A324458 - OEIS

%I #27 Oct 05 2024 18:12:05

%S 45,325,405,637,891,1729,2821,3751,4961,6517,7381,8125,8281,10625,

%T 13357,21141,26353,28033,29341,31213,33125,35443,46657,47081,58621,

%U 65341,74431,78625,81289,94501,98125,99937,123823,146461,231601,236321,252601,254221,294409

%N Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) = p.

%C The function s_p(m) gives the sum of the base-p digits of m.

%C The sequence contains the primary Carmichael numbers A324316.

%C Being a subsequence of A324460, a term m has the following properties:

%C m must have at least 2 prime factors. If m = p1^e1 * p2^e2 with two primes p1 and p2, then e1 + e2 >= 3.

%C Each prime factor p of m satisfies the inequalities p < m^(1/(ord_p(m)+1)) <= sqrt(m), where ord_p(m) gives the maximum exponent e such that p^e divides m.

%C In the terminology of A324460, the prime factorization of m equals a strict s-decomposition of m.

%C See Kellner 2019.

%C a(n) is squarefree iff a(n) is a primary Carmichael number A324316. - _Jonathan Sondow_, Mar 16 2019

%H Amiram Eldar, <a href="/A324458/b324458.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..117 from Bernd C. Kellner)

%H Bernd C. Kellner, <a href="https://doi.org/10.5281/zenodo.10963985">On primary Carmichael numbers</a>, Integers 22 (2022), Article #A38, 39 pp.; arXiv:<a href="https://arxiv.org/abs/1902.11283">1902.11283</a> [math.NT], 2019.

%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.

%e The number 45 has the prime factors 3 and 5. Since s_3(45) = 3 and s_5(45) = 5, 45 is a member.

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

%t f[n_] := AllTrue[Transpose[FactorInteger[n]][[1]], s[n, #] == # &];

%t Select[Range[10^7], f[#] &]

%Y Subsequence is A324316. Subsequence of A324457, A324459, and A324460.

%Y Cf. A324455, A324456.

%K nonn,base

%O 1,1

%A _Bernd C. Kellner_, Feb 28 2019