A066673 -id:A066673 - cp's OEIS frontend

A066669 Numbers m such that phi(m) = 2^k*prime for some k >= 0.

7, 9, 11, 13, 14, 18, 21, 22, 23, 25, 26, 28, 29, 33, 35, 36, 39, 41, 42, 44, 45, 46, 47, 50, 52, 53, 55, 56, 58, 59, 65, 66, 69, 70, 72, 75, 78, 82, 83, 84, 87, 88, 89, 90, 92, 94, 97, 100, 104, 105, 106, 107, 110, 112, 113, 115, 116, 118, 119, 123, 130, 132, 137, 138

Offset: 1

Labos Elemer, Dec 18 2001

nonn

Sequence is infinite, since 2n is in the sequence if and only if n is in the sequence. What is its density? - Charles R Greathouse IV, Feb 21 2013

Products of powers of 2, distinct terms (at least one) of A074781, and possibly (if all the factors from A074781 are Fermat primes, A019434) an additional Fermat prime (i.e., it can be divisible by a square of one Fermat prime, A330828). - Amiram Eldar, Feb 11 2025

			7 is a term because phi(7) = 6 divided by 2 is 3, a prime.
21 is a term because phi(21) = 12 divided by 4 is 3, a prime.
15 is not a term because phi(15) = 8 divided by 8 is 1, not a prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010, A066670, A066671, A066672, A066673, A065966.

Cf. A019434, A058500, A074781, A330828.

Select[Range@ 138, PrimeQ@ Last@ Most@ NestWhileList[#/2 &, EulerPhi@ #, IntegerQ@ # &] &] (* Michael De Vlieger, Mar 18 2017 *)

is(n)=n=eulerphi(n);isprime(n>>valuation(n,2)) \\ Charles R Greathouse IV, Feb 21 2013

A066670 Primes arising in A066669: the only odd prime divisor of phi(A066669(n)).

Original entry on oeis.org

3, 3, 5, 3, 3, 3, 3, 5, 11, 5, 3, 3, 7, 5, 3, 3, 3, 5, 3, 5, 3, 11, 23, 5, 3, 13, 5, 3, 7, 29, 3, 5, 11, 3, 3, 5, 3, 5, 41, 3, 7, 5, 11, 3, 11, 23, 3, 5, 3, 3, 13, 53, 5, 3, 7, 11, 7, 29, 3, 5, 3, 5, 17, 11, 3, 23, 3, 7, 37, 5, 3, 3, 13, 5, 5, 41, 83, 3, 43, 7, 5, 29, 11, 89, 3, 11, 5, 23, 3, 3

Offset: 1

Labos Elemer, Dec 18 2001

nonn

			A066669(9) = 23, phi(23) = 2*11, so a(9)=11.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A000265, A053575, A006530, A065966, A066669, A066671, A066672, A066673.

Select[Array[#/2^IntegerExponent[#, 2] &@ EulerPhi@ # &, 200], PrimeQ]  (* Michael De Vlieger, Dec 08 2018 *)

lista(nn) = {for (n=1, nn, en=eulerphi(n); if (isprime(p=en>>valuation(en, 2)), print1(p, ", ")); ); } \\ Michel Marcus, Dec 08 2018

From Amiram Eldar, Jul 18 2024: (Start)
a(n) = A053575(A066669(n)).
a(n) = A000265(A000010(A066669(n))) = A006530(A000010(A066669(n))). (End)

A066671 a(n) is the largest power of 2 that divides phi(A066669(n)).

Original entry on oeis.org

2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 4, 4, 4, 4, 8, 4, 8, 8, 4, 4, 8, 2, 2, 4, 8, 4, 8, 8, 4, 2, 16, 4, 4, 8, 8, 8, 8, 8, 2, 8, 8, 8, 8, 8, 4, 2, 32, 8, 16, 16, 4, 2, 8, 16, 16, 8, 8, 2, 32, 16, 16, 8, 8, 4, 16, 4, 16, 16, 4, 8, 32, 16, 8, 16, 16, 2, 2, 16, 4, 8, 16, 4, 8, 2, 16, 8, 32, 4, 64, 32, 32

Offset: 1

Labos Elemer, Dec 18 2001

nonn

			The first, 4th and 15th terms in A066669 are 7, 13 and 35; phi(7) = 2*3, phi(13) = 4*3, phi(35) = 24 = 8*3; the largest powers of 2 are 2, 4 and 8; so a(1) = 2, a(4) = 4, a(15) = 8.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A065966, A066669, A066670, A066672, A066673, A069177.

Select[Array[{#1/#2, #2} & @@ {#, 2^IntegerExponent[#, 2]} &@ EulerPhi@ # &, 200], PrimeQ@ First@ # &][[All, -1]] (* Michael De Vlieger, Dec 08 2018 *)

lista(nn) = {for (n=1, nn, en=eulerphi(n); if (isprime(p=en>>valuation(en, 2)), print1(en/p, ", ")););} \\ Michel Marcus, Jan 03 2017

From Amiram Eldar, Jul 18 2024: (Start)
a(n) = A069177(A066669(n)).
a(n) = 2^A066672(n). (End)

Name corrected by Amiram Eldar, Jul 18 2024

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A066669 Numbers m such that phi(m) = 2^k*prime for some k >= 0.

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A066670 Primes arising in A066669: the only odd prime divisor of phi(A066669(n)).

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A066671 a(n) is the largest power of 2 that divides phi(A066669(n)).

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