A347078 - OEIS

%I #24 Jul 30 2023 10:54:15

%S 3,5,11,17,197,257,59051,65537,149771,178931,227531,363611,3017171,

%T 4782971,5456897,7935491,10093331,15046643,15896171,20493731,26224643,

%U 27709697,28558337,29986577,51451931,55591937,71014331,72641531,83119691,87441203,98545331,116294657

%N Primes p such that tau(p - 1) = tau(p - 2) + 1, where tau(k) is the number of divisors of k (A000005).

%C The first 5 known Fermat primes from A019434 are in the sequence.

%C Triples of [tau(a(n)-2), tau(a(n)-1), tau(a(n))]: [1, 2, 2], [2, 3, 2], [3, 4, 2], [4, 5, 2], [8, 9, 2], [8, 9, 2], [11, 12, 2], [16, 17, 2], [15, 16, 2], ...

%C Exactly one of p-1 and p-2 is a perfect square as its number of divisors is odd. - _David A. Corneth_, Sep 05 2021

%H David A. Corneth, <a href="/A347078/b347078.txt">Table of n, a(n) for n = 1..13040</a> (terms <= 10^15)

%e Prime 65537 is in the sequence because tau(65536) = tau(65535) + 1 = 16 + 1 = 17.

%t s = {}; Do[If[PrimeQ[p] && DivisorSigma[0, p - 1] == DivisorSigma[0, p - 2] + 1, AppendTo[s, p]], {p, 3, 5*10^6}]; s (* _Amiram Eldar_, Aug 16 2021 *)

%t Select[Prime[Range[6650000]],DivisorSigma[0,#-1]==DivisorSigma[0,#-2]+1&] (* _Harvey P. Dale_, Jul 30 2023 *)

%o (Magma) [m: m in [3..10^7] | IsPrime(m) and #Divisors(m - 1) eq #Divisors(m - 2) + 1]

%o (PARI) isok(p) = isprime(p) && (numdiv(p-1) == numdiv(p-2)+1); \\ _Michel Marcus_, Aug 16 2021

%Y Cf. A000005, A019434, A075408.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Aug 15 2021

%E More terms from _Amiram Eldar_, Aug 16 2021