A336930 - OEIS

%I #22 Jul 05 2022 23:30:33

%S 1,3,4,9,11,12,13,16,23,25,27,31,33,36,37,39,44,47,48,49,52,59,64,69,

%T 71,75,81,83,89,92,93,97,99,100,107,108,109,111,117,121,124,131,132,

%U 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

%N 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.

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

%C 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.

%C 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.

%o (PARI)

%o A007814(n) = valuation(n, 2);

%o A336931(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))-1)); };

%o isA336930(n) = !A336931(n);

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o isA004613(n) = (1==(n%4) && 1==factorback(factor(n)[, 1]%4)); \\ After code in A004613.

%o isA336930(n) = isA004613(A003961(core(n)));

%o (Python)

%o from math import prod

%o from itertools import count, islice

%o from sympy import factorint, nextprime, divisor_count

%o def A336930_gen(startvalue=1): # generator of terms >= startvalue

%o     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)))

%o A336930_list = list(islice(A336930_gen(),30)) # _Chai Wah Wu_, Jul 05 2022

%Y Positions of zeros in A336931.

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

%K nonn

%O 1,2

%A _Antti Karttunen_, Aug 17 2020