A272060 -id:A272060 - cp's OEIS frontend

A272061 Primes p such that sigma((p-1)/2) + tau((p-1)/2) is prime.

3, 5, 17, 257, 65537, 453519617, 1372257937, 1927561217, 21320672257, 76001667857, 138388464037, 1216026685697, 2085136000001, 8503056000001, 30118144000001, 35427446793217, 37015056000001, 83037656250001, 87329473560577, 97850397828097, 222330465562501, 233952748524197

Offset: 1

Jaroslav Krizek, Apr 19 2016

nonn

Primes p such that A007503((p-1)/2) is a prime q.
Corresponding values of primes q: 2, 5, 19, 263, 65551, 496922891, ...
Prime terms from A272060.
The first 5 known Fermat primes from A019434 are in this sequence.
Primes of the form 2*m+1 with m a term of A064205. - Michel Marcus, Apr 25 2016

			sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..235

Cf. A000005, A000203, A007503, A055813, A064205, A272060.

[n: n in [3..1000000] | IsPrime(n) and IsPrime(NumberOfDivisors((n-1) div 2) + SumOfDivisors((n-1) div 2)) and (n-1) mod 2 eq 0];

with(numtheory): A272061:=n->`if`(isprime(n) and isprime(sigma((n-1)/2)+tau((n-1)/2)), n, NULL): seq(A272061(n), n=3..10^5); # Wesley Ivan Hurt, Apr 20 2016

Select[Prime[Range[10000]],PrimeQ[DivisorSigma[1,(#-1)/2] + DivisorSigma[0,(#-1)/2]] & ] (* Robert Price, Apr 21 2016 *)

isok(n) = isprime(sigma((n-1)/2) + numdiv((n-1)/2));
lista(nn) = forprime (p=3, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Apr 19 2016

is(n)=my(f=factor(n\2)); isprime(sigma(f)+numdiv(f)) && isprime(n) \\ Charles R Greathouse IV, Apr 28 2016

a(7)-a(8) from Michel Marcus, Apr 24 2016
a(9) from Charles R Greathouse IV, Apr 29 2016
a(10) from Charles R Greathouse IV, Apr 29 2016
a(11)-a(20), using A064205 bfile, added by Michel Marcus, Nov 23 2022
a(21)-a(22) from Amiram Eldar, Dec 06 2022

A358342 Lesser of twin primes p such that sigma((p-1)/2) + tau((p-1)/2) is a prime.

Original entry on oeis.org

3, 5, 17, 65537, 1927561217, 6015902625062501, 12370388895062501, 835920078368222501, 6448645485213008897, 50973659693056000001, 54332889713542767617, 64304984013657011717, 112112769248058062501, 147337258721536000001

Offset: 1

Jaroslav Krizek, Nov 10 2022

Lesser of twin primes p such that A000203((p-1)/2) + A000005((p-1)/2) is a prime q.
The first 4 terms are Fermat primes from A019434.
Corresponding values of primes q:  2, 5, 19, 65551, 2248681529, ...
Subsequence of A272060 and A272061.
Lesser of twin primes of the form 2*m+1 with m a term of A064205.
There are no other terms <= 10^14.
All the terms above 3 are in A145824. - Amiram Eldar, Jan 05 2023

			17 and 19 are twin primes; sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Intersection of A001359 and A272061.

Cf. A000005 (tau), A000203 (sigma), A019434, A064205, A145824, A272060.

[n: n in [3..10^7] | IsPrime(n) and IsPrime(n+2) and IsPrime(&+Divisors((n-1) div 2) + #Divisors((n-1) div 2))]

Join[{3}, Select[4*Range[25000]^2 + 1, PrimeQ[#] && PrimeQ[# + 2] && PrimeQ[DivisorSigma[1, (# - 1)/2] + DivisorSigma[0, (# - 1)/2]] &]]
(* or *)
A272061 = Cases[Import["https://oeis.org/A272061/b272061.txt", "Table"], {, }][[;; , 2]]; Select[A272061, PrimeQ[# + 2] &] (* Amiram Eldar, Jan 05 2023 *)

isok(p) = if (isprime(p) && isprime(p+2), my(f=factor((p-1)/2)); isprime(sigma(f)+numdiv(f))); \\ Michel Marcus, Nov 23 2022

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A272061 Primes p such that sigma((p-1)/2) + tau((p-1)/2) is prime.

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A358342 Lesser of twin primes p such that sigma((p-1)/2) + tau((p-1)/2) is a prime.

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Magma

Mathematica

PARI