

A129654


Number of different ways to represent n as general polygonal number P(m,r) = 1/2*r*((m2)*r(m4)) = 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
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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 mgonal number P(m,r) = 1/2*r*((m2)*r(m4)).
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/r4+2*r)/(r1) ;
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/r4+2*r)/(r1); If[IntegerQ[m], resul = resul+1 ] ] ]; resul); Table[a[n], {n, 2, 106}] (* JeanFrançois 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) % (d1) == 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



