A324316 - OEIS

%I #61 Jul 04 2022 01:32:54

%S 1729,2821,29341,46657,252601,294409,399001,488881,512461,1152271,

%T 1193221,1857241,3828001,4335241,5968873,6189121,6733693,6868261,

%U 7519441,10024561,10267951,10606681,14469841,14676481,15247621,15829633,17098369,17236801,17316001,19384289,23382529,29111881,31405501,34657141,35703361,37964809

%N Primary Carmichael numbers.

%C 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 Dickson's conjecture implies that the sequence is infinite, see Kellner 2019.

%C 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 The distribution of primary Carmichael numbers is A324317.

%C See Kellner and Sondow 2019 and Kellner 2019.

%C 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 The first term is the Hardy-Ramanujan number. - _Omar E. Pol_, Jan 09 2020

%H 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 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 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 p.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.

%H 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 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers.</a>

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

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

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

%t Select[Range[1, 10^7, 2], TestCP[#] &]

%o (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 (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**6) if ok(k)]) # _Michael S. Branicky_, Jul 03 2022

%Y Subsequence of A002997, A324315.

%Y Least primary Carmichael number with n prime factors is A306657.

%Y Cf. also A005117, A195441, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405, A324973, A324976, A001235.

%K nonn,base

%O 1,1

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