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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 hexagonal numbers (A000384) with index equal to their largest prime factor.
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%I #33 Jun 26 2022 12:56:08

%S 231,561,3655,5565,8911,10585,13695,23653,32131,45451,59685,74305,

%T 108345,115921,157641,243253,248865,302253,314821,334153,371091,

%U 392055,417241,458403,505515,546535,688551,702705,795691,821121,915981,932295,1004653,1145341,1181953

%N 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 hexagonal numbers (A000384) with index equal to their largest prime factor.

%C 561, 8911, and 10585 are also Carmichael numbers (A002997).

%C The smallest primary Carmichael number (A324316) in the sequence is 8801128801 = 181 * 733 * 66337 = A000384(66337).

%C See the section on polygonal numbers in Kellner and Sondow 2019.

%C Subsequence of the special polygonal numbers A324973. - _Jonathan Sondow_, Mar 27 2019

%H Amiram Eldar, <a href="/A324319/b324319.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 A324315(1) = 231 = 3 * 7 * 11 = 11 * (2 * 11 - 1) = A000384(11), so 231 is a member.

%t SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];

%t LP[n_] := Transpose[FactorInteger[n]][[1]];

%t HN[n_] := n(2n - 1);

%t TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];

%t Select[HN@ Prime[Range[100]], TestS[#] &]

%Y Cf. A000384, A002997, A195441, A324315, A324316, A324317, A324318, A324320, A324369, A324370, A324371, A324404, A324405, A324973.

%K nonn,base

%O 1,1

%A _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 23 2019

%E More terms from _Amiram Eldar_, Dec 05 2020