A324458 Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) = p.

45, 325, 405, 637, 891, 1729, 2821, 3751, 4961, 6517, 7381, 8125, 8281, 10625, 13357, 21141, 26353, 28033, 29341, 31213, 33125, 35443, 46657, 47081, 58621, 65341, 74431, 78625, 81289, 94501, 98125, 99937, 123823, 146461, 231601, 236321, 252601, 254221, 294409

Offset: 1

Author

Bernd C. Kellner, Feb 28 2019

Keywords

• nonn
• base

Comments

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

The sequence contains the primary Carmichael numbers A324316.

Being a subsequence of A324460, 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 A324460, the prime factorization of m equals a strict s-decomposition of m.

See Kellner 2019.

a(n) is squarefree iff a(n) is a primary Carmichael number A324316. - Jonathan Sondow, Mar 16 2019

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..117 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.

Crossrefs

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

Cf. A324455, A324456.

Programs

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[10^7], f[#] &]