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A324320 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.

%I A324320 #30 Jun 26 2022 12:58:33
%S A324320 1045,2465,2821,15841,20501,34133,51221,68101,89441,116033,118405,
%T A324320 162401,170885,216545,300833,364705,439301,472033,530881,642181,
%U A324320 687365,746005,970145,976981,997633,1104133,1148245,1193221,1231361,1239061,1398101,1654661,1971541
%N A324320 Terms of A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) that are also octagonal numbers (A000567) with index equal to their largest prime factor.
%C A324320 2465 is also a Carmichael number (A002997).
%C A324320 2821 is also a primary Carmichael number (A324316).
%C A324320 See the section on polygonal numbers in Kellner and Sondow 2019.
%C A324320 Subsequence of the special polygonal numbers A324973. - _Jonathan Sondow_, Mar 27 2019
%H A324320 Amiram Eldar, <a href="/A324320/b324320.txt">Table of n, a(n) for n = 1..10000</a>
%H A324320 Bernd C. Kellner and Jonathan Sondow, <a href="https://doi.org/10.4169/amer.math.monthly.124.8.695">Power-Sum Denominators</a>, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:<a href="https://arxiv.org/abs/1705.03857">1705.03857</a> [math.NT], 2017.
%H A324320 Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/v52/v52.pdf">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, #A52 Integers 21 (2021), 21 pp.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.
%e A324320 A324315(4) = 1045 = 5 * 11 * 19 = 19 * (3 * 19 - 2) = A000567(19), so 1045 is a member.
%t A324320 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
%t A324320 LP[n_] := Transpose[FactorInteger[n]][[1]];
%t A324320 ON[n_] := n(3n - 2);
%t A324320 TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
%t A324320 Select[ON@ Prime[Range[100]], TestS[#] &]
%Y A324320 Cf. A000567, A002997, A324315, A324316, A324317, A324318, A324319, A324369, A324370, A324371, A324404, A324405, A324973.
%K A324320 nonn,base
%O A324320 1,1
%A A324320 _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 23 2019
%E A324320 More terms from _Amiram Eldar_, Dec 05 2020