A123487 -id:A123487 - cp's OEIS frontend

A123488 Smallest prime of the form (q^p-1)/(q-1), where p = prime(n) and q is also prime (q = A123487(n)).

3, 7, 31, 127, 12207031, 8191, 131071, 524287, 1484520425576434196455942238665054573307722183, 10333327940128053377465393536117755205850522803739374960650929

Offset: 1

Alexander Adamchuk, Sep 30 2006

nonn

Corresponding smallest primes q are listed in A123487.

a(n) coincides with A084732(n) when A066180(n) is prime or 0.

Cf. A123487, A066180, A084732.

a(n) = (A123487(n)^prime(n) - 1) / (A123487(n) - 1).

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

Original entry on oeis.org

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

A123627 Smallest prime q such that (q^p+1)/(q+1) is prime, where p = prime(n); or 0 if no such prime q exists.

Original entry on oeis.org

0, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 19, 61, 2, 7, 839, 89, 2, 5, 409, 571, 2, 809, 227, 317, 2, 5, 79, 23, 4073, 2, 281, 89, 739, 1427, 727, 19, 19, 2, 281, 11, 2143, 2, 1013, 4259, 2, 661, 1879, 401, 5, 4099, 1579, 137, 43, 487, 307, 547, 1709, 43, 3, 463, 2161, 353, 443, 2

Offset: 1

Alexander Adamchuk, Oct 04 2006, Aug 05 2008

nonn

a(1) = 0 because such a prime does not exist, Mod[n^2+1,n+1] = 2 for n>1.
Corresponding primes (q^p+1)/(q+1), where prime q = a(n) and p = Prime[n], are listed in A123628[n] = {1,3,11,43,683,2731,43691,174763,2796203,402488219476647465854701,715827883,...}.
a(n) coincides with A103795[n] when A103795[n] is prime.
a(n) = 2 for n = PrimePi[A000978[k]] = {2,3,4,5,6,7,8,9,11,14,18,22,26,31,39,43,46,65,69,126,267,380,495,762,1285,1304,1364,1479,1697,4469,8135,9193,11065,11902,12923,13103,23396,23642,31850,...}.
Corresponding primes of the form (2^p + 1)/3 are the Wagstaff primes that are listed in A000979[n] = {3,11,43,683,2731,43691,174763,2796203,715827883,...}.

Robert Israel, Table of n, a(n) for n = 1..240

Cf. A123628, A103795, A123487, A123488, A000978, A000979.

f:= proc(n) local p,q;
  p:= ithprime(n);
  q:= 1;
  do
   q:= nextprime(q);
   if isprime((q^p+1)/(q+1)) then return q fi
  od
end proc:
f(1):= 0:
map(f, [$1..70]); # Robert Israel, Jul 31 2019

a(1) = 0, for n>1 Do[p=Prime[k]; n=1; q=Prime[n]; cp=(q^p+1)/(q+1); While[ !PrimeQ[cp], n=n+1; q=Prime[n]; cp=(q^p+1)/(q+1)]; Print[q], {k, 2, 61}]
Do[p=Prime[k]; n=1; q=Prime[n]; cp=(q^p+1)/(q+1); While[ !PrimeQ[cp], n=n+1; q=Prime[n]; cp=(q^p+1)/(q+1)]; Print[{k,q}], {k, 1, 134}]
spq[n_]:=Module[{p=Prime[n],q=2},While[!PrimeQ[(q^p+1)/(q+1)],q=NextPrime[ q]]; q]; Join[{0},Array[spq,70,2]] (* Harvey P. Dale, Mar 23 2019 *)

A123628(n) = (a(n)^prime(n) + 1) / (a(n) + 1).

A123628 Smallest prime of the form (q^p+1)/(q+1), where p = prime(n) and q is also prime (q = A123627(n)); or 1 if such a prime does not exist.

Original entry on oeis.org

1, 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 402488219476647465854701, 715827883, 10300379826060720504760427912621791994517454717, 254760179343040585394724919772965278539769280548173566545431025735121201

Offset: 1

Alexander Adamchuk, Oct 03 2006

nonn

a(1) = 1 because such a prime does not exist; (n^2+1) mod (n+1) = 2 for n > 1. a(n) = (A103795(n)^prime(n)+1)/(A103795(n)+1) when A103795(n) is prime. Corresponding smallest primes q such that (q^p+1)/(q+1) is prime, where p = prime(n), are listed in A123627(n) = {0, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 19, 61, 2, 7, 839, 1459, 2, 5, 409, 571, 2, ...}. All Wagstaff primes or primes of form (2^p + 1)/3 belong to a(n). Wagstaff primes are listed in A000979(n) = {3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, ...}. Corresponding indices n such that a(n) = (2^prime(n) + 1)/3 are PrimePi(A000978(n)) = {2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, 31, 39, 43, 46, 65, 69, 126, 267, 380, 495, 762, 1285, 1304, 1364, 1479, 1697, 4469, 8135, 9193, 11065, 11902, 12923, 13103, 23396, 23642, 31850, ...}. All primes with prime indices in the Jacobsthal sequence A001045(n) belong to a(n).

Cf. A123627, A103795, A123487, A123488, A000978, A000979, A001045, A049883, A107036.

a(n) = (A123627(n)^prime(n) + 1) / (A123627(n) + 1).

A278913 a(n) is the smallest number k with prime sum of divisors such that tau(k) = n-th prime.

Original entry on oeis.org

2, 4, 16, 64, 9765625, 4096, 65536, 262144, 1471383076677527699142172838322885948765175969, 10264895304762966931257013446474591264089923314972889033759201, 1073741824, 18701397461209715023927088008788055619800417991632621566284510161

Offset: 1

Jaroslav Krizek, Nov 30 2016

nonn

tau(n) = A000005(n) = the number of divisors of n.

a(11) = 1073741824; a(n) > A023194(10000) = 5896704025969 for n = 9, 10 and n >= 12.

			a(3) = 16 because 16 is the smallest number with prime values of sum of divisors (sigma(16) = 31) such that tau(16) = 5 = 3rd prime.

Davin Park, Table of n, a(n) for n = 1..50

Cf. A000005, A000203, A023194, A278914.

A278913:=func; [A278913(n): n in[1..8]];

A278913[n_] := NestWhile[NextPrime, 2, ! PrimeQ[Cyclotomic[Prime[n], #]] &]^(Prime[n] - 1) (* Davin Park, Dec 28 2016 *)

a(n) = {my(k=1); while(! (isprime(sigma(k)) && isprime(p=numdiv(k)) && (primepi(p) == n)), k++); k;} \\ Michel Marcus, Dec 03 2016

a(n) = A123487(n)^(prime(n)-1). - Davin Park, Dec 10 2016

More terms from Davin Park, Dec 08 2016

A252503 Smallest prime p such that Phi_n(p) is also prime, where Phi is the cyclotomic polynomial, or 0 if no such p exists.

Original entry on oeis.org

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 7, 2, 11, 3, 2, 113, 2, 43, 2, 2, 5, 151, 2, 2, 2, 2, 2, 179, 3, 61, 2, 23, 2, 53, 2, 89, 137, 11, 2, 5, 5, 2, 7, 73, 11, 307, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 0, 2, 53, 491, 197, 2, 3, 3, 3, 11, 19, 59, 7, 2, 2, 271, 2, 191, 61, 41, 7, 2, 2, 59, 5, 2, 2

Offset: 1

Eric Chen, Dec 18 2014

nonn

Conjecture: a(n) > 0 for all n != 2^k (k>5).
Clearly, if n is a power of 2, and Phi_n(2) is not prime, then a(n) = 0.
Records: 3, 5, 7, 11, 113, 151, 179, 307, 491, 839, 1427, 2411, 5987, 6389, 8933, 11813, 18587, 31721, 40763, 46349, ..., . - Robert G. Wilson v, May 21 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..1300 (first 1000 terms from Charles R Greathouse IV)

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

Do[n=1; p=Prime[n]; cp=Cyclotomic[k, p]; While[!PrimeQ[cp], n=n+1; p=Prime[n]; cp=Cyclotomic[k, p]]; Print[p], {k, 1, 300}]

a(n)=if(n>>valuation(n,2)==1 && n>32, if(ispseudoprime(2^(n/2)+1), 2, 0), my(P=polcyclo(n)); forprime(p=2,, if(ispseudoprime(subst(P,'x,p)), return(p)))) \\ Charles R Greathouse IV, Dec 18 2014

cp's OEIS Frontend

A123488 Smallest prime of the form (q^p-1)/(q-1), where p = prime(n) and q is also prime (q = A123487(n)).

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

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A123627 Smallest prime q such that (q^p+1)/(q+1) is prime, where p = prime(n); or 0 if no such prime q exists.

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A123628 Smallest prime of the form (q^p+1)/(q+1), where p = prime(n) and q is also prime (q = A123627(n)); or 1 if such a prime does not exist.

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A278913 a(n) is the smallest number k with prime sum of divisors such that tau(k) = n-th prime.

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Magma

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A252503 Smallest prime p such that Phi_n(p) is also prime, where Phi is the cyclotomic polynomial, or 0 if no such p exists.

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