A343901 a(n) is the number of primes p such that (p-1)|A000010(n).

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, 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, 4, 5, 4, 5, 8, 7, 5, 7, 8, 5, 4, 4, 4, 5, 3, 5, 4, 4, 4

Offset

1,3

Comments

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

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

Links

Table of n, a(n) for n=1..87.

Example

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.

Mathematica

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

Prog

(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
(PARI) a(n) = sumdiv(eulerphi(n), d, isprime(d+1)); \\ Amiram Eldar, Dec 06 2024

Crossrefs

Cf. A000010, A343902, A343903.

Sequence in context: A340862 A202472 A235613 * A376181 A322418 A019569

Adjacent sequences: A343898 A343899 A343900 * A343902 A343903 A343904

Keyword

nonn

Author

Felix Fröhlich, May 03 2021

Extensions

Data corrected by Amiram Eldar, Dec 06 2024

Status

approved