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

%I A272061 #39 Jun 20 2025 16:53:48
%S A272061 3,5,17,257,65537,453519617,1372257937,1927561217,21320672257,
%T A272061 76001667857,138388464037,1216026685697,2085136000001,8503056000001,
%U A272061 30118144000001,35427446793217,37015056000001,83037656250001,87329473560577,97850397828097,222330465562501,233952748524197
%N A272061 Primes p such that sigma((p-1)/2) + tau((p-1)/2) is prime.
%C A272061 Primes p such that A007503((p-1)/2) is a prime q.
%C A272061 Corresponding values of primes q: 2, 5, 19, 263, 65551, 496922891, ...
%C A272061 Prime terms from A272060.
%C A272061 The first 5 known Fermat primes from A019434 are in this sequence.
%C A272061 Primes of the form 2*m+1 with m a term of A064205. - _Michel Marcus_, Apr 25 2016
%H A272061 Amiram Eldar, <a href="/A272061/b272061.txt">Table of n, a(n) for n = 1..235</a>
%e A272061 sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.
%p A272061 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
%t A272061 Select[Prime[Range[10000]],PrimeQ[DivisorSigma[1,(#-1)/2] + DivisorSigma[0,(#-1)/2]] & ] (* _Robert Price_, Apr 21 2016 *)
%o A272061 (Magma) [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];
%o A272061 (PARI) isok(n) = isprime(sigma((n-1)/2) + numdiv((n-1)/2));
%o A272061 lista(nn) = forprime (p=3, nn, if (isok(p), print1(p, ", "))); \\ _Michel Marcus_, Apr 19 2016
%o A272061 (PARI) is(n)=my(f=factor(n\2)); isprime(sigma(f)+numdiv(f)) && isprime(n) \\ _Charles R Greathouse IV_, Apr 28 2016
%Y A272061 Cf. A000005, A000203, A007503, A055813, A064205, A272060.
%K A272061 nonn
%O A272061 1,1
%A A272061 _Jaroslav Krizek_, Apr 19 2016
%E A272061 a(7)-a(8) from _Michel Marcus_, Apr 24 2016
%E A272061 a(9) from _Charles R Greathouse IV_, Apr 29 2016
%E A272061 a(10) from _Charles R Greathouse IV_, Apr 29 2016
%E A272061 a(11)-a(20), using A064205 bfile, added by _Michel Marcus_, Nov 23 2022
%E A272061 a(21)-a(22) from _Amiram Eldar_, Dec 06 2022