A324457 Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) >= p.

24, 45, 48, 72, 96, 120, 144, 189, 192, 216, 224, 225, 231, 240, 288, 315, 320, 325, 336, 352, 360, 384, 405, 432, 450, 480, 525, 540, 560, 561, 567, 576, 594, 600, 637, 648, 672, 704, 720, 768, 792, 819, 825, 832, 850, 864, 891, 896, 924, 945, 960, 975, 980

Offset

1,1

Comments

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

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

Being a subsequence of A324459, a term m has the following properties:

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

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.

In the terminology of A324459, the prime factorization of m equals an s-decomposition of m.

See Kellner 2019.

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..477 from Bernd C. Kellner)

Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Example

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

Mathematica

s[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
f[n_] := AllTrue[Transpose[FactorInteger[n]][[1]], s[n, #] >= # &];
Select[Range[10^4], f[#] &]

Crossrefs

Subsequences are A002997, A324315, and A324458.

Subsequence of A324459 and A324857.

Cf. A324316, A324455, A324456, A324460.

Sequence in context: A322843 A211568 A324459 * A357876 A217080 A226327

Adjacent sequences: A324454 A324455 A324456 * A324458 A324459 A324460

Keyword

nonn,base

Author

Bernd C. Kellner, Feb 28 2019

Status

approved