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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]][[1]];

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 March 29 01:21 EDT 2023. Contains 361596 sequences. (Running on oeis4.)