A336839 - OEIS

%I #22 Aug 17 2020 20:51:46

%S 1,1,1,3,1,1,1,1,3,1,1,1,1,1,1,5,1,3,1,3,1,1,1,1,1,1,1,1,1,1,1,3,1,1,

%T 1,9,1,1,1,1,1,1,1,3,3,1,1,5,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,1,1,1,3,

%U 1,1,1,3,1,1,1,1,1,1,1,5,5,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,3,3,1,1,1,1,1

%N Denominator of the arithmetic mean of the divisors of A003961(n).

%C Also denominator of A336841(n) / A000005(n).

%C All terms are odd because A336932(n) = A007814(A003973(n)) >= A295664(n) for all n.

%H Antti Karttunen, <a href="/A336839/b336839.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = denominator(A003973(n)/A000005(n)).

%F a(n) = d(n)/A336856(n) = d(n)/gcd(d(n),A003973(n)) = d(n)/gcd(d(n),A336841(n)), where d(n) is the number of divisors of n, A000005(n).

%F a(n) = A057021(A003961(n)).

%F For all primes p, and e >= 0, a(A000225(e)) = a(p^((2^e) - 1)) = 1. [See A336856]

%F It seems that for all odd primes p, and with the exponents e=5, 11, 17 or 23 (at least these), a(p^e) = 1.

%F It seems that a(27^((2^n)-1)) = A052940(n-1) for all n >= 1.

%o (PARI)

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

%o A336839(n) = denominator(sigma(A003961(n))/numdiv(n));

%Y Cf. A000005, A000225, A003961, A003973, A007814, A052940, A057021, A295664, A336840, A336841, A336856, A336931, A336932.

%Y Cf. A336918 (positions of 1's), A336919 (of terms > 1).

%Y Cf. A336837 and A336838 (numerators).

%K nonn,frac

%O 1,4

%A _Antti Karttunen_, Aug 07 2020