A324973 Special polygonal numbers.

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

Offset

1,1

Comments

Squarefree polygonal numbers P(r,p) = (p^2*(r-2)-p*(r-4))/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

Table of n, a(n) for n=1..47.

Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

Wikipedia, Polygonal number

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/p-1)/(p-1)) && denominator(r)==1, 0); \\ Jinyuan Wang, Feb 18 2021

Crossrefs

Subsequence of A324972 = intersection of A005117 and A090466.

A002997, A324316, A324319 and A324320 are subsequences.

Cf. A324974, A324975, A324976.

Sequence in context: A069750 A233450 A298374 * A318555 A359923 A035077

Adjacent sequences: A324970 A324971 A324972 * A324974 A324975 A324976

Keyword

nonn

Author

Bernd C. Kellner and Jonathan Sondow, Mar 21 2019

Extensions

Several missing terms inserted by Jinyuan Wang, Feb 18 2021

Status

approved