cp's OEIS Frontend

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.

%I A336930 #22 Jul 05 2022 23:30:33
%S A336930 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 A336930 71,75,81,83,89,92,93,97,99,100,107,108,109,111,117,121,124,131,132,
%U A336930 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 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.
%C A336930 Numbers k for which A295664(k) is equal to A336932(k). Note that A295664(A003961(n)) = A295664(n).
%C A336930 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 A336930 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 A336930 (PARI)
%o A336930 A007814(n) = valuation(n, 2);
%o A336930 A336931(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))-1)); };
%o A336930 isA336930(n) = !A336931(n);
%o A336930 (PARI)
%o A336930 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A336930 isA004613(n) = (1==(n%4) && 1==factorback(factor(n)[, 1]%4)); \\ After code in A004613.
%o A336930 isA336930(n) = isA004613(A003961(core(n)));
%o A336930 (Python)
%o A336930 from math import prod
%o A336930 from itertools import count, islice
%o A336930 from sympy import factorint, nextprime, divisor_count
%o A336930 def A336930_gen(startvalue=1): # generator of terms >= startvalue
%o A336930     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 A336930_list = list(islice(A336930_gen(),30)) # _Chai Wah Wu_, Jul 05 2022
%Y A336930 Positions of zeros in A336931.
%Y A336930 Cf. A000290, A002144, A003961, A003973, A004613, A007814, A007913, A295664, A336918, A336932, A336937.
%K A336930 nonn
%O A336930 1,2
%A A336930 _Antti Karttunen_, Aug 17 2020