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 A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)) = n>1, for m,r>1. 18
 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 4, 3, 2, 3, 2, 2, 4, 3, 2, 3, 3, 2, 3, 4, 2, 3, 2, 2, 3, 3, 3, 5, 2, 2, 3, 3, 2, 3, 2, 2, 5, 3, 2, 3, 3, 2, 4, 3, 2, 3, 4, 2, 3, 3, 2, 3, 2, 2, 3, 4, 3, 5, 2, 2, 3, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 2, 5, 3, 2, 3, 3, 2, 3, 3, 2, 3, 4, 3, 3, 3, 3, 4, 2, 2, 3, 4, 2, 3, 2, 2, 5, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS The indices k of the first appearance of number n in a(k) are listed in A063778(n) = {2,3,6,15,36,225,...} = Least number k>1 such that k could be represented in n different ways as general m-gonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)). From Gus Wiseman, May 03 2019: (Start) Also the number of integer partitions of n whose augmented differences are all equal, where the augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k; for example aug(6,5,5,3,3,3) = (2,1,3,1,1,3). Equivalently, a(n) is the number of integer partitions of n whose differences are all equal to the last part minus one. The Heinz numbers of these partitions are given by A307824. For example, the a(35) = 5 partitions are:   (35)   (23,12)   (11,9,7,5,3)   (8,7,6,5,4,3,2)   (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) (End) LINKS Alois P. Heinz, Table of n, a(n) for n = 2..10000 E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45. Eric Weisstein's World of Mathematics, Polygonal Number FORMULA a(n) = A177025(n) + 1. G.f.: x * Sum_{k>=1} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 09 2020 EXAMPLE a(6) = 3 because 6 = P(2,6) = P(3,3) = P(6,2). MAPLE A129654 := proc(n) local resul, dvs, i, r, m ;    dvs := numtheory[divisors](2*n) ;    resul := 0 ;    for i from 1 to nops(dvs) do       r := op(i, dvs) ;       if r > 1 then          m := (2*n/r-4+2*r)/(r-1) ;          if is(m, integer) then             resul := resul+1 ;          fi ;       fi ;    od ;    RETURN(resul) ; end: # R. J. Mathar, May 14 2007 MATHEMATICA a[n_] := (dvs = Divisors[2*n]; resul = 0; For[i = 1, i <= Length[dvs], i++, r = dvs[[i]]; If[r > 1, m = (2*n/r-4+2*r)/(r-1); If[IntegerQ[m], resul = resul+1 ] ] ]; resul); Table[a[n], {n, 2, 106}] (* Jean-François Alcover, Sep 13 2012, translated from R. J. Mathar's Maple program *) Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]], {n, 2, 106}] (* Jonathan Sondow, May 09 2014 *) atpms[n_]:=Select[Join@@Table[i*Range[k, 1, -1], {k, n}, {i, 0, n}], Total[#+1]==n&]; Table[Length[atpms[n]], {n, 100}] (* Gus Wiseman, May 03 2019 *) PROG (PARI) a(n) = sumdiv(2*n, d, (d>1) && (2*n/d + 2*d - 4) % (d-1) == 0); \\ Daniel Suteu, Dec 22 2018 CROSSREFS Cf. A063778, A177025. Column k=0 of A239550. Cf. A007862, A049988, A307824, A325349, A325350, A325356, A325357, A325358, A325458, A325459. Sequence in context: A223942 A278597 A138789 * A116504 A186233 A226056 Adjacent sequences:  A129651 A129652 A129653 * A129655 A129656 A129657 KEYWORD nonn AUTHOR Alexander Adamchuk, Apr 27 2007 STATUS approved

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Last modified April 12 19:50 EDT 2021. Contains 342932 sequences. (Running on oeis4.)