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 A165718 Integers of the form k*(k+7)/6. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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). FORMULA 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 G.f.: x*(-3-2*x+x^2+x^3)/((1+x)^2 * (x-1)^3). - R. J. Mathar, Sep 25 2009 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) EXAMPLE 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. MATHEMATICA q=3; s=0; lst={}; Do[s+=((n+q)/q); If[IntegerQ[s], AppendTo[lst, s]], {n, 6!}]; lst PROG (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 CROSSREFS Cf. A165717. Sequence in context: A034746 A275219 A080931 * A031878 A265282 A160792 Adjacent sequences:  A165715 A165716 A165717 * A165719 A165720 A165721 KEYWORD nonn,easy AUTHOR Vladimir Joseph Stephan Orlovsky, Sep 24 2009 EXTENSIONS Definition simplified by R. J. Mathar, Sep 25 2009 STATUS approved

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