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A349273 Number of odd divisors of prime(n) - 1.

%I A349273 #21 Oct 11 2024 13:20:30
%S A349273 1,1,1,2,2,2,1,3,2,2,4,3,2,4,2,2,2,4,4,4,3,4,2,2,2,3,4,2,4,2,6,4,2,4,
%T A349273 2,6,4,5,2,2,2,6,4,2,3,6,8,4,2,4,2,4,4,4,1,2,2,8,4,4,4,2,6,4,4,2,8,4,
%U A349273 2,4,2,2,4,4,8,2,2,6,3,4,4,8,4,4,4,4,2,4,4,8,2,2,6,6,4
%N A349273 Number of odd divisors of prime(n) - 1.
%C A349273 a(n) is odd if and only if prime(n) is in A249410. - _Jianing Song_, Nov 14 2021
%H A349273 Robert Israel, <a href="/A349273/b349273.txt">Table of n, a(n) for n = 1..10000</a>
%F A349273 a(n) = A001227(A006093(n)).
%p A349273 nod:= n -> numtheory:-tau(n/2^padic:-ordp(n,2)):
%p A349273 map(nod, [seq(ithprime(i)-1,i=1..100)]); # _Robert Israel_, Oct 11 2024
%t A349273 a[n_] := DivisorSigma[0, (k = Prime[n] - 1)/2^IntegerExponent[k, 2]]; Array[a, 100] (* _Amiram Eldar_, Jun 03 2021 *)
%t A349273 Count[Divisors[#-1],_?OddQ]&/@Prime[Range[100]] (* _Harvey P. Dale_, Jan 22 2024 *)
%o A349273 (Magma) [NumberOfDivisors(p-1)/Valuation(2*p-2, 2): p in PrimesUpTo(500)];
%o A349273 (Python)
%o A349273 from sympy import divisors, prime
%o A349273 def a(n): return sum(d%2 for d in divisors(prime(n)-1))
%o A349273 print([a(n) for n in range(1, 96)]) # _Michael S. Branicky_, Jul 04 2021
%o A349273 (PARI) a(n) = sumdiv(prime(n)-1, d, d%2); \\ _Michel Marcus_, Dec 18 2021
%Y A349273 Cf. A000040, A001227, A006093, A249410.
%K A349273 nonn,easy
%O A349273 1,4
%A A349273 _Juri-Stepan Gerasimov_, Nov 12 2021