cp's OEIS Frontend

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.

Showing 1-2 of 2 results.

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

Views

Author

Altug Alkan, Dec 10 2015

Keywords

Comments

For the first nine Carmichael numbers of the form k^3 + 1, the values of k + 1 are 13, 37, 139, 271, 4801, 7561, 12841, 14701, 678481 and only 14701 is not a prime number.
The sequence also includes: 32490089562753934948660824001, 782293837499544845175052968001, 611009032634107957276386802479001. - Daniel Suteu, Dec 25 2020

Examples

			2628073 is a term because it is a Carmichael number and 2628073 = 138^3 + 1.

Links

Crossrefs

Cf. A002997, A265285, A265328.

Programs

  • 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(k=n^3+1), print1(k, ", ")))
  • PARI
    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

Extensions

More terms from Alois P. Heinz, Dec 10 2015
a(10)-a(13) from Daniel Suteu, Dec 25 2020
a(14) from Daniel Suteu confirmed by Amiram Eldar, May 02 2024

A265285 Carmichael numbers (A002997) k such that k-1 is a square.

Original entry on oeis.org

46657, 2433601, 67371265, 351596817937, 422240040001, 18677955240001, 458631349862401, 286245437364810001, 20717489165917230086401
Offset: 1

Views

Author

Altug Alkan, Dec 06 2015

Keywords

Comments

This sequence contains all Carmichael numbers n such that for all primes p dividing n, p-1 divides n-1 and furthermore, n-1 is a square.
Numbers sqrt(a(n)-1) form a subsequence of A135590. - Max Alekseyev, Apr 25 2024

Examples

			46657 is a term because 46657 - 1 = 46656 = 216^2.
2433601 is a term because 2433601 - 1 = 2433600 = 1560^2.

Links

Crossrefs

Subsequence of A265237 and of A265328.
Cf. A002997, A135590, A265237, A303791.

Programs

  • Maple
    isA002997:= proc(n) local F,p;
         if n::even or isprime(n)  then return false fi;
         F:= ifactors(n)[2];
         if max(seq(f[2],f=F)) > 1 then return false fi;
         andmap(f -> (n-1) mod (f[1]-1) = 0,  F)
end proc:
select(isA002997, [seq(4*i^2+1,i=1..10^6)]); # Robert Israel, Dec 08 2015
  • 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) && issquare(n-1), print1(n, ", ")))
  • PARI
    lista(kmax) = {my(m); for(k = 2, kmax, m = k^2 + 1; if(!isprime(m), f = factor(k); for(i = 1, #f~, f[i, 2] *= 2); fordiv(f, d, if(!(m % (d+1)) && isprime(d+1), m /= (d+1))); if(m == 1, print1(k^2 + 1, ", ")))); } \\ Amiram Eldar, May 02 2024

Extensions

a(4)-a(5), using A002997 b-file, from Michel Marcus, Dec 07 2015
a(6) and a(7) from Robert Israel, Dec 08 2015
a(8) from Max Alekseyev, Apr 30 2018
a(9) from Daniel Suteu confirmed by Max Alekseyev, Apr 25 2024
Showing 1-2 of 2 results.

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