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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.
3
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.
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
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 17 2020
STATUS
approved