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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.
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%I #44 Jul 04 2022 01:32:21

%S 231,561,1001,1045,1105,1122,1155,1729,2002,2093,2145,2465,2821,3003,

%T 3315,3458,3553,3570,3655,3927,4186,4199,4522,4774,4845,4862,5005,

%U 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 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 The sequence is infinite, because it contains all Carmichael numbers (A002997).

%C 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 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 m is a term if and only if m > 1 divides denominator(Bernoulli_m(x) - Bernoulli_m) = A195441(m-1).

%C 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 See Kellner and Sondow 2019.

%H Amiram Eldar, <a href="/A324315/b324315.txt">Table of n, a(n) for n = 1..10000</a>

%H Bernd C. Kellner and Jonathan Sondow, <a href="https://doi.org/10.4169/amer.math.monthly.124.8.695">Power-Sum Denominators</a>, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:<a href="https://arxiv.org/abs/1705.03857">1705.03857</a> [math.NT], 2017.

%H Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/v52/v52.pdf">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, #A52 Integers 21 (2021), 21 pp.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.

%F 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 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 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];

%t LP[n_] := Transpose[FactorInteger[n]][[1]];

%t TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];

%t Select[Range[10^4], TestS[#] &]

%o (Python)

%o from sympy import factorint

%o from sympy.ntheory import digits

%o def ok(n):

%o pf = factorint(n)

%o if n < 2 or max(pf.values()) > 1: return False

%o return all(sum(digits(n, p)[1:]) >= p for p in pf)

%o print([k for k in range(10**4) if ok(k)]) # _Michael S. Branicky_, Jul 03 2022

%Y Cf. A002997, A005117, A195441, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405.

%K nonn,base

%O 1,1

%A _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 21 2019