A204847 Primitive cofactor of n-th repunit A002275(n).

1, 11, 111, 101, 11111, 91, 1111111, 10001, 333667, 9091, 11111111111, 9901, 1111111111111, 909091, 90090991, 100000001, 11111111111111111, 999001, 1111111111111111111, 99009901, 900900990991, 826446281, 11111111111111111111111, 99990001, 100001000010000100001

Offset: 1

Author

N. J. A. Sloane, Jan 19 2012

Keywords

• nonn

Comments

Except for a(1) = 1 and a(3) = 111, this is the Zsigmondy numbers for a = 10, b = 1: Zs(n, 10, 1) is the greatest divisor of 10^n - 1^n that is coprime to 10^m - 1^m for all positive integers m < n. The prime terms are called unique primes or unique period primes (A007615).

Differs from A019328 for n = 1, 9, 22, 27, 42, ... - Jianing Song, Apr 30 2018

Links

• Makoto Kamada, Factorizations of 11...11 (Repunit).
• Samuel Yates, Cofactors of repunits, Journal of Recreational Mathematics, Vol. 8(2), pp. 99, 1975-76.
• Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Crossrefs

Cf. A002275, A019328, A102380, A204845, A204846.

Programs

PARI

lista(nn) = {vf = []; vfs = []; for (n=1, nn, if (n==1, print1(n, ", "), f = factor((10^n-1)/9)[,1]; vkeep = []; for (k = 1, #f~, if (!vecsearch(vfs, f[k]), vkeep = concat(vkeep, f[k]));); print1(prod(j=1, #vkeep, vkeep[j]), ", "); vf = concat(vf, vkeep); vfs = Set(vf);););} \\ Michel Marcus, May 18 2018

Formula

Equals A002275(n)/(product of terms in n-th row of A204845).

Extensions

a(11)-a(24) from Jianing Song, Apr 30 2018

a(25) from Jinyuan Wang, May 02 2021