A265628
Carmichael numbers (A002997) of the form k^3 + 1.
Original entry on oeis.org
1729, 46657, 2628073, 19683001, 110592000001, 432081216001, 2116874304001, 3176523000001, 312328165704192001, 12062716067698821000001, 211215936967181638848001, 411354705193473163968001, 14295706553536348081491001, 32490089562753934948660824001
Offset: 1
2628073 is a term because it is a Carmichael number and 2628073 = 138^3 + 1.
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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(k=n^3+1), print1(k, ", ")))
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lista(kmax) = {my(m); for(k = 2, kmax, f = factor(k); for(i = 1, #f~, f[i, 2] *= 3); m = k^3 + 1; fordiv(f, d, if(!(m % (d+1)) && isprime(d+1), m /= (d+1))); if(m == 1, print1(k^3 + 1, ", "))); } \\ Amiram Eldar, May 02 2024
A265328
Carmichael numbers (A002997) k such that k-1 is a perfect power (A001597).
Original entry on oeis.org
1729, 46657, 2433601, 2628073, 19683001, 67371265, 110592000001, 351596817937, 422240040001, 432081216001, 2116874304001, 3176523000001, 18677955240001, 458631349862401, 286245437364810001, 312328165704192001, 12062716067698821000001, 20717489165917230086401, 211215936967181638848001, 411354705193473163968001
Offset: 1
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.
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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 *)
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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, ", ")))
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use ntheory ":all"; foroddcomposites { say if is_power($-1) && is_carmichael($) } 1e8; # Dana Jacobsen, May 05 2017
A303791
Carmichael numbers (A002997) that are central polygonal numbers (A002061).
Original entry on oeis.org
5310721, 2278677961, 9593125081, 29859667201, 467593730289953281, 98538479002618905601, 146842414757227736821
Offset: 1
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