This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A265328 #47 Feb 16 2025 08:33:27
%S A265328 1729,46657,2433601,2628073,19683001,67371265,110592000001,
%T A265328 351596817937,422240040001,432081216001,2116874304001,3176523000001,
%U A265328 18677955240001,458631349862401,286245437364810001,312328165704192001,12062716067698821000001,20717489165917230086401,211215936967181638848001,411354705193473163968001
%N A265328 Carmichael numbers (A002997) k such that k-1 is a perfect power (A001597).
%C A265328 From _Antti Karttunen_, Dec 08 2015: (Start)
%C A265328 The prime factorizations of the first six terms are:
%C A265328 7*13*19, 13*37*97, 17*37*53*73, 7*37*73*139, 13*37*151*271, 5*13*37*109*257
%C A265328 and the prime factorizations of the corresponding perfect powers (numbers one smaller) are:
%C A265328 (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).
%C A265328 (End)
%C A265328 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
%C A265328 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
%H A265328 G. Tarry, I. Franel, A. Korselt, and G. Vacca, <a href="https://oeis.org/wiki/File:Probl%C3%A8me_chinois.pdf">Problème chinois</a>, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
%H A265328 Samuel S. Wagstaff, <a href="https://doi.org/10.1007/s11139-024-00846-1">Ramanujan's taxicab number and its ilk</a>, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; <a href="https://www.researchgate.net/publication/379960100_Ramanujan's_taxicab_number_and_its_ilk">ResearchGate link</a>, <a href="https://homes.cerias.purdue.edu/~ssw/taxi.pdf">author's copy</a>.
%H A265328 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>.
%H A265328 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%e A265328 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).
%e A265328 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).
%e A265328 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).
%t A265328 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 *)
%o A265328 (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}
%o A265328 for(n=1, 1e10, if(is_c(n) && ispower(n-1), print1(n, ", ")))
%o A265328 (Perl) use ntheory ":all"; foroddcomposites { say if is_power($_-1) && is_carmichael($_) } 1e8; # _Dana Jacobsen_, May 05 2017
%Y A265328 Contains A265285 as a subsequence.
%Y A265328 Cf. A001597, A002997, A135590, A270840.
%K A265328 nonn,hard
%O A265328 1,1
%A A265328 _Altug Alkan_, Dec 07 2015
%E A265328 More terms from _Dana Jacobsen_, May 05 2017
%E A265328 a(17) from _Daniel Suteu_ confirmed, a(18)-a(20) added by _Max Alekseyev_, Apr 25 2024