A278741 -id:A278741

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A249759

Primes p such that sigma(p-1) is a prime q.

+10
13

3, 5, 17, 65537

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Subsequence of {A023194(n)+1}.

Conjectures: 1) sequence is finite; 2) sequence is a subsequence of A019434 (Fermat primes); 3) sequence consists of Fermat primes p such that sigma(p-1) is a Mersenne prime; 4) a(n) = (A249761(n)+3)/2.

3 is the only prime p such that sigma(p+1) is prime, i.e., 3 is the only prime p such that sigma(p-1) and sigma(p+1) are both primes.

Conjecture: 3 is the only number n such that n and sigma(n+1) are both prime.

Primes p such that A051027(p-1) = sigma(sigma(p-1)) = 2*(p-1). Subsequence of A256438. - Jaroslav Krizek, Mar 29 2015

From Jaroslav Krizek, Mar 17 2016: (Start)

Primes p such that A000203(A000010(p)) = sigma(phi(p)) is a prime.

Prime terms from A062514 and A270413, A270414, A270415 and A270416. (End)

From Jaroslav Krizek, Nov 27 2016: (Start)

Corresponding values of primes q are in A249761: 3, 7, 31, 131071, ...

Conjecture: subsequence of A256438 and A278741.

Conjecture: also primes p such that tau(p-1) is a prime q; corresponding values of primes q are 2, 3, 5, 17. (End)

LINKS

Table of n, a(n) for n=1..4.

FORMULA

a(n) = A249760(n) + 1.

Sigma(a(n)-1) = A249761(n).

EXAMPLE

Prime 17 is in the sequence because sigma(17-1) = sigma(16) = 31 (prime).

MAPLE

with(numtheory): A249759:=n->`if`(isprime(n) and isprime(sigma(n-1)), n, NULL): seq(A249759(n), n=1..6*10^5); # Wesley Ivan Hurt, Nov 14 2014

MATHEMATICA

Select[Range[10^5], PrimeQ[#]&& PrimeQ[DivisorSigma[1, # - 1]] &] (* Vincenzo Librandi, Nov 14 2014 *)

Select[Prime[Range[7000]], PrimeQ[DivisorSigma[1, #-1]]&] (* Harvey P. Dale, Jun 14 2020 *)

PROG

(Magma) [p: p in PrimesUpTo(1000000) | IsPrime(SumOfDivisors(p-1))]

(PARI) lista(nn) = {forprime(p=1, nn, if (isprime(sigma(p-1)), print1(p, ", ")); ); } \\ Michel Marcus, Nov 14 2014

CROSSREFS

Cf. A000203, A000668, A019434, A023194, A249760, A249761.

KEYWORD

nonn,more

AUTHOR

Jaroslav Krizek, Nov 13 2014

STATUS

approved

A359134

a(n) = Sum_{d|n} (2*d)^(n/d - 1).

+10
2

1, 3, 5, 13, 17, 55, 65, 201, 293, 779, 1025, 3365, 4097, 12303, 17781, 49681, 65537, 204547, 262145, 791549, 1095429, 3145751, 4194305, 12897625, 16787217, 50331675, 68788805, 201591509, 268435457, 815505231, 1073741825, 3223326753, 4355433957, 12884901923

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..1000

FORMULA

G.f.: Sum_{k>0} x^k / (1 - 2 * k * x^k).

If p is prime, a(p) = 1 + 2^(p-1).

MATHEMATICA

a[n_] := DivisorSum[n, (2*#)^(n/# - 1) &]; Array[a, 30] (* Amiram Eldar, Aug 14 2023 *)

PROG

(PARI) a(n) = sumdiv(n, d, (2*d)^(n/d-1));

(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-2*k*x^k)))

CROSSREFS

Cf. A087909, A278741, A349970, A359733, A359796.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jan 13 2023

STATUS

approved

A279882

a(n) = 2^(prime(n) + 1) - 1.

+10
1

7, 15, 63, 255, 4095, 16383, 262143, 1048575, 16777215, 1073741823, 4294967295, 274877906943, 4398046511103, 17592186044415, 281474976710655, 18014398509481983, 1152921504606846975, 4611686018427387903, 295147905179352825855, 4722366482869645213695

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers whose binary representation is 1 repeated (prime(n)+1) times.

The only prime term is 7.

LINKS

Table of n, a(n) for n=1..20.

FORMULA

a(n) = A101304(n) - 2.

a(n) = A000225(A008864(n)). - Felix Fröhlich, Dec 21 2016

EXAMPLE

For n=3; a(3) = 2^(prime(3) + 1) - 1 = 2^(5 + 1) - 1 = 2^6 - 1 = 63.

MAPLE

A279882:=n->2^(ithprime(n)+1)-1: seq(A279882(n), n=1..30); # Wesley Ivan Hurt, Jan 23 2017

MATHEMATICA

f[n_] := 2^(Prime[n]+1)-2; Array[f, 20] (* Robert G. Wilson v, Dec 21 2016 *)

2^(Prime[Range[20]]+1)-1 (* Harvey P. Dale, Jul 29 2024 *)

PROG

(Magma) [2^(NthPrime(n)+1)-1: n in[1..50]]

(PARI) a(n) = 2^(prime(n)+1)-1 \\ Felix Fröhlich, Dec 21 2016

CROSSREFS

Cf. A101304 (2^(prime(n)+1)+1), A098102 (2^(prime(n)-1)-1), A278741 (2^(prime(n)-1)+1).

KEYWORD

nonn,easy

AUTHOR

Jaroslav Krizek, Dec 21 2016

STATUS

approved

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