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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.

%I A129654 #39 Feb 16 2025 08:33:05
%S A129654 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,
%T A129654 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,
%U A129654 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
%N 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.
%C A129654 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)).
%C A129654 From _Gus Wiseman_, May 03 2019: (Start)
%C A129654 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:
%C A129654   (35)
%C A129654   (23,12)
%C A129654   (11,9,7,5,3)
%C A129654   (8,7,6,5,4,3,2)
%C A129654   (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)
%C A129654 (End)
%H A129654 Alois P. Heinz, <a href="/A129654/b129654.txt">Table of n, a(n) for n = 2..10000</a>
%H A129654 E. Deza and M. Deza, <a href="http://www.worldscientific.com/doi/suppl/10.1142/8188/suppl_file/8188_chap01.pdf">Figurate Numbers</a>, World Scientific, 2012; see p. 45.
%H A129654 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>
%H A129654 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%F A129654 a(n) = A177025(n) + 1.
%F A129654 G.f.: x * Sum_{k>=1} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - _Ilya Gutkovskiy_, Apr 09 2020
%e A129654 a(6) = 3 because 6 = P(2,6) = P(3,3) = P(6,2).
%p A129654 A129654 := proc(n) local resul, dvs, i, r, m ;
%p A129654    dvs := numtheory[divisors](2*n) ;
%p A129654    resul := 0 ;
%p A129654    for i from 1 to nops(dvs) do
%p A129654       r := op(i, dvs) ;
%p A129654       if r > 1 then
%p A129654          m := (2*n/r-4+2*r)/(r-1) ;
%p A129654          if is(m, integer) then
%p A129654             resul := resul+1 ;
%p A129654          fi ;
%p A129654       fi ;
%p A129654    od ;
%p A129654    RETURN(resul) ;
%p A129654 end: # _R. J. Mathar_, May 14 2007
%t A129654 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 *)
%t A129654 Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]], {n, 2, 106}] (* _Jonathan Sondow_, May 09 2014 *)
%t A129654 atpms[n_]:=Select[Join@@Table[i*Range[k,1,-1],{k,n},{i,0,n}],Total[#+1]==n&];
%t A129654 Table[Length[atpms[n]],{n,100}] (* _Gus Wiseman_, May 03 2019 *)
%o A129654 (PARI) a(n) = sumdiv(2*n, d, (d>1) && (2*n/d + 2*d - 4) % (d-1) == 0); \\ _Daniel Suteu_, Dec 22 2018
%Y A129654 Cf. A063778, A177025.
%Y A129654 Column k=0 of A239550.
%Y A129654 Cf. A007862, A049988, A307824, A325349, A325350, A325356, A325357, A325358, A325458, A325459.
%K A129654 nonn
%O A129654 2,2
%A A129654 _Alexander Adamchuk_, Apr 27 2007