A249541 -id:A249541 - cp's OEIS frontend

A254576 Primes p such that phi(p-2) divides p-1 where phi is Euler's totient function (A000010).

3, 5, 17, 257, 65537, 83623937

Offset: 1

Jaroslav Krizek, Feb 25 2015

The first 5 known Fermat primes from A019434 are terms.
Conjecture: also primes p such that 2*phi(p-2) = p-1 (i.e., primes in A232720).
a(7) > 10^25. - Max Alekseyev, Feb 02 2024

Subsequence of A249541.

Cf. A000010, A019434, A050474, A203966, A232720.

[n: n in [3..10000000] | IsPrime(n) and (n-1) mod EulerPhi(n-2) eq 0];

A263810 Numbers k such that k = tau(k) * phi(k-2) + 1.

Original entry on oeis.org

3, 4, 5, 17, 257, 65537, 83623937

Offset: 1

Jaroslav Krizek, Oct 27 2015

Numbers k such that k = A000005(k) * A000010(k-2) + 1.
Sequence deviates from A249541; numbers 4294967297 and 6992962672132097 are not terms of this sequence.
The first 5 known Fermat primes from A019434 are in this sequence.
Conjecture: primes from this sequence are in A254576.
a(8) > 10^25. If k = tau(k) * phi(k-2) + 1 then phi(k-2) must divide k-1, thus k-2 must be a term of A203966, which has already been searched up to 10^25. - Giovanni Resta, Feb 21 2020; updated by Max Alekseyev, Feb 21 2025

			17 is in this sequence because 17 = tau(17)*phi(15) + 1 = 2*8 + 1.

Cf. A000005, A000010, A019434, A249541, A254576, A203966.

Cf. A263811 (numbers k such that k = tau(k) * phi(k-1) + 1).

[n: n in [3..1000000] |  n eq NumberOfDivisors(n) * EulerPhi(n-2) + 1];

Select[Range@ 100000, # == DivisorSigma[0, #] EulerPhi[# - 2] + 1 &] (* Michael De Vlieger, Oct 27 2015 *)

for(n=3, 1e8, if(numdiv(n)*eulerphi(n-2) == n-1, print1(n ", "))) \\ Altug Alkan, Oct 28 2015

lista(na, nb) = {my(f1 = factor(na-2), f2 = factor(na-1), f3); for(n=na, nb, f3 = factor(n); if (numdiv(f3)*eulerphi(f1) == n-1, print1(n ", ")); f1 = f2; f2 = f3;);}; \\ Michel Marcus, Feb 21 2020

cp's OEIS Frontend

A254576 Primes p such that phi(p-2) divides p-1 where phi is Euler's totient function (A000010).

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