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%I #18 Dec 16 2020 06:18:21
%S 3,5,10,13,20,24,33,38,49,55,68,75,90,98,115,124,143,153,174,185,208,
%T 220,245,258,285,299,328,343,374,390,423,440,475,493,530,549,588,608,
%U 649,670,713,735,780,803,850,874,923,948,999,1025,1078,1105,1160,1188
%N Integers of the form k*(k+7)/6.
%C 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
%C Only 3 terms are prime numbers (3,5,13). Are all the rest composite?
%C 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
%H Colin Barker, <a href="/A165718/b165718.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - _R. J. Mathar_, Sep 25 2009
%F G.f.: x*(-3-2*x+x^2+x^3)/((1+x)^2 * (x-1)^3). - _R. J. Mathar_, Sep 25 2009
%F a(n) = Sum_{i=1..n} numerator(i/2) + denominator(i/2). - _Wesley Ivan Hurt_, Feb 26 2017
%F From _Colin Barker_, Feb 26 2017: (Start)
%F a(n) = (3*n^2 + 14*n) / 8 for n even.
%F a(n) = (3*n^2 + 16*n + 5) / 8 for n odd.
%F (End)
%F From _Peter Bala_, Dec 15 2020: (Start)
%F a(n) = A001318(n+2) - 2.
%F 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)
%e 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.
%t q=3;s=0;lst={};Do[s+=((n+q)/q);If[IntegerQ[s],AppendTo[lst,s]],{n,6!}];lst
%o (PARI) Vec(x*(-3-2*x+x^2+x^3) / ((1+x)^2*(x-1)^3) + O(x^60)) \\ _Colin Barker_, Feb 26 2017
%o (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
%Y Cf. A001318, A165717.
%K nonn,easy
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Sep 24 2009
%E Definition simplified by _R. J. Mathar_, Sep 25 2009
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