A005422 -id:A005422 - cp's OEIS frontend

A366718 Largest prime factor of 12^n - 1.

11, 13, 157, 29, 22621, 157, 4943, 233, 80749, 22621, 266981089, 20593, 20369233, 13063, 22621, 260753, 74876782031, 80749, 29043636306420266077, 85403261, 8177824843189, 57154490053, 321218438243, 2227777, 12629757106815551, 20369233, 86769286104133

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..310
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. A024140, A006530, A005420, A074477, A274906, A074479, A274907, A074249, A274908, A274909, A005422, A274910, A366707, A366708, A366709, A366710, A366711, A366717, A366720.

[Maximum(PrimeDivisors(12^n-1)): n in [1..40]];

Table[FactorInteger[12^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024140(n)).

A081318 Integers n such that the reciprocal of the largest prime factor of 10^n-1 is not a repeating decimal fraction with a period of n.

Original entry on oeis.org

6, 18, 22, 30, 32, 38, 42, 46, 54, 66, 74, 78, 82, 90, 94, 96, 110, 118, 132, 138, 146, 154, 162, 174, 186, 194, 198, 206, 210, 218, 228, 231, 240, 242, 254, 258, 260, 264, 266, 268, 274, 282, 284, 286, 298, 300, 306, 310, 318, 322, 334, 338, 344, 348

Offset: 1

Hugo Pfoertner, Mar 18 2003

For all but three of the terms through a(41)=274, the reciprocal of the largest prime factor of 10^a(n)-1 is a decimal fraction with a period of a(n)/2. Of the three exceptions, there are two (a(32)=231 and a(38)=264) where the period is a(n)/3, and one (a(19)=132) where the period is a(n)/4. - Jon E. Schoenfield, Jun 27 2010

			30 is in the sequence because the factorization of 10^30-1 is 3^3*7*11*13*31*37*41*211*241*271*2161*9091*2906161 and 2906161 occurs already in 10^15-1=3^3*31*37*41*271*2906161 producing a decimal fraction with a period of 15, (1/2906161=0.000000344096559000000344096559000000344...)

Makoto Kamada, Factorizations of 11...11 (repunit).

Cf. A081317.

Integers n such that A061075(n) is not equal to A005422(n).

More terms from Hans Havermann, May 31 2003
Terms a(31)-a(37) from Jon E. Schoenfield, Jun 19 2010
Terms a(38)-a(41) added, link added, and earlier comment expanded by Jon E. Schoenfield, Jun 27 2010
a(42)-a(54) from Max Alekseyev, Aug 17 2013, Apr 26 2022

A177928 Let n be the number whose square n^2 has the decimal expansion { d(1) d(2) ... d(D) }, and let q be the corresponding number whose decimal expansion is { d(2) d(3) ... d(D) d(1)}. Sequence lists numbers n dividing q.

Original entry on oeis.org

1, 2, 3, 9, 27, 33, 66, 99, 123, 246, 271, 333, 351, 407, 429, 462, 481, 518, 546, 567, 666, 693, 702, 715, 777, 814, 819, 924, 936, 999, 1434, 2151, 2868, 3333, 4521, 4818, 6666, 7227, 7373, 7535, 8631, 9042, 9999, 33333, 53658, 54546, 66666, 80487, 81819

Offset: 1

Michel Lagneau, May 15 2010

A178028 is a subsequence of this sequence.

When n divides q, n divides d(D)*(10^D - 1) because q = 10*n^2 - d(D)*(10^D - 1). If n is prime, n divides (10^D - 1); for example, the prime term 271 divides 10^5 - 1 = 99999 = 271*369.

			429 is in the sequence because 429^2 = 184041 and 840411/429 = 1959.

Eric Weisstein's World of Mathematics, Repunits

Cf. A002275, A003020, A005422, A067063, A102380, A178028.

for n from 1 to 10^6 do: d:=convert(n^2, base, 10):n1:=nops(d):s:=sum('d[i]*10^i','i'=1..n1-1)+d[n1]:if irem(s,n)=0 then printf(`%d, `,n):else fi:od:

Select[Range[100000], Mod[FromDigits[RotateLeft[IntegerDigits[#^2]]], #] == 0 &] (* T. D. Noe, Jul 27 2012 *)

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A366718 Largest prime factor of 12^n - 1.

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A081318 Integers n such that the reciprocal of the largest prime factor of 10^n-1 is not a repeating decimal fraction with a period of n.

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A177928 Let n be the number whose square n^2 has the decimal expansion { d(1) d(2) ... d(D) }, and let q be the corresponding number whose decimal expansion is { d(2) d(3) ... d(D) d(1)}. Sequence lists numbers n dividing q.

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