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

1,2

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

Table of n, a(n) for n=1..25.

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.

Formula

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

Prog

(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

Crossrefs

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

Sequence in context: A061493 A093788 A327992 * A098759 A273977 A358441

Adjacent sequences: A204844 A204845 A204846 * A204848 A204849 A204850

Keyword

nonn

Author

N. J. A. Sloane, Jan 19 2012

Extensions

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

a(25) from Jinyuan Wang, May 02 2021

Status

approved