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 A324458 Numbers m > 1 such that every prime divisor p of m satisfies s_p(m) = p. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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, 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, 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]][], s[n, #] == # &]; Select[Range[10^7], f[#] &] CROSSREFS Subsequence is A324316. Subsequence of A324457, A324459, and A324460. Cf. A324455, A324456. Sequence in context: A291177 A341568 A093761 * A073873 A272850 A129153 Adjacent sequences: A324455 A324456 A324457 * A324459 A324460 A324461 KEYWORD nonn,base AUTHOR Bernd C. Kellner, Feb 28 2019 STATUS approved

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Last modified December 10 18:07 EST 2023. Contains 367717 sequences. (Running on oeis4.)