A265328 Carmichael numbers (A002997) k such that k-1 is a perfect power (A001597).

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

Table of n, a(n) for n=1..20.

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.

Index entries for sequences related to Carmichael numbers.

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.

Cf. A001597, A002997, A135590, A270840.

Sequence in context: A272885 A272935 A284671 * A265628 A306479 A272798

Adjacent sequences: A265325 A265326 A265327 * A265329 A265330 A265331

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