A084738 -id:A084738 - cp's OEIS frontend

A084740 Least k such that (n^k-1)/(n-1) is prime, or 0 if no such prime exists.

2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, 5, 5, 3, 41, 3, 2, 5, 3, 0, 2, 5, 17, 5, 11, 7, 2, 3, 3, 4421, 439, 7, 5, 7, 2, 17, 13, 3, 2, 3, 2, 19, 97, 3, 2, 17, 2, 17, 3, 3, 2, 23, 29, 7, 59

Offset: 2

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 15 2003

nonn

When (n^k-1)/(n-1) is prime, k must be prime. As mentioned by Dubner, when n is a perfect power, then (n^k-1)/(n-1) will usually be composite for all k, which is the case for n = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, ... - T. D. Noe, Jan 30 2004
a(152) > prime(1100) or 0. - Derek Orr, Nov 29 2014
a(n)=2 if and only if n=p-1, where p is an odd prime; that is, n belongs to A006093, except 2. - Thomas Ordowski, Sep 19 2015
Probably a(152) = 270217 since (152^270217-1)/(152-1) has been shown to be probably prime. - Michael Stocker, Jan 24 2019

			a(7) = 5 as (7^5 - 1 )/(7 - 1) = 2801 = 1 + 7 + 7^2 + 7^3 + 7^4 is a prime but no smaller partial sum yields a prime.

Eric Chen, Table of known a(n) up to a(360) [with a(316) corrected by Jon E. Schoenfield]
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Andy Steward, Titanic Prime Generalized Repunits
Eric Weisstein's World of Mathematics, Repunit
Robert G. Wilson v, Letter to N. J. A. Sloane, circa 1991.
Robert G. Wilson v, Table of known a(n) from n = 2 to 1000.

Cf. A065507, A065854, A084738, A084742, A096059, A126659, A128164, A246005.

Cf. A066180, A085398, A103795, A123487, A123627.

a(n) = {l=List([9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343]); for(q=1, #l, if(n==l[q], return(0))); k=1; while(k, s=(n^prime(k)-1)/(n-1); if(ispseudoprime(s), return(prime(k))); k++)}
n=2; while(n<361, print1(a(n), ", "); n++) \\ Derek Orr, Jul 13 2014

More terms from T. D. Noe, Jan 23 2004

A128164 Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.

Original entry on oeis.org

3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, 25667, 19, 3, 3, 5, 5, 3, 0, 7, 3, 5, 5, 5, 7, 0, 3, 13, 313, 0, 13, 3, 349, 5, 3, 1319, 5, 5, 19, 7, 127, 19, 0, 3, 4229, 103, 11, 3, 17, 7, 3, 41, 3, 7, 7, 3, 5, 0, 19, 3, 19, 5, 3, 29, 3, 7, 5, 5, 3, 41, 3, 3, 5, 3, 0, 23, 5, 17, 5, 11, 7, 61, 3, 3

Offset: 2

Alexander Adamchuk, Feb 20 2007

nonn

a(n) = A084740(n) for all n except n = p-1, where p is an odd prime, for which A084740(n) = 2.
All nonzero terms are odd primes.
a(n) = 0 for n = {4,9,16,25,32,36,49,64,81,100,121,125,144,...}, which are the perfect powers with exceptions of the form n^(p^m) where p>2 and (n^(p^(m+1))-1)/(n^(p^m)-1) are prime and m>=1 (in which case a(n^(p^m))=p). - Max Alekseyev, Jan 24 2009
a(n) = 3 for n in A002384, i.e., for n such that n^2 + n + 1 is prime.
a(152) > 20000. - Eric Chen, Jun 01 2015
a(n) is the least number k such that (n^k - 1)/(n-1) is a Brazilian prime, or 0 if no such Brazilian prime exists. - Bernard Schott, Apr 23 2017
These corresponding Brazilian primes are in A285642. - Bernard Schott, Aug 10 2017
a(152) = 270217, see the top PRP link. - Eric Chen, Jun 04 2018
a(184) = 16703, a(200) = 17807, a(210) = 19819, a(306) = 26407, a(311) = 36497, a(326) = 26713, a(331) = 25033; a(185) > 66337, a(269) > 63659, a(281) > 63421, and there are 48 unknown a(n) for n <= 1024. - Eric Chen, Jun 04 2018
Six more terms found: a(522)=20183, a(570)=12907, a(684)=22573, a(731)=15427, a(820)=12043, a(996)=14629. - Michael Stocker, Apr 09 2020

			a(7) = 5 because (7^5 - 1)/6 = 2801 = 11111_7 is prime and (7^k - 1)/6 = 1, 8, 57, 400 for k = 1, 2, 3, 4. - _Bernard Schott_, Apr 23 2017

Max Alekseyev and Eric Chen, Table of n, a(n) for n = 2..184 (terms 2..151 from Max Alekseyev)
Eric Chen, Table of n, a(n) for n = 2..1024 status (updated by Jinyuan Wang)
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Richard Fischer, Generalized repunit primes of the form (B^N-1)/(B-1)
Top PRPs, Search by (152^n-1)/(152-1)
Top PRPs, Search by (b^n-1)/a
Eric Weisstein's World of Mathematics, Repunit

Cf. A084738, A065854, A084740, A084741, A065507, A084742, A066180, A084732, A285642, A085104.
Cf. A002384, A049409, A100330, A162862, A217070-A217089. (numbers b such that (b^p-1)/(b-1) is prime for prime p = 3 to 97)
Cf. A000043, A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A133857, A006035, A127995, A127996, A127997, A204940, A127998, A127999, A128000, A181979, A098438, A128002, A209120, A185073, A128003, A128004, A181987, A128005, A239637, A240765, A294722, A242797, A243279, A267375, A245237, A245442, A173767. (numbers n such that (b^n-1)/(b-1) is prime for b = 2 to 53)
A126589 gives locations of zeros.

Table[Function[m, If[m > 0, k = 3; While[! PrimeQ[(m^k - 1)/(m - 1)], k++]; k, 0]]@ If[Set[e, GCD @@ #[[All, -1]]] > 1, {#, IntegerExponent[n, #]} &@ Power[n, 1/e] /. {{k_, m_} /; Or[Not[PrimePowerQ@ m], Prime@ m, FactorInteger[m][[1, 1]] == 2] :> 0, {k_, m_} /; m > 1 :> n}, n] &@ FactorInteger@ n, {n, 2, 17}] (* Michael De Vlieger, Apr 24 2017 *)

a052409(n) = my(k=ispower(n)); if(k, k, n>1)
a052410(n) = if (ispower(n, , &r), r, n)
is(n) = issquare(n) || (ispower(n) && !ispseudoprime((n^a052410(a052409(n))-1)/(n-1)))
a(n) = if(is(n), 0, forprime(p=3, 2^16, if(ispseudoprime((n^p-1)/(n-1)), return(p)))) \\ Eric Chen, Jun 01 2015, corrected by Eric Chen, Jun 04 2018, after Charles R Greathouse IV in A052409 and Michel Marcus in A052410

a(18) = 25667 found by Henri Lifchitz, Sep 26 2007

A074386 Numbers k such that sigma(k) is the square of a prime.

Original entry on oeis.org

3, 81, 400

Offset: 1

Labos Elemer, Aug 22 2002

The next term, if it exists, is > 10^11. - Donovan Johnson, Aug 24 2012
a(4), if it exists, satisfies sigma(a(4)) > 10^36.  - Hiroaki Yamanouchi, Sep 10 2014
If n belongs to this sequence, it may have at most two distinct prime divisors. If n=p^k, then sigma(p^k) = (p^(k+1)-1)/(p-1) = r^2 for some prime r. For k=1, it trivially has the only solution n=3; while for k>1 it is a partial case of the Nagell-Ljunggren equation and has the only prime solution r=11 (see Bennett-Levin 2015) corresponding to n=3^4=81. If n=p^k*q^m, then sigma(n) = (p^(k+1)-1)/(p-1) * (q^(m+1)-1)/(q-1) = r^2 for some prime r, implying that (p^(k+1)-1)/(p-1) = (q^(m+1)-1)/(q-1) = r. Here k+1 and m+1 must be odd distinct primes. The Goormaghtigh conjecture would imply that its only solution is n=400 with (p,k,q,m)=(5,2,2,4). - Max Alekseyev, Apr 24 2015

			sigma[{3,81,400}]={4,121,961}.

M. A. Bennett and A. Levin, The Nagell-Ljunggren equation via Runge’s method, Monatshefte für Mathematik 177:1 (2015), 15-31.
Wikipedia, Goormaghtigh conjecture

Cf. A000203, A001248, A008848, A023194, A028982.
Cf. A084738, A065854, A128164.
Subsequence of A006532.

Do[s=DivisorSigma[1, n]; If[PrimeQ[Sqrt[s]], Print[n]], {n, 1, 1000000}] (* Corrected by N. J. A. Sloane, May 26 2008 *)

Definition corrected by Juan Lopez, May 26 2008

Edited by N. J. A. Sloane, May 26 2008

A285642 Smallest Brazilian prime in base n, or 0 if no such prime exists.

Original entry on oeis.org

7, 13, 0, 31, 43, 2801, 73, 0, 1111111111111111111, 50544702849929377, 157, 30941, 211, 241, 0, 307

Offset: 2

Bernard Schott, Apr 23 2017

nonn

Also the smallest prime of the form (n^k - 1)/(n - 1) with k > 2. The corresponding values of k are in A128164.
For n = 18, a(n) = (18^25667 - 1)/17 as explained in the extension of A128164, but it is too large to write in the Data field.
Differs from A084738: in A084738, the primes of the form (n^2 - 1)/(n - 1) = n + 1 are included, for instance 7 = 6 + 1 = 11_6 but not included here, so a(6) = 43 = 111_6.
As mentioned by Dubner, see link, when n is a power of a prime ( >= 2 ), the number (n^k - 1)/(n - 1) with k > 2 is usually composite, so a(4) = a(9) = a(16) = a(25) = 0 for instance, exception a(8) = 73.
Values of a(19)-a(31): {109912203092239643840221, 421, 463, 245411, 292561, 601, 0, 321272407, 757, 637421, 732541, 837931, 917087137}. - Michael De Vlieger, Apr 24 2017

			a(7) = (7^5 - 1)/6 = 11111_7 =  1 + 7 + 7^2 + 7^3 + 7^4 = 2801.
a(10) is the repunit R_19 which is a string of nineteen 1's.

H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.

Cf. A084738, A084740, A085104, A128164.

Table[Function[m, If[m > 0, k = 3; While[! PrimeQ[Set[x, (m^k - 1)/(m - 1)]], k++]; x, 0]]@ If[Set[e, GCD @@ #[[All, -1]]] > 1, {#, IntegerExponent[n, #]} &@ Power[n, 1/e] /. {{k_, m_} /; Or[Not[PrimePowerQ@ m], Prime@ m, FactorInteger[m][[1, 1]] == 2] :> 0, {k_, m_} /; m > 1 :> n}, n] &@ FactorInteger@ n, {n, 2, 17}] (* Michael De Vlieger, Apr 24 2017 *)

A126589 Numbers n>1 such that prime of the form (n^k-1)/(n-1) does not exist for k>2; or A128164(n) = 0.

Original entry on oeis.org

4, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025

Offset: 1

Alexander Adamchuk, Mar 13 2007

nonn

Appears to be the union of the perfect squares k^2 (for k>1) and the prime powers p^k (for k>1) with some exceptions, such as 2^3, 3^3, 2^7, etc.

The perfect powers except those of the form n^(p^m) where p and (n^(p^(m+1))-1)/(n^(p^m)-1) are primes, p>2 and m>=1. - Max Alekseyev, Mar 09 2009

			A128164 begins with offset 2: {3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, ...}. Thus a(1) = 4, a(2) = 9, a(3) = 16.

H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Eric Weisstein's World of Mathematics, Repunit.

Cf. A128164, A084738, A065854, A084740, A084741, A065507, A084742.

Extended by Max Alekseyev, Mar 09 2009

A096059 Numbers n such that for all k, (n^k-1)/(n-1) is not prime.

Original entry on oeis.org

9, 25, 32, 49, 64, 81, 121, 125, 144

Offset: 1

Amarnath Murthy, Jun 17 2004

Indices of 0 in A084740.
Possibly a subset of A001597.
The first unknown value is n=152, checked up to k = prime(1100). - Derek Orr, Nov 29 2014
It is known 169, 216, 225, 243, 289, 324, 343 are also members of this sequence. - Derek Orr, Nov 29 2014

Cf. A001597, A084738, A084740.

Edited by Don Reble, Mar 19 2007

A326609 Largest minimal prime in base n (written in base 10).

Original entry on oeis.org

3, 13, 5, 3121, 5209, 2801, 76695841, 811, 66600049, 29156193474041220857161146715104735751776055777, 388177921

Offset: 2

Richard N. Smith, Jul 13 2019

a(13) is (probably) 13^32020*8+183, it has 35670 digits, a(14) = 14^85*4+65, it has 99 digits, a(15) = (15^106*66-619)/7, it has 126 digits, a(16) = 16^3544*9+145, it has 4269 digits.
a(17) is the smallest prime of the form (4105*17^k-9)/16 if it exists, otherwise (probably) (73*17^111333-9)/16 (136991 digits), a(18) = 18^31*304+1 (42 digits).
Other known terms: a(20) = (20^449*16-2809)/19 (585 digits), a(22) = 22^763*20+7041 (1026 digits), a(23) is (probably) (23^800873*106-7)/11 (1090573 digits), a(24) = (24^99*512-121)/23 (138 digits), a(30) = 30^1023*12+1 (1513 digits), a(42) = (42^487*27-1093)/41 (791 digits).
a(19) is the smallest prime of the form (15964*19^k-1)/3 if it exists, otherwise (probably) (904*19^110984-1)/3 (141924 digits), a(21) is the smallest prime of the form 16*21^k+335 if it exists, otherwise (probably) (51*21^479149-1243)/4 (633542 digits).

Richard N. Smith, Table of n, a(n) for n = 2..16
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, preprint, 2014.
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, Experimental Mathematics, Volume 25, 2016 - Issue 3.
C. K. Caldwell, The Prime Glossary, minimal prime
Richard N. Smith, List of all known terms
Top PRPs, Search for 8*13^n+183
Top PRPs, Search for (73*17^n-9)/16
Top PRPs, Search for (106*23^n-7)/11
Wikipedia, Minimal prime (recreational mathematics)

Cf. A071062 (base 10 minimal primes), A110600 (base 12 minimal primes).

Cf. A293142 (largest non-repunit permutable prime), A317689 (largest non-repunit circular prime), A103443 (largest left-truncatable prime), A023107 (largest right-truncatable prime), A323137 (largest two-sided prime), A084738 (smallest repunit prime), A186995 (smallest weakly prime).

A347821 Smallest prime p such that n*p+1 is a perfect power, or 0 if no such p exists.

Original entry on oeis.org

3, 13, 5, 2, 3, 2801, 5, 3, 7, 50544702849929377, 13, 2, 2, 241, 13, 3, 19, 19, 17, 463, 3, 11, 89, 2, 23, 757, 29, 732541, 31, 917087137, 29, 7, 3

Offset: 1

Eric Chen, Sep 25 2021

nonn

For every n, all sufficiently large primes p such that n*p+1 is a perfect power are of the form ((n+1)^q-1)/n with q prime.
a(34) = (35^313-1)/34 is too large to include; it has 482 decimal digits.
a(35) - a(37) = {37, 61, 1483}.
a(38) = (39^349-1)/38 is too large to include; it has 554 decimal digits.
a(39) - a(100) = {5, 2, 43, 3500201, 5, 71, 43, 3851, 178481, 11, 47, 3221, 5, 178250690949465223, 2971, 127, 53, 3, 7, 3541, 61, 2, 59, 2, 61, 17, 3, 751410597400064602523400427092397, 21700501, 4831, 7, 19, 73, 5, 7, 5701, 73, 6007, 79, 39449441, 6481, 19, 79, 48037081, 6218272796370530483675222621221, 2, 3, 438668366137, 89, 5, 23, 331, 89, 654022685443, 11, 1001523179, 97, 3, 792806586866086631668831, 9901, 97, 10303}.
If n*p+1 = m^k, then n*p = m^k-1 = (m-1)*(m^(k-1) + m^(k-2) + ... + m + 1). If p >= n, then m^k = n*p+1 >= n^2+1 > n^2, and we have these three cases: Case 1: m-1 > n, then p can't be prime. Case 2: m-1 = n, this is A084738. Case 3: m-1 < n. If gcd(n, m-1) != m-1, then because m^(k-1) + m^(k-2) + ... + m + 1 > n, p can't be prime. This implies m-1 | n. The three cases means that we only need to check p < n and numbers m such that m-1 | n.
The first numbers n such that a(n) = 0 are {124, 215, 224, 242, ...}. a(268) is unknown; it is the smallest prime of the form (269^q - 1)/268 with prime q if such a prime exists (in which case it must be greater than (269^63659-1)/268), otherwise 0.

Eric Chen, Table of n, a(n) for n = 1..300 (with unknown term a(268)).

Cf. A001597, A084738.

a(n)=forprime(p=2,2^32,if(ispower(n*p+1),return(p)))

b(n)=forprime(p=2,2^16,if(ispseudoprime(q=((n+1)^p-1)/n),return(q)))
a(n)=forprime(p=2,2^30,if(ispower(n*p+1),return(p)));b(n) \\ this program might be incorrect beyond a(300)

a(n) <= A084738(n+1) if A084738(n+1) > 0.

cp's OEIS Frontend

A084740 Least k such that (n^k-1)/(n-1) is prime, or 0 if no such prime exists.

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A128164 Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.

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A074386 Numbers k such that sigma(k) is the square of a prime.

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A285642 Smallest Brazilian prime in base n, or 0 if no such prime exists.

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A126589 Numbers n>1 such that prime of the form (n^k-1)/(n-1) does not exist for k>2; or A128164(n) = 0.

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A096059 Numbers n such that for all k, (n^k-1)/(n-1) is not prime.

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A326609 Largest minimal prime in base n (written in base 10).

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A347821 Smallest prime p such that n*p+1 is a perfect power, or 0 if no such p exists.

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