

A090466


Regular figurative or polygonal numbers of order greater than 2.


15



6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 117, 118
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

The sorted kgonal numbers of order greater than 2. If one were to include either the rank 2 or the 2gonal numbers, then every number would appear.
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
The nth kgonal number is 1 + k*n(n1)/2  (n1)^2 = A057145(k,n).
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
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
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 dth (r+2)gonal number.  Jianing Song, Mar 14 2021


REFERENCES

Albert H. Beiler, Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains, Dover, NY, 1964, pp. 185199.


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
Eric Weisstein's World of Mathematics, Figurate Number
Index to sequences related to polygonal numbers


FORMULA

Integer n is in this sequence iff A176774(n) < n.  Max Alekseyev, Apr 24 2018


MAPLE

isA090466 := proc(n)
local nsearch, ksearch;
for nsearch from 3 do
if A057145(nsearch, 3) > n then
return false;
end if;
for ksearch from 3 do
if A057145(nsearch, ksearch) = n then
return true;
elif A057145(nsearch, ksearch) > n then
break;
end if;
end do:
end do:
end proc:
for n from 1 to 1000 do
if isA090466(n) then
printf("%d, ", n) ;
end if;
end do: # R. J. Mathar, Jul 28 2016


MATHEMATICA

Take[Union[Flatten[Table[1+k*n (n1)/2(n1)^2, {n, 3, 100}, {k, 3, 40}]]], 67] (* corrected by Ant King, Sep 19 2011 *)
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 *)


PROG

(PARI) list(lim)=my(v=List()); lim\=1; for(n=3, sqrtint(8*lim+1)\2, for(k=3, 2*(lim2*n+n^2)\n\(n1), listput(v, 1+k*n*(n1)/2(n1)^2))); Set(v); \\ Charles R Greathouse IV, Jan 19 2017
(PARI) is(n)=for(s=3, n\3+1, ispolygonal(n, s)&&return(s)); \\ M. F. Hasler, Jan 19 2017
(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


CROSSREFS

Cf. A057145, A001248. Complement is A090467.
Sequence A090428 (excluding 1) is a subset of this sequence.  T. D. Noe, Jun 14 2012
Sequence in context: A053869 A085275 A177201 * A090428 A039725 A262362
Adjacent sequences: A090463 A090464 A090465 * A090467 A090468 A090469


KEYWORD

easy,nonn


AUTHOR

Robert G. Wilson v, Dec 01 2003


EXTENSIONS

Verified by Don Reble, Mar 12 2006


STATUS

approved



