A256438 -id:A256438 - cp's OEIS frontend

A249759 Primes p such that sigma(p-1) is a prime q.

3, 5, 17, 65537

Offset: 1

Jaroslav Krizek, Nov 13 2014

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)

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

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

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

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

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 *)

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

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

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

A270413 Numbers m such that sigma(m-1) is a prime.

Original entry on oeis.org

3, 5, 10, 17, 26, 65, 290, 730, 1682, 2402, 3482, 4097, 5042, 7922, 10202, 15626, 17162, 27890, 28562, 29930, 65537, 83522, 85850, 146690, 262145, 279842, 458330, 491402, 531442, 552050, 579122, 597530, 683930, 703922, 707282, 734450, 829922, 1190282, 1203410

Offset: 1

Jaroslav Krizek, Mar 16 2016

nonn

Prime terms are in A249759.
Corresponding values of sigma(n-1): 3, 7, 13, 31, 31, 127, 307, 1093, ...
Conjecture: supersequence of A256438.
Conjecture: 31 is the only prime p such that p = sigma(x-1) = sigma(y-1) for distinct numbers x and y; 31 = sigma(17-1) = sigma(26-1).
Supersequence of A270414 and A270415.

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

Cf. A000203, A023194, A249759, A256438, A270414, A270415.

[n: n in [2..2000000] |  IsPrime(SumOfDivisors(n-1))];

Select[Range[10^6], PrimeQ@ DivisorSigma[1, # - 1] &] (* Michael De Vlieger, Mar 17 2016 *)

isok(n) = isprime(sigma(n-1)); \\ Michel Marcus, Mar 17 2016

a(n) = A023194(n) + 1.

A278741 Odd numbers k such that tau(k-1) is a prime.

Original entry on oeis.org

3, 5, 17, 65, 1025, 4097, 65537, 262145, 4194305, 268435457, 1073741825, 68719476737, 1099511627777, 4398046511105, 70368744177665, 4503599627370497, 288230376151711745, 1152921504606846977, 73786976294838206465, 1180591620717411303425, 4722366482869645213697

Offset: 1

Jaroslav Krizek, Nov 27 2016

nonn

tau(k) = A000005(k) = the number of divisors of k.
Conjecture: prime terms are in A249759: 3, 5, 17, 65537, ...
Supersequence of A256438 and A249759. Subsequence of {A009087(n) + 1}.

			Odd number 65 is in the sequence because tau(64) = 7 (prime).

Cf. A000005, A000040, A001348, A009087, A061286, A249759, A256438.

[n: n in[2..10000000] |  IsOdd(n) and IsPrime(NumberOfDivisors(n-1))];

isok(n) = (n % 2) && isprime(numdiv(n-1)); \\ Michel Marcus, Nov 27 2016

a(n) = A061286(n) + 1.
sigma(a(n)-1) = A001348(n), i.e., Mersenne numbers.
tau(a(n)-1) = A000040(n), i.e., all primes; a(n) = the smallest odd number k such that tau(a(n)-1) = prime(n) = A000040(n).

A270414 Numbers m such that sigma(m-1) and sigma(phi(m)) are both primes.

Original entry on oeis.org

3, 5, 10, 17, 65537

Offset: 1

Jaroslav Krizek, Mar 16 2016

Numbers n such that A000203(n-1) and A062402(n) are both primes.
There are no other terms <= 10^7.
Intersection of A270413 and A062514.
Prime terms are in A249759.
Corresponding values of sigma(n-1): 3, 7, 13, 31, 131071, ...
Corresponding values of sigma(phi(n)): 3, 7, 7, 31, 131071, ...
Conjecture: union of number 10 and A249759.

			10 is in the sequence because sigma(10-1) = sigma(9) = 13 and sigma(phi(10)) = sigma(4) = 7 (both primes).

Cf. A000010, A000203, A062514, A249759, A256438, A270413.

[n: n in [2..100000] |  IsPrime(SumOfDivisors(n-1)) and IsPrime(SumOfDivisors(EulerPhi(n)))];

Select[Range[10^6], And[PrimeQ@ DivisorSigma[1, # - 1], PrimeQ@ DivisorSigma[1, EulerPhi@ #]] &] (* Michael De Vlieger, Mar 17 2016 *)

isok(n) = isprime(sigma(n-1)) && isprime(sigma(eulerphi(n))); \\ Michel Marcus, Mar 17 2016

A265425 Numbers n such that n+2 and sigma(n-1) are both primes.

Original entry on oeis.org

3, 5, 17, 65, 4097, 65537, 262145, 1073741825

Offset: 1

Jaroslav Krizek, Dec 08 2015

If a(9) exists, it must be larger than A023194(10000) = 5896704025969.
Prime terms: 3, 5, 17, 65537, ...
Any prime present must be one of the lesser twin primes (A001359) and also a Fermat prime (A019434), at least. See comments in A023194. - Antti Karttunen, Dec 08 2015
Sequence is different from A256438; numbers 1152921504606846977, 309485009821345068724781057, 81129638414606681695789005144065 and 85070591730234615865843651857942052865 are not terms of this sequence.
Numbers 2^m+1 such that 2^m + 3 and 2^(m+1) - 1 are both prime. - Hiroaki Yamanouchi, Jan 04 2016

			Number 17 is in the sequence because 17 + 2 = 19 and sigma(17-1) = sigma(16) = 31; 17 and 31 are primes.

Cf. A000203, A001359, A019434, A023194, A256438.

[n: n in [2..1000000] | IsPrime(n+2) and IsPrime(SumOfDivisors(n-1))]

Select[Range[10^7], And[PrimeQ[# + 2], PrimeQ[DivisorSigma[1, # - 1]]] &] (* Michael De Vlieger, Dec 09 2015 *)

for(n=2, 10^7, if(ispseudoprime(n+2) && ispseudoprime(sigma(n-1)), print1(n, ", "))) \\ Altug Alkan, Dec 08 2015

A281622 Numbers k such that sigma(k-1) is a Mersenne prime (A000668).

Original entry on oeis.org

3, 5, 17, 26, 65, 4097, 65537, 262145, 1073741825

Offset: 1

Jaroslav Krizek, Jan 25 2017

Conjecture 1: the next terms are: 1152921504606846977, 309485009821345068724781057, 81129638414606681695789005144065, 85070591730234615865843651857942052865.
Conjecture 2: Union of 26 and A256438.
Conjecture 3: Mersenne prime 31 is the only prime p such that p = sigma(x-1) = sigma(y-1) for distinct numbers x and y; 31 = sigma(17-1) = sigma(26-1).

			65 is a term because sigma(64) = 127 (Mersenne prime).

Union of 26 and odd terms of A270413.
Prime terms are in A249759.
Subsequence of A270413.
Cf. A000203, A000668, A256438.

[n: n in[2..1000000], k in [1..20] | SumOfDivisors(n-1) eq 2^k-1 and IsPrime(2^k-1)];

isok(n) = my(s = sigma(n-1)); isprime(s) && ispower(s+1,,&p) && (p==2); \\ Michel Marcus, Jan 27 2017

Conjecture: a(n) = 2^A090748(n) + 1. - Daniel Suteu, Feb 08 2017

cp's OEIS Frontend

A249759 Primes p such that sigma(p-1) is a prime q.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A270413 Numbers m such that sigma(m-1) is a prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A278741 Odd numbers k such that tau(k-1) is a prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

PARI

Formula

A270414 Numbers m such that sigma(m-1) and sigma(phi(m)) are both primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

A265425 Numbers n such that n+2 and sigma(n-1) are both primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

A281622 Numbers k such that sigma(k-1) is a Mersenne prime (A000668).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

PARI

Formula