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

%I A063778 #35 Feb 16 2025 08:32:45
%S A063778 3,6,15,36,225,561,1225,11935,11781,27405,220780,203841,3368925,
%T A063778 4921840,7316001,33631521,142629201,879207616,1383958576,3800798001,
%U A063778 12524486976,181285005825,118037679760,239764947345,738541591425,1289707733601,1559439365121
%N A063778 a(n) = the least integer that is polygonal in exactly n ways.
%C A063778 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.
%C A063778 a(28) > 4*10^12. - _Donovan Johnson_, Dec 08 2010
%C A063778 From _Husnain Raza_, Jan 01 2024: (Start)
%C A063778 a(28) <= 14189300403201
%C A063778 a(29) <= 100337325689601
%C A063778 a(30) <= 1735471549713825
%C A063778 a(31) <= 334830950355825
%C A063778 a(32) <= 1473426934890625
%C A063778 a(33) <= 5409964920838401
%C A063778 (End)
%H A063778 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>.
%e A063778 a(3) = 15 because 15 is the least integer which is polygonal in 3 ways (15 is n-gonal for n = 3, 6, 15).
%p A063778 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
%t A063778 P[m_, r_] := P[m, r] = r*(4 + m*(r - 1) - 2*r)/2;
%t A063778 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];
%t A063778 Table[a[n], {n, 1, 5}] (* _Robert P. P. McKone_, Jan 02 2024 *)
%o A063778 (PARI) a(n) = my(k=3); while (sum(p=3, k, ispolygonal(k, p)) != n, k++); k; \\ _Michel Marcus_, Aug 17 2024
%Y A063778 Cf. A177025 (number of different ways to represent n as a polygonal).
%Y A063778 Cf. A129654 (number of different ways to represent n as general polygonal).
%K A063778 nonn,nice
%O A063778 1,1
%A A063778 _David W. Wilson_, Aug 16 2001
%E A063778 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, May 23 2007
%E A063778 a(22)-a(27) from _Donovan Johnson_, Dec 08 2010