

A324316


Primary Carmichael numbers.


23



1729, 2821, 29341, 46657, 252601, 294409, 399001, 488881, 512461, 1152271, 1193221, 1857241, 3828001, 4335241, 5968873, 6189121, 6733693, 6868261, 7519441, 10024561, 10267951, 10606681, 14469841, 14676481, 15247621, 15829633, 17098369, 17236801, 17316001, 19384289, 23382529, 29111881, 31405501, 34657141, 35703361, 37964809
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Squarefree integers m > 1 such that if prime p divides m, then the sum of the basep digits of m equals p. It follows that m is then a Carmichael number (A002997).
Conjecture: the sequence is infinite.
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.
The distribution of primary Carmichael numbers is A324317.
See Kellner and Sondow 2019 and Kellner 2019.
Primary Carmichael numbers are special polygonal numbers A324973. The rank of the nth primary Carmichael number is A324976(n). See Kellner and Sondow 2019.  Jonathan Sondow, Mar 26 2019
The first term is the HardyRamanujan number.  Omar E. Pol, Jan 09 2020


LINKS

Bernd C. Kellner, Table of n, a(n) for n = 1..10000 (computed by using Pinch's database, see link below)
Bernd C. Kellner and Jonathan Sondow, PowerSum Denominators, Amer. Math. Monthly, 124 (2017), 695709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of basep digits, arXiv:1902.10672 [math.NT], 2019.
Bernd C. Kellner, On primary Carmichael numbers, arXiv:1902.11283 [math.NT], 2019.
R. G. E. Pinch, The Carmichael numbers up to 10^18, 2008.
Index entries for sequences related to Carmichael numbers.


FORMULA

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 <= p1 for i = 1, 2, ..., k (note that a_0 = 0).


EXAMPLE

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.


MATHEMATICA

SD[n_, p_] := If[n < 1  p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestCP[n_] := (n > 1) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] == # &];
Select[Range[1, 10^7, 2], TestCP[#] &]


PROG

(Perl) use ntheory ":all"; my $m; forsquarefree { $m=$_; say if @_ > 2 && is_carmichael($m) && vecall { $_ == vecsum(todigits($m, $_)) } @_; } 1e7; # Dana Jacobsen, Mar 28 2019


CROSSREFS

Subsequence of A002997, A324315.
Least primary Carmichael number with n prime factors is A306657.
Cf. also A005117, A195441, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405, A324973, A324976, A001235.
Sequence in context: A154729 A083737 A182208 * A182207 A138129 A242880
Adjacent sequences: A324313 A324314 A324315 * A324317 A324318 A324319


KEYWORD

nonn,base


AUTHOR

Bernd C. Kellner and Jonathan Sondow, Feb 21 2019


STATUS

approved



