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A336931
Difference between the 2-adic valuation of A003973(n) [= the sum of divisors of the prime shifted n] and the 2-adic valuation of the number of divisors of n.
4
0, 1, 0, 0, 2, 1, 1, 1, 0, 3, 0, 0, 0, 2, 2, 0, 1, 1, 2, 2, 1, 1, 0, 1, 0, 1, 0, 1, 4, 3, 0, 1, 0, 2, 3, 0, 0, 3, 0, 3, 1, 2, 3, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 5, 0, 2, 1, 1, 1, 0, 2, 1, 2, 1, 0, 4, 0, 1, 3, 1, 0, 2, 1, 1, 1, 2, 0, 2, 0, 1, 3, 4, 4, 1, 0, 3, 1, 0, 0, 1, 4, 1, 0, 1, 0, 0, 2, 2, 1, 1, 3
OFFSET
1,5
COMMENTS
Note that A295664(n) = A295664(A003961(n)).
FORMULA
Additive with a(p^e) = 0 when e is even, a(p^e) = A007814(1+A003961(p))-1 when e is odd.
a(n) = A336932(n) - A295664(n).
a(n) = a(A007913(n)).
PROG
(PARI)
A007814(n) = valuation(n, 2);
A336931(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))-1)); };
(PARI)
A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A007814(n) = valuation(n, 2);
A336931(n) = (A007814(A003973(n)) - A007814(numdiv(n)));
(Python)
from math import prod
from sympy import factorint, nextprime, divisor_count
def A336931(n): return (~(m:=prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p, e in factorint(n).items()))& m-1).bit_length()-(~(k:=int(divisor_count(n))) & k-1).bit_length() # Chai Wah Wu, Jul 05 2022
CROSSREFS
Cf. A003961, A003973, A007814, A007913, A295664, A336930 (positions of zeros), A336932, A336937.
Sequence in context: A249303 A361167 A319081 * A363953 A182662 A308778
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 17 2020
STATUS
approved