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 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 30 21:59 EDT 2023. Contains 365812 sequences. (Running on oeis4.)