cp's OEIS Frontend

A336838 Numerator of the arithmetic mean of the divisors of A003961(n).

%I A336838 #26 Aug 20 2020 21:24:22
%S A336838 1,2,3,13,4,6,6,10,31,8,7,13,9,12,12,121,10,62,12,52,18,14,15,30,19,
%T A336838 18,39,26,16,24,19,182,21,20,24,403,21,24,27,40,22,36,24,91,124,30,27,
%U A336838 363,133,38,30,39,30,78,28,60,36,32,31,52,34,38,62,1093,36,42,36,130,45,48,37,310,40,42,57,52,42,54,42
%N A336838 Numerator of the arithmetic mean of the divisors of A003961(n).
%C A336838 Ratio r(n) = a(n)/A336839(n) is multiplicative. For example r(3) = 3/1, r(4) = 13/3, thus r(12) = r(3)*r(4) = 13/1.
%C A336838 Conjecture: For all primes p with an odd exponent e, a(p^e) is a multiple of A048673(p). Note that q+1 is a divisor of (q+1)^e - sigma(q^e) = (q+1)^e - (1 + q + q^2 + ... + q^e) when e is odd, thus also A048673(p) = (q+1)/2 is, where q = A003961(p), thus the conjecture holds, unless the denominator (A336839) has enough prime factors of A048673(p).
%H A336838 Antti Karttunen, <a href="/A336838/b336838.txt">Table of n, a(n) for n = 1..16383</a>
%H A336838 Antti Karttunen, <a href="/A336838/a336838.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A336838 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A336838 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A336838 a(n) = A057020(A003961(n)).
%F A336838 a(n) = numerator(A003973(n)/A000005(n)).
%F A336838 a(n) = A003973(n) / A336856(n) = A003973(n) / gcd(A000005(n), A003973(n)).
%F A336838 a(p) = A048673(p) for all primes p.
%F A336838 a(p^3) = 2*A048673(p)^3 - 2*A048673(p)^2 + A048673(p). [The denominator A336839(p^3) = 1 for all p]
%o A336838 (PARI)
%o A336838 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A336838 A336838(n) = numerator(sigma(A003961(n))/numdiv(n));
%Y A336838 Cf. A000005, A003961, A003973, A048673, A057020, A336840, A336918.
%Y A336838 Cf. A336839 (denominators).
%K A336838 nonn,frac
%O A336838 1,2
%A A336838 _Antti Karttunen_, Aug 07 2020