A270413 -id:A270413 - 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).

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

A270415 Numbers n such that sigma(n-1) and sigma(n) - 1 are both primes.

Original entry on oeis.org

3, 5, 10, 17, 26, 65, 65537, 146690, 703922, 1481090, 1885130, 2036330, 2211170, 2430482, 2505890, 5470922, 9840770, 11607650, 17783090, 24137570, 74425130, 76615010, 77563250, 133379402, 138697730, 138980522, 142396490, 155575730, 177715562, 181899170

Offset: 1

Jaroslav Krizek, Mar 16 2016

nonn

Numbers n such that A000203(n-1) and A039653(n) are both primes.
Intersection of A270413 and A248792.
Prime terms are in A249759.
Corresponding values of sigma(n-1): 3, 7, 13, 31, 31, 127, 131071, ...
Corresponding values of sigma(n) - 1: 3, 5, 17, 17, 41, 83, 65537, ...

			17 is in the sequence because sigma(17-1) = sigma(16) = 31 and sigma(10) - 1 = 18 - 1 = 17 (both primes).

Cf. A000203, A039653, A248792, A249759, A270413.

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

Select[Range[10^7], And[PrimeQ@ DivisorSigma[1, # - 1], PrimeQ[DivisorSigma[1, #] - 1]] &] (* Michael De Vlieger, Mar 17 2016 *)

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

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

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

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

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A270415 Numbers n such that sigma(n-1) and sigma(n) - 1 are both primes.

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

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Magma

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