A023512 - OEIS

A023512

Exponent of 2 in prime factorization of prime(n) + 1.

0, 2, 1, 3, 2, 1, 1, 2, 3, 1, 5, 1, 1, 2, 4, 1, 2, 1, 2, 3, 1, 4, 2, 1, 1, 1, 3, 2, 1, 1, 7, 2, 1, 2, 1, 3, 1, 2, 3, 1, 2, 1, 6, 1, 1, 3, 2, 5, 2, 1, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 3, 4, 1, 2, 7, 1, 1, 1, 1, 2, 1, 4, 1, 3, 2, 1, 1, 1, 4, 2, 5, 3, 2, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 2

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OFFSET

1,2

COMMENTS

2^a(n) is the largest power of 2 dividing (prime(n)+1).

By Dirichlet's theorem on arithmetic progressions, the asymptotic density of primes p such that p == 2^k-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

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2261 from K. G. Stier)

FORMULA

a(n) = A007814(A008864(n)). - Amiram Eldar, Mar 14 2025

EXAMPLE

a(9) = 3 because the 9th prime is 23 and the largest power of 2 dividing 24 is 3.

MAPLE

with(numtheory): a:=proc(n) local div, s, j, c: div:=divisors(1+ithprime(n)): s:=nops(div): for j from 1 to s do if type(simplify(log[2](div[j])), integer)=true then c[j]:=simplify(log[2](div[j])) else c[j]:=0 fi od: max(seq(c[j], j=1..s)) end: seq(a(n), n=1..120); # most probably not the simplest Maple program - Emeric Deutsch, Jul 20 2005

MATHEMATICA

Join[{0}, Table[FactorInteger[Prime[n] + 1][[1]][[2]], {n, 2, 100}]] (* Clark Kimberling, Oct 01 2013 *)

IntegerExponent[Prime[Range[100]] + 1, 2] (* Zak Seidov, Apr 25 2014 *)

PROG

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

(PARI) a(n)=valuation(prime(n)+1, 2);

vector(100, n, a(n)) \\ Joerg Arndt, Mar 11 2014

CROSSREFS

Cf. A007814, A008864, A023506, A239114.

Sequence in context: A255191 A277818 A324607 * A322027 A088192 A218459

Adjacent sequences: A023509 A023510 A023511 * A023513 A023514 A023515

KEYWORD