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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. 16
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

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

FORMULA

a(n) = A177025(n) + 1.

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-Fran├žois 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 November 13 15:39 EST 2019. Contains 329106 sequences. (Running on oeis4.)