1,1

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.

Eric Weisstein's World of Mathematics, Polygonal Number.

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

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: 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 (mathar(AT)strw.leidenuniv.nl), May 14 2007

Cf. A129654 = number of different ways to represent n as general polygonal number n(m, r) = 1/2*r*((m-2)*r-(m-4)) = n>1, for m, r>1.

Sequence in context: A005043 A099323 A058534 this_sequence A087124 A086326 A098701

Adjacent sequences: A063775 A063776 A063777 this_sequence A063779 A063780 A063781

nonn,more

David W. Wilson (davidwwilson(AT)comcast.net), Aug 16 2001

Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Andrew Plewe, May 23 2007