

A090466


Regular figurative or polygonal numbers of order greater than 2.


19



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
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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



FORMULA



MAPLE

isA090466 := proc(n)
local nsearch, ksearch;
for nsearch from 3 do
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;


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

Sequence A090428 (excluding 1) is a subsequence of this sequence.  T. D. Noe, Jun 14 2012


KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



