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 A324315 Squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p. 15
 231, 561, 1001, 1045, 1105, 1122, 1155, 1729, 2002, 2093, 2145, 2465, 2821, 3003, 3315, 3458, 3553, 3570, 3655, 3927, 4186, 4199, 4522, 4774, 4845, 4862, 5005, 5187, 5565, 5642, 5681, 6006, 6118, 6270, 6279, 6545, 6601, 6670, 6734, 7337, 7395, 7735, 8177, 8211, 8265, 8294, 8323, 8463, 8645, 8789, 8855, 8911, 9282, 9361, 9435, 9690, 9867 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The sequence is infinite, because it contains all Carmichael numbers (A002997). If m is a term and p is a prime factor of m, then p <= a*sqrt(m) with a = sqrt(11/21) = 0.7237..., where the bound is sharp. A term m must have at least 3 prime factors if m is odd, and must have at least 4 prime factors if m is even. m is a term if and only if m > 1 divides denominator(Bernoulli_m(x) - Bernoulli_m) = A195441(m-1). A term m is a Carmichael number iff s_p(m) == 1 (mod p-1) whenever prime p divides m, where s_p(m) is the sum of the base p digits of m. See Kellner and Sondow 2019. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017. Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019. FORMULA a_1 + a_2 + ... + a_k >= p for m = a_1 * p + a_2 * p^2 + ... + a_k * p^k with 0 <= a_i <= p-1 for i = 1, 2, ..., k (note that a_0 = 0). EXAMPLE 231 = 3 * 7 * 11 is squarefree, and 231 in base 3 is 22120_3 = 2 * 3^4 + 2 * 3^3 + 1 * 3^2 + 2 * 3 + 0 with 2+2+1+2+0 = 7 >= 3, and 231 = 450_7 with 4+5+0 = 9 >= 7, and 231 = 1a0_11 with 1+a+0 = 1+10+0 = 11 >= 11, so 231 is a member. MATHEMATICA SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]]; LP[n_] := Transpose[FactorInteger[n]][]; TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &]; Select[Range[10^4], TestS[#] &] PROG (Python) from sympy import factorint from sympy.ntheory import digits def ok(n): pf = factorint(n) if n < 2 or max(pf.values()) > 1: return False return all(sum(digits(n, p)[1:]) >= p for p in pf) print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Jul 03 2022 CROSSREFS Cf. A002997, A005117, A195441, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405. Sequence in context: A117223 A160355 A211712 * A276832 A324319 A360214 Adjacent sequences: A324312 A324313 A324314 * A324316 A324317 A324318 KEYWORD nonn,base AUTHOR Bernd C. Kellner and Jonathan Sondow, Feb 21 2019 STATUS approved

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Last modified November 30 02:55 EST 2023. Contains 367452 sequences. (Running on oeis4.)