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Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) >= p.
6

%I #30 Oct 05 2024 16:29:38

%S 24,45,48,72,96,120,144,189,192,216,224,225,231,240,288,315,320,325,

%T 336,352,360,384,405,432,450,480,525,540,560,561,567,576,594,600,637,

%U 648,672,704,720,768,792,819,825,832,850,864,891,896,924,945,960,975,980

%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 is infinite, since it contains A324315, and thus the Carmichael numbers A002997.

%C Being a subsequence of A324459, 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 A324459, the prime factorization of m equals an s-decomposition of m.

%C See Kellner 2019.

%C a(n) is a Carmichael number A002997 iff a(n) is squarefree and s_p(a(n)) == 1 (mod p-1) for every prime factor p of a(n). See Kellner and Sondow 2019. - _Jonathan Sondow_, Mar 16 2019

%H Amiram Eldar, <a href="/A324457/b324457.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..477 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[10^4], f[#] &]

%Y Subsequences are A002997, A324315, and A324458.

%Y Subsequence of A324459 and A324857.

%Y Cf. A324316, A324455, A324456, A324460.

%K nonn,base

%O 1,1

%A _Bernd C. Kellner_, Feb 28 2019