# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a324315 Showing 1-1 of 1 %I A324315 #44 Jul 04 2022 01:32:21 %S A324315 231,561,1001,1045,1105,1122,1155,1729,2002,2093,2145,2465,2821,3003, %T A324315 3315,3458,3553,3570,3655,3927,4186,4199,4522,4774,4845,4862,5005, %U A324315 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 %N 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. %C A324315 The sequence is infinite, because it contains all Carmichael numbers (A002997). %C A324315 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. %C A324315 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. %C A324315 m is a term if and only if m > 1 divides denominator(Bernoulli_m(x) - Bernoulli_m) = A195441(m-1). %C A324315 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. %C A324315 See Kellner and Sondow 2019. %H A324315 Amiram Eldar, Table of n, a(n) for n = 1..10000 %H A324315 Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017. %H A324315 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. %F A324315 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). %e A324315 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. %t A324315 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]]; %t A324315 LP[n_] := Transpose[FactorInteger[n]][[1]]; %t A324315 TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &]; %t A324315 Select[Range[10^4], TestS[#] &] %o A324315 (Python) %o A324315 from sympy import factorint %o A324315 from sympy.ntheory import digits %o A324315 def ok(n): %o A324315 pf = factorint(n) %o A324315 if n < 2 or max(pf.values()) > 1: return False %o A324315 return all(sum(digits(n, p)[1:]) >= p for p in pf) %o A324315 print([k for k in range(10**4) if ok(k)]) # _Michael S. Branicky_, Jul 03 2022 %Y A324315 Cf. A002997, A005117, A195441, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405. %K A324315 nonn,base %O A324315 1,1 %A A324315 _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 21 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE