cp's OEIS Frontend

A090466 Regular figurative or polygonal numbers of order greater than 2.

%I A090466 #105 Feb 16 2025 08:32:51
%S A090466 6,9,10,12,15,16,18,21,22,24,25,27,28,30,33,34,35,36,39,40,42,45,46,
%T A090466 48,49,51,52,54,55,57,58,60,63,64,65,66,69,70,72,75,76,78,81,82,84,85,
%U A090466 87,88,90,91,92,93,94,95,96,99,100,102,105,106,108,111,112,114,115,117,118
%N A090466 Regular figurative or polygonal numbers of order greater than 2.
%C A090466 The sorted k-gonal numbers of order greater than 2. If one were to include either the rank 2 or the 2-gonal numbers, then every number would appear.
%C A090466 Number of terms less than or equal to 10^k for k = 1,2,3,...: 3, 57, 622, 6357, 63889, 639946, 6402325, 64032121, 640349979, 6403587409, 64036148166, 640362343980, ..., . - _Robert G. Wilson v_, May 29 2014
%C A090466 The n-th k-gonal number is 1 + k*n(n-1)/2 - (n-1)^2 = A057145(k,n).
%C A090466 For all squares (A001248) of primes p >= 5 at least one a(n) exists with p^2 = a(n) + 1. Thus the subset P_s(3) of rank 3 only is sufficient. Proof: For p >= 5, p^2 == 1 (mod {3,4,6,8,12,24}) and also P_s(3) + 1 = 3*s - 2 == 1 (mod 3). Thus the set {p^2} is a subset of {P_s(3) + 1}; Q.E.D. - _Ralf Steiner_, Jul 15 2018
%C A090466 For all primes p > 5, at least one polygonal number exists with P_s(k) + 1 = p when k = 3 or 4, dependent on p mod 6. - _Ralf Steiner_, Jul 16 2018
%C A090466 Numbers m such that r = (2*m/d - 2)/(d - 1) is an integer for some d, where 2 < d < m is a divisor of 2*m. If r is an integer, then m is the d-th (r+2)-gonal number. - _Jianing Song_, Mar 14 2021
%D A090466 Albert H. Beiler, Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains, Dover, NY, 1964, pp. 185-199.
%H A090466 Robert G. Wilson v, <a href="/A090466/b090466.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms are from T. D. Noe)
%H A090466 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FigurateNumber.html">Figurate Number</a>
%H A090466 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A090466 Integer k is in this sequence iff A176774(k) < k. - _Max Alekseyev_, Apr 24 2018
%p A090466 isA090466 := proc(n)
%p A090466     local nsearch,ksearch;
%p A090466     for nsearch from 3 do
%p A090466         if A057145(nsearch,3) > n then
%p A090466             return false;
%p A090466         end if;
%p A090466         for ksearch from 3 do
%p A090466             if A057145(nsearch,ksearch) = n then
%p A090466                 return true;
%p A090466             elif A057145(nsearch,ksearch) > n then
%p A090466                 break;
%p A090466             end if;
%p A090466         end do:
%p A090466     end do:
%p A090466 end proc:
%p A090466 for n from 1 to 1000 do
%p A090466     if isA090466(n) then
%p A090466         printf("%d,",n) ;
%p A090466     end if;
%p A090466 end do: # _R. J. Mathar_, Jul 28 2016
%t A090466 Take[Union[Flatten[Table[1+k*n (n-1)/2-(n-1)^2,{n,3,100},{k,3,40}]]],67] (* corrected by _Ant King_, Sep 19 2011 *)
%t A090466 mx = 150; n = k = 3; lst = {}; While[n < Floor[mx/3]+2, a = PolygonalNumber[n, k]; If[a < mx+1, AppendTo[ lst, a], (n++; k = 2)]; k++]; lst = Union@ lst (* _Robert G. Wilson v_, May 29 2014 and updated Jul 23 2018; PolygonalNumber requires version 10.4 or higher *)
%o A090466 (PARI) list(lim)=my(v=List()); lim\=1; for(n=3,sqrtint(8*lim+1)\2, for(k=3,2*(lim-2*n+n^2)\n\(n-1), listput(v, 1+k*n*(n-1)/2-(n-1)^2))); Set(v); \\ _Charles R Greathouse IV_, Jan 19 2017
%o A090466 (PARI) is(n)=for(s=3,n\3+1,ispolygonal(n,s)&&return(s)); \\ _M. F. Hasler_, Jan 19 2017
%o A090466 (PARI) isA090466(m) = my(v=divisors(2*m)); for(i=3, #v, my(d=v[i]); if(d==m, return(0)); if((2*m/d - 2)%(d - 1)==0, return(1))); 0 \\ _Jianing Song_, Mar 14 2021
%Y A090466 Cf. A057145, A001248, A177028 (A342772, A342805), A177201, A316676, A364693 (characteristic function).
%Y A090466 Complement is A090467.
%Y A090466 Sequence A090428 (excluding 1) is a subsequence of this sequence. - _T. D. Noe_, Jun 14 2012
%Y A090466 Other subsequences: A324972 (squarefree terms), A324973, A342806, A364694.
%Y A090466 Cf. also A275340.
%K A090466 easy,nonn
%O A090466 1,1
%A A090466 _Robert G. Wilson v_, Dec 01 2003
%E A090466 Verified by _Don Reble_, Mar 12 2006