A023506 Exponent of 2 in prime factorization of prime(n) - 1.

0, 1, 2, 1, 1, 2, 4, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 5, 2, 1, 1, 2, 4, 1, 1, 3, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 6, 2, 1, 1, 1, 1, 2, 3, 1, 4, 1, 8, 1, 2, 1, 2, 3, 1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 5, 1, 1, 2, 1, 1, 2, 2, 4, 3, 1, 2, 1, 4, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 3, 1

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Also the number of steps to reach an integer starting with prime(n)/2 and iterating the map x->x*ceiling(x). - Benoit Cloitre, Sep 06 2002

Also exponent of 2 in -1 + prime(n)^s for odd exponents s because (-1 + prime(n)^s)/(prime(n) - 1) is odd. - Labos Elemer, Jan 20 2004

First occurrence of 0,1,2,3,4,...: 1, 2, 3, 13, 7, 25, 44, 116, 55, 974, 1581, 2111, 1470, 4289, 10847, 15000, 6543, 91466, 62947, 397907, 498178, ..., for primes 2, 3, 5, 41, 17, 97, 193, 641, 257, 7681, 13313, 18433, 12289, 40961, 114689, 163841, 65537, 1179649, 786433, 5767169, 7340033, .... - Robert G. Wilson v, May 28 2009

By Dirichlet's theorem on arithmetic progressions, the asymptotic density of primes p such that p == 1 (mod 2^k) within all the primes is 1/2^(k-1), for k >= 1. This is also the asymptotic density of terms in this sequence that are >= k. Therefore, the asymptotic density of the occurrences of k in this sequence is d(k) = 1/2^(k-1) - 1/2^k = 1/2^k, and the asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 2. - Amiram Eldar, Mar 14 2025

Examples

			For n = 25, prime(25) = 97, A006093(25) = 96 = 2*2*2*2*2*3, so a(25) = 5.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007814, A000010, A000040, A006093, A023512, A057023, A057773 (partial sums).

Subsequence of A001511 (except 1st term).

Programs

Magma

[Valuation(NthPrime(n)-1, 2): n in [1..110]]; // Bruno Berselli, Aug 05 2013

Maple

A023506:= x -> padic[ordp](ithprime(x)-1,2):
seq(A023506(x),x=1..1000); # Robert Israel, May 06 2014

Mathematica

f[n_] := Block[{fi = First@ FactorInteger[ Prime@n - 1]}, If[ fi[[1]] == 2, fi[[2]], 0]]; Array[f, 105] (* Robert G. Wilson v, May 28 2009 *)
Table[IntegerExponent[Prime[n] - 1, 2], {n, 110}] (* Bruno Berselli, Aug 05 2013 *)

PARI

A023506(n) = {local(m,r);r=0;m=prime(n)-1;while(m%2==0,m=m/2;r++);r} \\ Michael B. Porter, Jan 26 2010

PARI

forprime(p=2, 700, print1(valuation(p-1,2),", ")); \\ Bruno Berselli, Aug 05 2013

Python

from sympy import prime
def A023506(n): return (~(m:=prime(n)-1)& m-1).bit_length() # Chai Wah Wu, Jul 07 2022

Formula

a(n) = A007814(A000010(A000040(n))) = A007814(A006093(n)).