

A324973


Special polygonal numbers.


8



6, 15, 66, 70, 91, 190, 231, 435, 561, 703, 715, 782, 861, 946, 1045, 1105, 1426, 1653, 1729, 1770, 1785, 1794, 1891, 2035, 2278, 2465, 2701, 2821, 2926, 3059, 3290, 3367, 3486, 3655, 4371, 4641, 4830, 5005, 5083, 5151, 5365, 5551, 5565, 5995, 6441, 6545, 6601
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OFFSET

1,1


COMMENTS

Squarefree polygonal numbers P(r,p) = (p^2*(r2)p*(r4))/2 whose greatest prime factor is p >= 3, and whose rank (or order) is r >= 3 (see A324974).
The Carmichael numbers A002997 and primary Carmichael numbers A324316 are subsequences. See Kellner and Sondow 2019.


LINKS



EXAMPLE

P(3,5) = 15 is squarefree, and its greatest prime factor is 5, so 15 is a member.
More generally, if p is an odd prime and P(3,p) is squarefree, then P(3,p) is a member, since P(3,p) = (p^2+p)/2 = p*(p+1)/2, so p is its greatest prime factor.
CAUTION: P(6,7) = 91 = 7*13 is a member even though 7 is NOT its greatest prime factor, as P(6,7) = P(3,13) and 13 is its greatest prime factor.


MATHEMATICA

GPF[n_] := Last[Select[Divisors[n], PrimeQ]];
T = Select[Flatten[Table[{p, (p^2*(r  2)  p*(r  4))/2}, {p, 3, 150}, {r, 3, 100}], 1], SquareFreeQ[Last[#]] && First[#] == GPF[Last[#]] &];
Take[Union[Table[Last[t], {t, T}]], 47]


PROG

(PARI) is(k) = if(issquarefree(k) && k>1, my(p=vecmax(factor(k)[, 1]), r); p>2 && (r=2*(k/p1)/(p1)) && denominator(r)==1, 0); \\ Jinyuan Wang, Feb 18 2021


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



