|
%I #48 Jan 01 2023 20:22:57
%S 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,
%T 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,
%U 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
%N Exponent of 2 in prime factorization of prime(n) - 1.
%C 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
%C 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
%C 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
%H Robert Israel, <a href="/A023506/b023506.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A007814(A000010(A000040(n))) = A007814(A006093(n)).
%e n=25, p(25)=97, A006093(25) = 96 = 2*2*2*2*2*3, so a(25)=5.
%p A023506:= x -> padic[ordp](ithprime(x)-1,2):
%p seq(A023506(x),x=1..1000); # _Robert Israel_, May 06 2014
%t 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 *)
%t Table[IntegerExponent[Prime[n] - 1, 2], {n, 110}] (* _Bruno Berselli_, Aug 05 2013 *)
%o (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
%o (PARI) forprime(p=2, 700, print1(valuation(p-1,2),", ")); \\ _Bruno Berselli_, Aug 05 2013
%o (Magma) [Valuation(NthPrime(n)-1, 2): n in [1..110]]; // _Bruno Berselli_, Aug 05 2013
%o (Python)
%o from sympy import prime
%o def A023506(n): return (~(m:=prime(n)-1)& m-1).bit_length() # _Chai Wah Wu_, Jul 07 2022
%Y Cf. A007814, A000010, A000040, A006093, A057023, A057773 (partial sums).
%Y Subsequence of A001511 (except 1st term).
%K nonn,easy
%O 1,3
%A _Clark Kimberling_
|
