A336930 Numbers k such that the 2-adic valuation of A003973(k), the sum of divisors of the prime shifted k is equal to the 2-adic valuation of the number of divisors of k.

1, 3, 4, 9, 11, 12, 13, 16, 23, 25, 27, 31, 33, 36, 37, 39, 44, 47, 48, 49, 52, 59, 64, 69, 71, 75, 81, 83, 89, 92, 93, 97, 99, 100, 107, 108, 109, 111, 117, 121, 124, 131, 132, 139, 141, 143, 144, 147, 148, 151, 156, 167, 169, 176, 177, 179, 188, 191, 192, 193, 196, 207, 208, 213, 225, 227, 229, 236, 239, 243, 249, 251

Offset

1,2

Comments

Numbers k for which A295664(k) is equal to A336932(k). Note that A295664(A003961(n)) = A295664(n).

Numbers k such that A003961(A007913(k)) [or equally, A007913(A003961(k))] is in A004613, i.e., has only prime divisors of the form 4k+1.

Subsequences include squares (A000290), and also primes p which when prime-shifted [as A003961(p)] become primes of the form 4k+1 (A002144), and all their powers as well as the products between these.

Links

Table of n, a(n) for n=1..72.

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)); };
isA336930(n) = !A336931(n);
(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA004613(n) = (1==(n%4) && 1==factorback(factor(n)[, 1]%4)); \\ After code in A004613.
isA336930(n) = isA004613(A003961(core(n)));
(Python)
from math import prod
from itertools import count, islice
from sympy import factorint, nextprime, divisor_count
def A336930_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(~(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(), count(max(startvalue, 1)))
A336930_list = list(islice(A336930_gen(), 30)) # Chai Wah Wu, Jul 05 2022

Crossrefs

Positions of zeros in A336931.

Cf. A000290, A002144, A003961, A003973, A004613, A007814, A007913, A295664, A336918, A336932, A336937.

Sequence in context: A126269 A319618 A172980 * A035252 A227939 A047459

Adjacent sequences: A336927 A336928 A336929 * A336931 A336932 A336933

Keyword

nonn

Author

Antti Karttunen, Aug 17 2020

Status

approved