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A324319
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.
12
231, 561, 3655, 5565, 8911, 10585, 13695, 23653, 32131, 45451, 59685, 74305, 108345, 115921, 157641, 243253, 248865, 302253, 314821, 334153, 371091, 392055, 417241, 458403, 505515, 546535, 688551, 702705, 795691, 821121, 915981, 932295, 1004653, 1145341, 1181953
OFFSET
1,1
COMMENTS
561, 8911, and 10585 are also Carmichael numbers (A002997).
The smallest primary Carmichael number (A324316) in the sequence is 8801128801 = 181 * 733 * 66337 = A000384(66337).
See the section on polygonal numbers in Kellner and Sondow 2019.
Subsequence of the special polygonal numbers A324973. - Jonathan Sondow, Mar 27 2019
LINKS
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.
EXAMPLE
A324315(1) = 231 = 3 * 7 * 11 = 11 * (2 * 11 - 1) = A000384(11), so 231 is a member.
MATHEMATICA
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
HN[n_] := n(2n - 1);
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[HN@ Prime[Range[100]], TestS[#] &]
KEYWORD
nonn,base
AUTHOR
EXTENSIONS
More terms from Amiram Eldar, Dec 05 2020
STATUS
approved