Bala, Nov 20 2024

Examples

			For k=1, 2, 3, ..., k*(k+7)/6 is 4/3, 3, 5, 22/3, 10, 13, 49/3, 20, 24, 85/3, 33, ..., and the integer values out of these become the sequence.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000217, A001318, A007494, A165717.

Programs

Mathematica

q=3;s=0;lst={};Do[s+=((n+q)/q);If[IntegerQ[s],AppendTo[lst,s]],{n,6!}];lst

PARI

Vec(x*(-3-2*x+x^2+x^3) / ((1+x)^2*(x-1)^3) + O(x^60)) \\ Colin Barker, Feb 26 2017

PARI

a(n)=if(n%2, 3*n^2 + 16*n + 5, 3*n^2 + 14*n)/8 \\ Charles R Greathouse IV, Feb 27 2017

Formula

From R. J. Mathar, Sep 25 2009: (Start)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

G.f.: x*(-3-2*x+x^2+x^3)/((1+x)^2 * (x-1)^3). (End)

a(n) = Sum_{i=1..n} numerator(i/2) + denominator(i/2). - Wesley Ivan Hurt, Feb 26 2017

From Colin Barker, Feb 26 2017: (Start)

a(n) = (3*n^2 + 14*n) / 8 for n even.

a(n) = (3*n^2 + 16*n + 5) / 8 for n odd. (End)

From Peter Bala, Dec 15 2020: (Start)

a(n) = A001318(n+2) - 2.

Exponents in the expansion of Sum_{n >= 0} x^n * Product_{k = 1..n+1} (1 - x^k) = 1 - x^3 - x^5 + x^10 + x^13 - x^20 - x^24 + + - - .... (End)

Sum_{n>=1} 1/a(n) = 159/98 - 2*Pi/(7*sqrt(3)). - Amiram Eldar, Jul 26 2024

E.g.f.: (x*(19 + 3*x)*cosh(x) + (5 + 17*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Dec 07 2024

Extensions

Definition simplified by R. J. Mathar, Sep 25 2009