A000978 -id:A000978 - cp's OEIS frontend

A089655 a(1)=1 and for n>=2 a(n) is the denominator of A(n) (see comment for A(n) definition).

1, 1, 4, 1, 4, 1, 8, 3, 8, 3, 4, 1, 4, 1, 16, 1, 48, 1, 12, 1, 4, 1, 8, 5, 8, 45, 4, 9, 4, 1, 32, 1, 32, 1, 12, 1, 12, 1, 8, 1, 8, 1, 4, 3, 4, 3, 16, 7, 80, 7, 20, 1, 36, 1, 72, 1, 8, 1, 4, 1, 4, 3, 64, 3, 64, 1, 4, 1, 4, 1, 24, 1, 24, 5, 4, 5, 4, 1, 16, 27, 16, 27, 4, 1, 4, 1, 8, 1, 24, 1, 12, 1, 4, 1

Offset: 1

Benoit Cloitre, Jan 03 2004

nonn

For n>=2, A(n) is the least rational value >1 such that A(n)*(n^(2k)-1)*B(2k) is an integer value for k=1 up to 200, where B(2k) is the 2k-th Bernoulli number. It appears that sequence of numerators of A(n) coincide with A007947 (terms were computed by W. Edwin Clark). We conjecture : A(n)*(n^(2k)-1)*B(2k) is an integer value for all k>0.

Cf. A007947.

It appears that if p is prime and 2^p-1 and (2^p+1)/3 are both primes (i.e. p is in A000043 and in A000978), then a(2^p)=(4^p-1)/3 (converse doesn't hold).

For n>1 a(n)=(n^2-1)/rad(n^2-1) where rad(k) is the squarefree kernel of k; a(n)=A003557(n^2-1) - Benoit Cloitre, Oct 26 2004

A123176 Numbers n such that (2^p + 1)/3 is prime, where p is the n-th prime.

Original entry on oeis.org

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, 77509, 285228

Offset: 1

Alexander Adamchuk, Oct 03 2006

Also prime(a(n)) are the indices of prime Jacobsthal numbers (A001045) with prime indices. Primes in the Jacobsthal sequence are listed in A049883.

C. Caldwell's The Top Twenty, Wagstaff.

Cf. A000978, A000979, A001045, A049883, A107036, A123214.

Select[Range[500],PrimeQ[(2^Prime[#]+1)/3]&] (* The program generates the first 23 terms of the sequence. To generate more, increase the Range constant. *) (* Harvey P. Dale, Mar 09 2022 *)

a(n) = A000720( A000978(n) ).

Two more terms computed from A000978 by Max Alekseyev, Mar 03 2010

A215801 Prime numbers p such that (2^p + 1)/3 can be written in the form a^2 + 3*b^2.

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 109, 127, 139, 151, 199, 277, 313, 433, 457, 547, 613, 619, 643, 739, 967

Offset: 1

V. Raman, Aug 23 2012

nonn

These (2^p + 1)/3 numbers have no prime factors of the form 2 (mod 3) to an odd power.

Samuel S. Wagstaff, Jr., The Cunningham Project, Factorizations of 2^n-1, for odd n's < 1200

Cf. A000051, A000978, A215800.

forprime(i=2, 100, a=factorint(2^i+1)~; has=0; for(j=1, #a, if(a[1, j]%3==2&&a[2, j]%2==1, has=1; break)); if(has==0, print(i" -\t"a[1, ])))

9 more terms from V. Raman, Aug 28 2012

A227171 Numbers n such that (18^n + 17^n)/35 is prime.

Original entry on oeis.org

3, 47, 53, 2411, 4057, 7963, 10273, 15737, 53299

Offset: 1

Jean-Louis Charton, Jul 03 2013

All terms are prime.

Cf. A000978, A057469, A128066, A128335, A128336, A187805, A181141, A187819, A217095.

is(n)=ispseudoprime((18^n+17^n)/35) \\ Charles R Greathouse IV, Jun 13 2017

a(7), a(8) from Richard Fischer, Aug 18 2013

a(9) from Robert Price, Aug 25 2013

A227979 Integers not of the form (a^k+b^k)/(a+b) for any positive integer values of a, b, k with b > a.

Original entry on oeis.org

2, 4, 6, 8, 9, 14, 16, 18, 22, 23, 24, 32, 33, 36, 38, 42, 44, 46, 47, 54, 56, 59, 62, 64, 66, 69, 71, 72, 77, 81, 83, 86, 88, 92, 94, 96, 98, 99, 107, 114, 118, 121, 126, 128, 131, 132, 134, 138, 141, 142, 144, 152, 154, 158, 161, 162, 166, 167, 168, 177

Offset: 1

Robert Price, Sep 30 2013

nonn

This form, (a^k+b^k)/(a+b), is a generalization of the Fermat numbers.
Not all integers are in this set.
See A229791 for the complement of this sequence.

Robert Price, Table of n, a(n) for n = 1..66
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

A few of the sequences using this form that identify primes are A000978, A007658, A057469, A128066, A057171, A082387, A122853, A128335.

limit=200; lst = {}; Do[p = (a^k + b^k)/(a + b); If[p <= limit && IntegerQ[p], AppendTo[lst, p]], {k, Log[2,3*limit+1]}, {b, 2, limit*2}, {a, b-1}]; Complement[Range[limit], Union[lst]]

A229791 Integers generated by (a^k+b^k)/(a+b) for all possible positive integer values of a,b,k with b>a.

Original entry on oeis.org

1, 3, 5, 7, 10, 11, 12, 13, 15, 17, 19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 34, 35, 37, 39, 40, 41, 43, 45, 48, 49, 50, 51, 52, 53, 55, 57, 58, 60, 61, 63, 65, 67, 68, 70, 73, 74, 75, 76, 78, 79, 80, 82, 84, 85, 87, 89, 90, 91, 93, 95, 97, 100, 101, 102, 103

Offset: 1

Robert Price, Sep 29 2013

nonn

This form, (a^k+b^k)/(a+b), is a generalization of the Fermat numbers.
Not all integers are in this set.
See A227979 for the complement of this sequence.

Robert Price, Table of n, a(n) for n = 1..134
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

A few of the sequences using this form that identify primes are A000978, A007658, A057469, A128066, A057171, A082387, A122853, A128335.

limit=105; lst = {}; Do[p = (a^k + b^k)/(a + b); If[p <= limit && IntegerQ[p], AppendTo[lst, p]], {k, Log[2,3*limit+1]}, {b, 2, limit*2}, {a, b-1}]; Union[lst]

A236167 Numbers k such that (47^k + 1)/48 is prime.

Original entry on oeis.org

5, 19, 23, 79, 1783, 7681

Offset: 1

Robert Price, Jan 19 2014

a(7) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field.
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 = numbers k such that (2^k + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604, A231865, A235683.

Do[ p=Prime[n]; If[ PrimeQ[ (47^p + 1)/48 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((47^n+1)/48) \\ Charles R Greathouse IV, Jun 06 2017

from sympy import isprime
def afind(startat=0, limit=10**9):
  pow47 = 47**startat
  for k in range(startat, limit+1):
    q, r = divmod(pow47+1, 48)
    if r == 0 and isprime(q): print(k, end=", ")
    pow47 *= 47
afind(limit=300) # Michael S. Branicky, May 19 2021

A286566 Compound filter (prime signature of n & prime signature of the n-th Jacobsthal number): a(n) = P(A101296(n), A286566(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 5, 9, 5, 19, 5, 26, 18, 19, 5, 51, 5, 19, 40, 73, 5, 72, 5, 72, 40, 40, 5, 113, 31, 19, 83, 111, 8, 129, 5, 101, 32, 19, 32, 221, 8, 19, 40, 179, 8, 199, 5, 84, 159, 40, 8, 312, 13, 84, 82, 84, 8, 239, 49, 261, 32, 82, 23, 419, 5, 19, 159, 224, 82, 334, 8, 84, 32, 334, 8, 543, 8, 32, 84, 84, 82, 285, 5, 243, 332, 32, 57, 478, 40, 32, 32, 218, 23, 419, 82

Offset: 1

Antti Karttunen, May 21 2017

nonn

Here, instead of A046523 and A278165 we use as the components of a(n) their rgs-versions A101296 and A286565 because of the latter sequences' more moderate growth rates.

Antti Karttunen, Table of n, a(n) for n = 1..1032 (computed using the b-file of A278165 provided by Hans Havermann)

Cf. A001045, A046523, A101296, A278165, A286565.
Cf. A000978 (positions of 5's).
Cf. A286467 (similar filter).

(define (A286566 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286565 n)) 2) (- (A101296 n)) (- (* 3 (A286565 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n)+A286565(n))^2) - A101296(n) - 3*A286565(n)).

A109799 Primes p such that 2^p - 1 is a Chen prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 61, 127

Offset: 1

Jason Earls, Aug 15 2005

For p in this sequence, 2^p - 1 is called a Mersenne-Chen prime.
Conjecture: 2^127 - 1 is the largest Mersenne-Chen prime.
Except for the initial term 2, this sequence is the intersection of A000043 and A000978 given by A107360. - Max Alekseyev, Oct 28 2008, Jan 28 2010

			a(5)=13 because 2^13 - 1 = 8191 is prime and 2^13 + 1 = 3*2731 is semiprime.

Cf. A000043, A000978, A107360, A109611.

Select[Prime[Range[40]],PrimeQ[2^#-1]&&PrimeOmega[2^#+1]<3&] (* James C. McMahon, Mar 30 2024 *)

A120334 Odd primes of the form p = 2^k +- 1 or p = 4^k +- 3 or such that 2^p - 1 is prime or (2^p + 1)/3 is prime or PRP.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 67, 79, 89, 101, 107, 127, 167, 191, 199, 257, 313, 347, 521, 607, 701, 1021, 1279, 1709, 2203, 2281, 2617, 3217, 3539, 4093, 4099, 4253, 4423, 5807, 8191, 9689, 9941

Offset: 1

Jorge Coveiro, Sep 11 2007

nonn

Odd primes satisfying at least one of the criteria considered in the New Mersenne Prime Conjecture (cf. link).

Gord Palameta, Table of n, a(n) for n = 1..83
Chris K. Caldwell, New Mersenne Prime Conjecture, Prime Pages.

Supersequence of A000978, A122834.

Almost a supersequence of A000043 (excluding only A000043(1) = 2).

cp's OEIS Frontend

A089655 a(1)=1 and for n>=2 a(n) is the denominator of A(n) (see comment for A(n) definition).

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A123176 Numbers n such that (2^p + 1)/3 is prime, where p is the n-th prime.

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A215801 Prime numbers p such that (2^p + 1)/3 can be written in the form a^2 + 3*b^2.

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A227171 Numbers n such that (18^n + 17^n)/35 is prime.

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A227979 Integers not of the form (a^k+b^k)/(a+b) for any positive integer values of a, b, k with b > a.

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A229791 Integers generated by (a^k+b^k)/(a+b) for all possible positive integer values of a,b,k with b>a.

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A236167 Numbers k such that (47^k + 1)/48 is prime.

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A286566 Compound filter (prime signature of n & prime signature of the n-th Jacobsthal number): a(n) = P(A101296(n), A286566(n)), where P(n,k) is sequence A000027 used as a pairing function.

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Formula

A109799 Primes p such that 2^p - 1 is a Chen prime.

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