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A265328
Carmichael numbers (A002997) k such that k-1 is a perfect power (A001597).
2
1729, 46657, 2433601, 2628073, 19683001, 67371265, 110592000001, 351596817937, 422240040001, 432081216001, 2116874304001, 3176523000001, 18677955240001, 458631349862401, 286245437364810001, 312328165704192001, 12062716067698821000001, 20717489165917230086401, 211215936967181638848001, 411354705193473163968001
OFFSET
1,1
COMMENTS
From Antti Karttunen, Dec 08 2015: (Start)
The prime factorizations of the first six terms are:
7*13*19, 13*37*97, 17*37*53*73, 7*37*73*139, 13*37*151*271, 5*13*37*109*257
and the prime factorizations of the corresponding perfect powers (numbers one smaller) are:
(2^6 * 3^3), (2^6 * 3^6), (2^6 * 3^2 * 5^2 * 13^2), (2^3 * 3^3 * 23^3), (2^3 * 3^9 * 5^3), (2^8 * 3^6 * 19^2).
(End)
For each k in {22934100, 59553720, 74371320, 242699310, 3190927740, 9214178820, 84855997590}, which is a subset of A270840, k^3+1 is a Carmichael number. - Daniel Suteu, Aug 24 2019
LINKS
G. Tarry, I. Franel, A. Korselt, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Eric Weisstein's World of Mathematics, Carmichael Number.
EXAMPLE
1729 = 7*13*19 is a term because 1729 - 1 = 1728 = 12^3, and 7-1 = 6, 13-1 = 12 and 19-1 = 18 can be all constructed from the primes available in 1728 = (2^6 * 3^3).
2433601 = 17*37*53*73 is a term because 2433601 - 1 = 2433600 = 1560^2, and 16, 36, 52 and 72 can be all constructed from the primes available in 2433600 = (2^6 * 3^2 * 5^2 * 13^2).
67371265 = 5*13*37*109*257 is a term because 67371264 = 8208^2, and 4 (= 2*2), 12 (= 2*2*3), 36 (= 2*2*3*3), 108 (= 2*2*3*3*3) and 256 (= 2^8) can be all constructed from the primes available in 67371264 = (2^8 * 3^6 * 19^2).
MATHEMATICA
Select[Cases[Range[1, 10^7, 2], n_ /; Mod[n, CarmichaelLambda@ n] == 1 && ! PrimeQ@ n], GCD @@ FactorInteger[# - 1][[All, 2]] > 1 &] (* Michael De Vlieger, Dec 14 2015, after Ant King at A001597 and Artur Jasinski at A002997 *)
PROG
(PARI) is_c(n)={my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
for(n=1, 1e10, if(is_c(n) && ispower(n-1), print1(n, ", ")))
(Perl) use ntheory ":all"; foroddcomposites { say if is_power($_-1) && is_carmichael($_) } 1e8; # Dana Jacobsen, May 05 2017
CROSSREFS
Contains A265285 as a subsequence.
Sequence in context: A272885 A272935 A284671 * A265628 A306479 A272798
KEYWORD
nonn,hard
AUTHOR
Altug Alkan, Dec 07 2015
EXTENSIONS
More terms from Dana Jacobsen, May 05 2017
a(17) from Daniel Suteu confirmed, a(18)-a(20) added by Max Alekseyev, Apr 25 2024
STATUS
approved