A343901 - OEIS

%I #13 Dec 06 2024 11:28:36

%S 1,1,2,2,3,2,3,3,3,3,3,3,5,3,3,3,4,3,4,3,5,3,3,3,4,5,4,5,4,3,5,4,4,4,

%T 5,5,7,4,5,4,5,5,4,4,5,3,3,4,4,4,4,5,4,4,5,5,7,4,3,4,8,5,7,4,6,4,5,4,

%U 4,5,4,5,8,7,5,7,8,5,4,4,4,5,3,5,4,4,4

%N a(n) is the number of primes p such that (p-1)|A000010(n).

%C Conjecture: a(n) > 0 for n > 2.

%C a(n) > 0 for all n >= 1 since (2-1)|phi(n). Furthermore, because phi(n) is even for n > 2 and therefore divisible by (3-1), a(n) > 1 for n > 2. - _Amiram Eldar_, Dec 06 2024

%e For n = 13: A000010(13) = 12 and for p = 2, 3, 5, 7, 13 we have p-1 = 1, 2, 4, 6, 12 and 12 is divisible by each value of p-1, so a(13) = 5.

%t a[n_] := DivisorSum[EulerPhi[n], 1 &, PrimeQ[# + 1] &]; Array[a, 100] (* _Amiram Eldar_, Dec 06 2024 *)

%o (PARI) a(n) = my(e=eulerphi(n), i=0); forprime(p=2, e+1, if(e%(p-1)==0, i++)); i \\ corrected by _Amiram Eldar_, Dec 06 2024

%o (PARI) a(n) = sumdiv(eulerphi(n), d, isprime(d+1)); \\ _Amiram Eldar_, Dec 06 2024

%Y Cf. A000010, A343902, A343903.

%K nonn

%O 1,3

%A _Felix Fröhlich_, May 03 2021

%E Data corrected by _Amiram Eldar_, Dec 06 2024