%I #26 Oct 06 2024 09:44:47
%S 3003,3315,5187,7395,8463,14763,19803,26733,31755,47523,50963,58035,
%T 62403,88023,105339,106113,123123,139971,152643,157899,166611,178923,
%U 183183,191919
%N Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 3 (mod p-1), where s_p(m) is the sum of the base p digits of m.
%C For d >= 1 define S_d = (terms m in A324315 such that s_p(m) == d (mod p-1) if prime p divides m). Then S_1 is precisely the Carmichael numbers (A002997), S_2 is A324404, S_3 is A324405, and the union of all S_d for d >= 1 is A324315.
%C Subsequence of the 3-Knödel numbers (A033553). Generally, for d > 1 the terms of S_d that are greater than d form a subsequence of the d-Knödel numbers.
%C See Kellner and Sondow 2019.
%H Amiram Eldar, <a href="/A324405/b324405.txt">Table of n, a(n) for n = 1..2000</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="https://doi.org/10.5281/zenodo.10816833">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, Integers 21 (2021), Article #A52, 21 pp.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019-2021.
%e 3003 = 3*7*11*13 is squarefree and equals 11010020_3, 11520_7, 2290_11, and 14a0_13 in base p = 3, 7, 11, and 13. Then s_3(3003) = 1+1+1+2 = 5 >= 3, s_7(3003) = 1+1+5+2 = 9 >= 7, s_11(3003) = 2+2+9 = 13 >= 11, and s_13(3003) = 1+4+a = 1+4+10 = 15 >= 13. Also, s_3(3003) = 5 == 3 (mod 2), s_7(3003) = 9 == 3 (mod 6), s_11(3003) = 13 == 3 (mod 10), and s_13(3003) = 15 == 3 (mod 12), so 3003 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 TestSd[n_, d_] := (n > 1) && (d > 0) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # && Mod[SD[n, #] - d, # - 1] == 0 &];
%t Select[Range[200000], TestSd[#, 3] &]
%Y Cf. A002997, A033553, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404.
%K nonn,base
%O 1,1
%A _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 26 2019