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A324316 Primary Carmichael numbers.

%I A324316 #61 Jul 04 2022 01:32:54
%S A324316 1729,2821,29341,46657,252601,294409,399001,488881,512461,1152271,
%T A324316 1193221,1857241,3828001,4335241,5968873,6189121,6733693,6868261,
%U A324316 7519441,10024561,10267951,10606681,14469841,14676481,15247621,15829633,17098369,17236801,17316001,19384289,23382529,29111881,31405501,34657141,35703361,37964809
%N A324316 Primary Carmichael numbers.
%C A324316 Squarefree integers m > 1 such that if prime p divides m, then the sum of the base-p digits of m equals p. It follows that m is then a Carmichael number (A002997).
%C A324316 Dickson's conjecture implies that the sequence is infinite, see Kellner 2019.
%C A324316 If m is a term and p is a prime factor of m, then p <= a*sqrt(m) with a = sqrt(66337/132673) = 0.7071..., where the bound is sharp.
%C A324316 The distribution of primary Carmichael numbers is A324317.
%C A324316 See Kellner and Sondow 2019 and Kellner 2019.
%C A324316 Primary Carmichael numbers are special polygonal numbers A324973. The rank of the n-th primary Carmichael number is A324976(n). See Kellner and Sondow 2019. - _Jonathan Sondow_, Mar 26 2019
%C A324316 The first term is the Hardy-Ramanujan number. - _Omar E. Pol_, Jan 09 2020
%H A324316 Bernd C. Kellner, <a href="/A324316/b324316.txt">Table of n, a(n) for n = 1..10000</a> (computed by using Pinch's database, see link below)
%H A324316 Bernd C. Kellner, <a href="http://math.colgate.edu/~integers/w38/w38.pdf">On primary Carmichael numbers</a>, #A38 Integers 22 (2022), 39 p.; arXiv:<a href="https://arxiv.org/abs/1902.11283">1902.11283</a> [math.NT], 2019.
%H A324316 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 A324316 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 p.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.
%H A324316 R. G. E. Pinch, <a href="http://www.s369624816.websitehome.co.uk/rgep/cartable.html">The Carmichael numbers up to 10^18</a>, 2008.
%H A324316 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers.</a>
%F A324316 a_1 + a_2 + ... + a_k = p if p is prime and 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 A324316 1729 = 7 * 13 * 19 is squarefree, and 1729 in base 7 is 5020_7 = 5 * 7^3 + 0 * 7^2 + 2 * 7 + 0 with 5+0+2+0 = 7, and 1729 in base 13 is a30_13 with a+3+0 = 10+3+0 = 13, and 1729 in base 19 is 4f0_19 with 4+f+0 = 4+15+0 = 19, so 1729 is a member.
%t A324316 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
%t A324316 LP[n_] := Transpose[FactorInteger[n]][[1]];
%t A324316 TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
%t A324316 Select[Range[1, 10^7, 2], TestCP[#] &]
%o A324316 (Perl) use ntheory ":all"; my $m; forsquarefree { $m=$_; say if @_ > 2 && is_carmichael($m) && vecall { $_ == vecsum(todigits($m,$_)) } @_; } 1e7; # _Dana Jacobsen_, Mar 28 2019
%o A324316 (Python)
%o A324316 from sympy import factorint
%o A324316 from sympy.ntheory import digits
%o A324316 def ok(n):
%o A324316     pf = factorint(n)
%o A324316     if n < 2 or max(pf.values()) > 1: return False
%o A324316     return all(sum(digits(n, p)[1:]) == p for p in pf)
%o A324316 print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Jul 03 2022
%Y A324316 Subsequence of A002997, A324315.
%Y A324316 Least primary Carmichael number with n prime factors is A306657.
%Y A324316 Cf. also A005117, A195441, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405, A324973, A324976, A001235.
%K A324316 nonn,base
%O A324316 1,1
%A A324316 _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 21 2019