

A324457


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


6



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
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OFFSET

1,1


COMMENTS

The function s_p(m) gives the sum of the basep 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 sdecomposition of m.
See Kellner 2019.
a(n) is a Carmichael number A002997 iff a(n) is squarefree and s_p(a(n)) == 1 (mod p1) for every prime factor p of a(n). See Kellner and Sondow 2019.  Jonathan Sondow, Mar 16 2019


LINKS



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



KEYWORD

nonn,base


AUTHOR



STATUS

approved



