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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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[#] &]

Crossrefs

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

Sequence in context: A211712 A324315 A276832 * A360214 A246886 A258167

Adjacent sequences: A324316 A324317 A324318 * A324320 A324321 A324322

Keyword

nonn,base

Author

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

Extensions

More terms from Amiram Eldar, Dec 05 2020

Status

approved