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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
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 n-th k-gonal number is 1 + k*n(n-1)/2 - (n-1)^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 d-th (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. 185-199.
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
FORMULA
Integer k is in this sequence iff A176774(k) < k. - 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 (n-1)/2-(n-1)^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*(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
(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 subsequence of this sequence. - T. D. Noe, Jun 14 2012
Sequence in context: A053869 A085275 A177201 * A090428 A039725 A262362
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Dec 01 2003
EXTENSIONS
Verified by Don Reble, Mar 12 2006
STATUS
approved

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Last modified May 21 02:29 EDT 2024. Contains 372720 sequences. (Running on oeis4.)