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Primitive cofactor of n-th repunit A002275(n).
4

%I #31 May 05 2021 14:06:43

%S 1,11,111,101,11111,91,1111111,10001,333667,9091,11111111111,9901,

%T 1111111111111,909091,90090991,100000001,11111111111111111,999001,

%U 1111111111111111111,99009901,900900990991,826446281,11111111111111111111111,99990001,100001000010000100001

%N Primitive cofactor of n-th repunit A002275(n).

%C 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).

%C Differs from A019328 for n = 1, 9, 22, 27, 42, ... - _Jianing Song_, Apr 30 2018

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit">Factorizations of 11...11 (Repunit)</a>.

%H Samuel Yates, <a href="http://www.geocities.jp/ma85003/math/repdigit.pdf">Cofactors of repunits</a>, Journal of Recreational Mathematics, Vol. 8(2), pp. 99, 1975-76.

%H Samuel Yates, <a href="http://www.jstor.org/stable/2689643">The Mystique of Repunits</a>, Math. Mag. 51 (1978), 22-28.

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

%o (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

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

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jan 19 2012

%E a(11)-a(24) from _Jianing Song_, Apr 30 2018

%E a(25) from _Jinyuan Wang_, May 02 2021