A270840 -id:A270840 - cp's OEIS frontend

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

Altug Alkan, Dec 07 2015

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
Wagstaff (2024) found that there are no Carmichael numbers k below 10^21 such that k+1 is a perfect power. - Amiram Eldar, Dec 29 2024

			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).

G. Tarry, I. Franel, A. Korselt, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Samuel S. Wagstaff, Ramanujan's taxicab number and its ilk, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; ResearchGate link, author's copy.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Contains A265285 as a subsequence.

Cf. A001597, A002997, A135590, A270840.

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 *)

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, ", ")))

use ntheory ":all"; foroddcomposites { say if is_power($-1) && is_carmichael($) } 1e8; # Dana Jacobsen, May 05 2017

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

A265653 Integers k such that (k-1)^3 + 1 is a Fermat pseudoprime to base 2 (A001567).

Original entry on oeis.org

13, 37, 139, 271, 547, 4801, 7561, 12841, 14701, 358201, 678481, 16139971, 22934101, 55058581, 59553721, 74371321, 113068381, 116605861, 242699311, 997521211, 1592680321, 1652749201, 3190927741, 5088964801, 6974736757, 9214178821

Offset: 1

Altug Alkan, Dec 12 2015

nonn

Corresponding Fermat pseudoprimes to base 2 are 1729, 46657, 2628073, 19683001, 162771337, 110592000001, 432081216001, ...

There is only one composite term up to 10^10: 14701. It also appears in A265628 (see comments). Can we say that if there is a Fermat pseudoprime to base 2 of the form (k-1)^3 + 1, k is a prime number most of the time? Are there other composite terms like 14701?

			13 is a term because (13-1)^3 + 1 = 1729, which is a Fermat pseudoprime to base 2.
37 is a term because (37-1)^3 + 1 = 46657, which is a Fermat pseudoprime to base 2.

Cf. A000040, A001567, A265628, A270840.

Select[Range[10^6], ! PrimeQ@ # && PowerMod[2, (# - 1), #] == 1 &@((# - 1)^3 + 1) &] (* Michael De Vlieger, Dec 12 2015, after Farideh Firoozbakht at A001567 *)

is(n) = {Mod(2, n)^n==2 & !isprime(n) & n>1};
for(n=1, 1e10, if(is((n-1)^3+1), print1(n, ", ")));

a(n) = A270840(n) + 1.

cp's OEIS Frontend

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

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