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A324315
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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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15
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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.
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FORMULA
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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).
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EXAMPLE
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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.
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MATHEMATICA
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SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestS[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # &];
Select[Range[10^4], TestS[#] &]
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,base
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AUTHOR
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Bernd C. Kellner and Jonathan Sondow, Feb 21 2019
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STATUS
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approved
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