A114207 -id:A114207 - cp's OEIS frontend

A014950 Numbers m such that m divides 10^m - 1.

1, 3, 9, 27, 81, 111, 243, 333, 729, 999, 2187, 2997, 4107, 6561, 8991, 12321, 13203, 19683, 20439, 26973, 36963, 39609, 59049, 61317, 80919, 110889, 118827, 151959, 177147, 183951, 242757, 332667, 356481, 455877, 488511, 531441, 551853, 728271

Offset: 1

Olivier Gérard

nonn

Also, m such that m | R(m) = A002275(m). - Lekraj Beedassy, Mar 25 2005
For n > 1, 3 divides a(n). If m is in the sequence and d divides m then for each positive integer k, d^k*m is in the sequence. So if m is in the sequence then m^k is in the sequence for each positive integer k. In particular, 3^k is in this sequence for all k. - Farideh Firoozbakht, Apr 14 2010
Numbers m such that m divides s(m), where s(1) = 1, s(k) = s(k-1) + k*10^(k-1).
Number of terms <= 10^k, beginning with k = 0: 1, 3, 5, 10, 15, 25, 41, 68, 108, 178, 291, ... - Robert G. Wilson v, Nov 30 2013
Numbers m such that m divides A033713(m). - Hans Havermann, Jan 25 2014

J. D. E. Konhauser et al., Which Way Did The Bicycle Go? Problem 80 pp. 26; 133, Dolciani Math. Exp., No. 18, MAA, Washington DC, 1996.

Hans Havermann, Table of n, a(n) for n = 1..1600 (first 800 terms from Robert G. Wilson v)
C. Cooper and R. E. Kennedy, Niven Repunits and 10^n = 1 (mod n), The Fibonacci Quarterly, pp. 139-143, vol 27, May 02 1989.
Hans Havermann, A014950 factorized and atomized.

Cf. A066364, A114207, A122787, A127100, A232769.

Select[ Range[3, 1000000, 6], PowerMod[10, #, #] == 1 &] (* modified by Robert G. Wilson v, Dec 03 2013 *)
k = 3; A014950 = {1}; While[k < 1000000, If[ PowerMod[ 10, k, k] == 1, AppendTo[ A014950, k]; Print@ k]; k += 6]; A014950 (* Robert G. Wilson v, Nov 29 2013 *)

is(n)=Mod(10,n)^n==1 \\ Charles R Greathouse IV, Nov 29 2013

Solutions to 10^m == 1 (mod m). - Vladeta Jovovic

More terms from Vladeta Jovovic, Dec 18 2001
More terms from Larry Reeves (larryr(AT)acm.org), Jan 06 2005
Edited by Max Alekseyev, May 20 2011

A066364 Prime divisors of solutions to 10^n == 1 (mod n).

Original entry on oeis.org

3, 37, 163, 757, 1999, 5477, 8803, 9397, 13627, 15649, 36187, 40879, 62597, 106277, 147853, 161839, 215893, 231643, 281683, 295759, 313471, 333667, 338293, 478243, 490573, 607837, 647357, 743933, 988643, 1014877, 1056241, 1168711, 1353173, 1390757, 1487867, 1519591, 1627523, 1835083, 1912969, 2028119, 2029759, 2064529

Offset: 1

Vladeta Jovovic, Dec 21 2001

nonn

			10^27-1 = 3^5*37*757*333667*440334654777631 is a solution to the congruence.

Max Alekseyev and Hans Havermann (Max Alekseyev to 501), Table of n, a(n) for n = 1..2060
Rüdiger Jehn and Kester Habermann, Properties of terms of OEIS A342810, arXiv:2106.05866 [math.GM], 2021.
Makoto Kamada, Factorizations of 11...11 (Repunit).

Cf. A014950, A001270, A027889, A007138, A114207.

fQ[p_] := Block[{fi = First@# & /@ FactorInteger[ MultiplicativeOrder[ 10, p]]}, Union[ MemberQ[ lst, #] & /@ fi] == {True}]; k = 4; lst = {3}; While[k < 180000, If[ p = Prime@ k; fQ@ p, AppendTo[ lst, p]; Print@ p]; k++]; lst (* Robert G. Wilson v, Nov 30 2013 *)

S=Set([3]); forprime(p=7,10^6, v=factorint(znorder(Mod(10,p)))[,1]; if(length(setintersect(S,Set(v)))==length(v), S=setunion(S,[p])) ); print(vecsort(eval(S))) \\ Max Alekseyev, Nov 16 2005

A prime p is a term iff all prime divisors of ord_p(10) are terms, where ord_p(10) is the order of 10 modulo p. - Max Alekseyev, Nov 16 2005

Edited by Max Alekseyev, Nov 16 2005

Edited by Hans Havermann, Jul 11 2014

A171981 Smallest multiples from A129066 of primes from A171980.

Original entry on oeis.org

5, 75025, 9006076025, 332813125, 54036081025, 162108093025, 12304690625, 3662109765625, 1238325212525, 225150026875625, 8562180281412026525, 309581286250625, 15197626762525, 4520507828125, 2059936966552758203125

Offset: 1

Max Alekseyev, Jan 20 2010

nonn

A129066 lists integers n such that n divides n-th Fibonacci number A000045(n) with multiples of 12 excluded, while A171980 lists possible prime divisors of elements of A129066 in the increasing order. This sequence lists smallest multiples from A129066 of primes from A171980.

Cf. A114207, A140258

a(n) = min { A129066(m) : A171980(n)|A129066(m) }

A354027 Smallest k dividing 3^k + 4^k that have divisor A354026(n).

Original entry on oeis.org

7, 2653, 4941601, 236474833, 20930936750059, 4899815786803, 330219333283, 4455186224209, 9770481627816301, 351829430300299, 26064955344333473991073, 192144131632327, 3006539977771582357, 2059085249993107, 13468375028434716454309, 1753571527639980559, 2338095370082599813

Offset: 1

Max Alekseyev, May 15 2022

nonn

Smallest term of A045584 divisible by prime A354026(n).

Cf. A045584, A354026, A114207, A140258, A171981.

cp's OEIS Frontend

A014950 Numbers m such that m divides 10^m - 1.

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A066364 Prime divisors of solutions to 10^n == 1 (mod n).

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A171981 Smallest multiples from A129066 of primes from A171980.

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Formula

A354027 Smallest k dividing 3^k + 4^k that have divisor A354026(n).

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