A066364 -id:A066364 - cp's OEIS frontend

A114207 Smallest solution to 10^m == 1 (mod m) having the prime divisor A066364(n).

3, 111, 13203, 20439, 1997001, 22494039, 116226009, 761157, 278522253, 206613747, 17677747557, 835525881, 12933400720959, 228717562653, 5465090439, 13095850041, 431138536893, 4734551277, 58199580096201, 59875330325409, 228520359, 3003003, 257494085001, 1029221499627, 136635497220969

Offset: 1

Max Alekseyev, Nov 17 2005

nonn

			a(6)=m(5477)=22494039 since it is the smallest m such that 10^m == 1 (mod m) and 5477|m.

Ray Chandler, Table of n, a(n) for n = 1..2060 (first 501 terms from Max Alekseyev)

Cf. A066364.

{ m(p) = my(f,l,q); f=factorint(p)[,1]; l=p; for(i=1,length(f),q=znorder(Mod(10,f[i])); l=lcm(l,q); l=lcm(l,m(q)) ); l }

a(n)=m(p), where p=A066364(n) and m(p)=lcm(p, ord_p(10), m(q)) with q going over all prime divisors of ord_p(10).

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

Original entry on oeis.org

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

A057719 Prime factors of numbers in A006521 (numbers k that divide 2^k + 1).

Original entry on oeis.org

3, 19, 163, 571, 1459, 8803, 9137, 17497, 41113, 52489, 78787, 87211, 135433, 139483, 144667, 164617, 174763, 196579, 274081, 370009, 370387, 478243, 760267, 941489, 944803, 1041619, 1220347, 1236787, 1319323, 1465129, 1663579, 1994659

Offset: 1

Ignacio Larrosa Cañestro, Oct 26 2000

nonn

A prime p is in this sequence iff all prime divisors of ord_p(2)/2 are in this sequence, where ord_p(2) is the order of 2 modulo p. - Max Alekseyev, Jul 30 2006

			2^171 + 1 == 0 (mod 171), 171 = 3^2*19, 2^13203+1 == 0 (mod 13203), 13203 = 3^4*163.

Joerg Arndt, Table of n, a(n) for n = 1..220 (terms up to 10^9, terms for n = 1..100 from T. D. Noe)
Alexander Kalmynin, On Novák numbers, arXiv:1611.00417 [math.NT], 2016. See Chapter 4 p. 7 Novák primes.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Cf. A006521, A066364.

Cf. A136474, A136473.

S = {2}; Reap[For[p = 3, p < 2 10^6, p = NextPrime[p], f = FactorInteger[ MultiplicativeOrder[2, p]]; If[f[[1, 1]] != 2 || f[[1, 2]] != 1, Continue[]]; f = f[[All, 1]]; If[Length[Intersection[S, f]] == Length[f], S = Union[S, {p}]; Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Nov 11 2018, from PARI *)

{ A057719() = local(S,f); S=Set([2]); forprime(p=3,10^7, f=factorint(znorder(Mod(2,p))); if(f[1,1]!=2||f[1,2]!=1,next); f=f[,1]; if(length(setintersect(S,Set(f)))==length(f), S=setunion(S,[p]); print1(p,", "))) }

Edited by Max Alekseyev, Jul 30 2006

A171980 Prime divisors of elements of A129066.

Original entry on oeis.org

5, 3001, 120041, 532501, 720241, 2160721, 3937501, 9375001, 16505501, 120040001, 158453021, 165055001, 202567501, 289312501, 562500061, 900307501, 985937501, 1500512501, 1512504701, 3169060421, 3301100021, 3908604433, 3993757501

Offset: 1

Max Alekseyev, Jan 20 2010

nonn

Corresponding smallest multiples from A129066 are given in A171981.

Prime p>5 is in this sequence if the multiplicative order of (sqrt(5)-3)/2 modulo p is the product of smaller terms of this sequence.

Max Alekseyev, Table of n, a(n) for n = 1..25 (complete up to 10^10)

Cf. A057719, A066364, A129729, A171981

A087807 Prime factors of solutions to 24^n == 1 (mod n).

Original entry on oeis.org

23, 47, 14759, 49727, 124799, 304751, 497261, 609503, 1828507, 2685259, 10741037, 12872687, 13877879, 23462213, 23652649, 27755759, 29134267, 31908959, 53753807, 65205263, 132771091, 218148653, 341965703, 551361983, 734951759

Offset: 1

Thomas Baruchel, Oct 14 2003

Primes that divide at least one term of A014960.

Prime p is in this sequence iff the multiplicative order of 24 modulo p is the product of smaller terms of this sequence. - Max Alekseyev, May 26 2010

			A014960(12) = 2870377 = 23 * 124799

Cf. A014960, A057719, A171980, A134360, A136372, A136373, A129729, A066364.

Corrected and extended by Max Alekseyev, May 26 2010

Edited by Max Alekseyev, Nov 16 2019

A342810 Numbers k that divide the smallest number whose sum of digits is k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 27, 81, 191, 243, 729, 999, 2187, 2997, 6561, 8991, 19683, 26973, 33321, 36963, 39049, 59049, 80919, 100389, 110889, 118827, 177147, 177897, 183951, 242757, 332667, 356481, 531441, 551853, 728271, 998001, 1069443, 1367631, 1594323, 1655559, 2184813

Offset: 1

Ruediger Jehn, Mar 22 2021

By definition, if k divides A051885(k), then k is a term of this sequence.
From Ruediger Jehn, Jun 17 2021: (Start)
None of the terms is divisible by 2*5*11*13.
If a term x has the form 3^m * y where m > 1 (which is the case for the overwhelming number of terms of this sequence), then all prime factors of y are terms of A066364.
If a term x has the form 3^m * p * q where m > 1, where p is a term of A066364 and where q is the product of all other factors of the prime factorization of x, then all numbers 3^m * p^i * q are also terms for any integer i. (End)

			21 is a term because the smallest number with a digital sum of 21 is 399 (A051885(21) = 399) which is divisible by 21.

Kester Habermann, Table of n, a(n) for n = 1..792 (first 200 terms from Chai Wah Wu)
Rüdiger Jehn and Kester Habermann, Properties of terms of OEIS A342810, arXiv:2106.05866 [math.GM], 2021.

Cf. A007953, A051885, A066364.

MAX=10000; for (e = 0, MAX, for (d = 1, 9, k =(d+1)*10^e - 1; x = d+9*e; if (k%x==0, print1(x, ", ");)))

A342810_list = [n for n in range(1,10**6) if n==1 or ((n % 9)+1)*pow(10,n//9,n) % n == 1] # Chai Wah Wu, Apr 04 2021

Name clarified by Jon E. Schoenfield, Apr 27 2021

A122787 a(n) is the smallest prime p such that the multiplicative order of 10 modulo p is 3^n.

Original entry on oeis.org

3, 37, 333667, 757, 163, 411361786890737698932559, 313471, 2558791, 618846643, 2238862519, 396319276163359, 34720813

Offset: 0

Farideh Firoozbakht, Oct 06 2006

For n>0, a(n) is the smallest prime p>3 such that 3^n*p but not 3^(n-1)*p is a solution to 10^x==1 (mod x). A014950 gives solutions of this equation. It's obvious that if n is a term of A014950 then 3n is also a term of A014950. So according to the definition of a(n), for each m>n-1, 3^m*a(n) is in the sequence A014950.
a(n) is the smallest prime divisor of \Phi_{3^n}(10)/3, where \Phi_k(x) is k-th cyclotomic polynomial. a(n) is congruent to 1 modulo 3^n and 1, 3, 9, 13, 27, 31, 37, or 39 modulo 40. - Max Alekseyev, Nov 18 2014
a(12)>10^17, a(13)=796884087799, a(14)=86093443, a(15)=70367039929, a(16)>8*10^18, a(17)=662489036191, a(18)>10^19, a(19)>10^19, a(20)=38180289190951, a(21)=28305715767319, a(22)>10^20, a(23)>10^20, a(24)=63829075244707. - Ray Chandler, Dec 25 2013

			p=333667 is the smallest prime such that multiplicative order of 10 modulo p is 3^2, so a(2)=333667.

J. Brillhart et al., Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 2002.

Cf. A014950, A066364.

a(n) = factor(polcyclo(3^n,10)/3)[1,1] \\ Max Alekseyev, Nov 18 2014

Revised and extended by Max Alekseyev, Apr 25 2009

a(13), a(15), a(17), a(20), a(21), a(24) from Ray Chandler, Dec 25 2013

A164816 Prime factors in a divisibility sequence of the Lucas sequence v(P=3,Q=5) of the second kind.

Original entry on oeis.org

3, 17, 103, 163, 373, 487, 1733, 3469, 4373, 8803, 10259, 15607, 16069, 26237, 26297, 31193, 31517, 35153, 37987, 38047, 38149, 39367, 52817, 60427, 60589, 61553, 74357, 76837, 78713, 100733, 103979, 114377, 119891, 152189, 181277, 231131, 235891, 238307, 239783, 280927, 289243, 316903, 338581

Offset: 1

Jonathan Vos Post, Aug 26 2009

nonn

This is the last sequence on p. 15 of Smyth. [WARNING: Smyth lists 2 as a possible prime factor, which, in fact, is not possible. - Max Alekseyev, Sep 17 2024]
The Lucas sequence with P = 3, Q = 5 is defined as v=2,3,-1,-18,-49,-57,.. where v(n) = P*v(n-1)-Q*v(n-2), with g.f. (2-3x)/(1-3x+5x^2).
The indices n such that n|v(n) define the sequence T = 1,3,9,27,81,153,243,459,... as listed by Smyth.
The OEIS sequence shows all distinct prime factors of elements of T.

Max Alekseyev, Table of n, a(n) for n = 1..627
Richard André-Jeannin, Divisibility of generalized Fibonacci and Lucas numbers by their subscripts, Fibonacci Quart., 29(4) (1991) 364-366.
Yu. Bilu, G. Hanrot, and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539 (2001) 75-122.
R. D. Carmichael, On the numerical factors of the arithmetic forms alpha*n+-beta*n, Annals of Math., 2nd ser., 15 (1/4) (1913/14) 30-48.
Chris Smyth, The Terms in Lucas Sequences Divisible by their Indices, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.4. Preprint: arXiv:0908.3832 [math.NT], 2009.

Cf. A057719, A066364, A087807.

More detailed definition, comments rephrased, non-ascii characters in URL's removed - R. J. Mathar, Sep 09 2009
a(8)-a(9), a(11), a(18) from Jean-François Alcover, Dec 08 2017
Incorrect codes (depending on a search limit) removed, prime 2 removed, terms a(10), (12)-a(17), and a(19) onward added by Max Alekseyev, Sep 17 2024

A354026 Primes that divide some k dividing 4^k + 3^k (A045584).

Original entry on oeis.org

7, 379, 14407, 689431, 4235659, 41647747, 137534083, 239900179, 242121643, 349909477, 1245283747, 1478065891, 1605314383, 2500276549, 2748751303, 5618210347, 7490947129, 11236420693, 11260421089, 16948514941, 29440659361, 74163546829, 75093609319, 82188727303

Offset: 1

Max Alekseyev, May 15 2022

nonn

Prime p > 3 is in this sequence iff all prime factors of the multiplicative order of -3/4 modulo p belong to this sequence.

Cf. A023394, A045584, A066364, A087807, A129729, A134360, A171980, A354027.

S=[]; forprime(p=5,oo, f=Set(factor(znorder(Mod(-3/4,p)))[,1]); if(#setintersect(S,f)==#f, S=setunion(S,[p]); print1(p,", ")));

a(18)-a(24) from Jinyuan Wang, Jan 29 2025

cp's OEIS Frontend

A114207 Smallest solution to 10^m == 1 (mod m) having the prime divisor A066364(n).

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

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A057719 Prime factors of numbers in A006521 (numbers k that divide 2^k + 1).

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A171980 Prime divisors of elements of A129066.

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A087807 Prime factors of solutions to 24^n == 1 (mod n).

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A342810 Numbers k that divide the smallest number whose sum of digits is k.

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A122787 a(n) is the smallest prime p such that the multiplicative order of 10 modulo p is 3^n.

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A164816 Prime factors in a divisibility sequence of the Lucas sequence v(P=3,Q=5) of the second kind.

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