A063778 a(n) = the least integer that is polygonal in exactly n ways.

3, 6, 15, 36, 225, 561, 1225, 11935, 11781, 27405, 220780, 203841, 3368925, 4921840, 7316001, 33631521, 142629201, 879207616, 1383958576, 3800798001, 12524486976, 181285005825, 118037679760, 239764947345, 738541591425, 1289707733601, 1559439365121

Offset: 1

Author

David W. Wilson, Aug 16 2001

Keywords

• nonn
• nice

Comments

a(n) has exactly n representations as an m-gonal number P(m,r) = r*((m-2)*r-(m-4))/2, with m>2, r>1.

a(28) > 4*10^12. - Donovan Johnson, Dec 08 2010

From Husnain Raza, Jan 01 2024: (Start)

a(28) <= 14189300403201

a(29) <= 100337325689601

a(30) <= 1735471549713825

a(31) <= 334830950355825

a(32) <= 1473426934890625

a(33) <= 5409964920838401

(End)

Examples

			a(3) = 15 because 15 is the least integer which is polygonal in 3 ways (15 is n-gonal for n = 3, 6, 15).
		

Links

• Eric Weisstein's World of Mathematics, Polygonal Number.

Crossrefs

Cf. A177025 (number of different ways to represent n as a polygonal).

Cf. A129654 (number of different ways to represent n as general polygonal).

Programs

Maple

A063778 := proc(nmax) local a,n,ps ; a := [seq(0,i=1..nmax)] ; n := 1 ; while true do ps := A129654(n) ; if ps > 0 and ps <= nmax and n > 1 then if op(ps,a) = 0 then a := subsop(ps=n,a) ; print(a) ; fi ; fi ; n := n+1 ; end: RETURN(a) ; end: A063778(30) ; # R. J. Mathar, May 14 2007

Mathematica

P[m_, r_] := P[m, r] = r*(4 + m*(r - 1) - 2*r)/2;
a[n_Integer] := a[n] = Module[{c, r, m, p, f}, p = 0; f = False; While[!f, p++; c = 0; For[m = 3, m <= p, m++, For[r = 1, r <= p, r++, If[p == P[m, r], c++;];];]; If[c == n, f = True;];]; p];
Table[a[n], {n, 1, 5}] (* Robert P. P. McKone, Jan 02 2024 *)

PARI

a(n) = my(k=3); while (sum(p=3, k, ispolygonal(k, p)) != n, k++); k; \\ Michel Marcus, Aug 17 2024

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 23 2007

a(22)-a(27) from Donovan Johnson, Dec 08 2010