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

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, 18, 39, 26, 16, 24, 19, 182, 21, 20, 24, 403, 21, 24, 27, 40, 22, 36, 24, 91, 124, 30, 27, 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

Offset

1,2

Comments

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.

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..16383

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

Formula

a(n) = A057020(A003961(n)).

a(n) = numerator(A003973(n)/A000005(n)).

a(n) = A003973(n) / A336856(n) = A003973(n) / gcd(A000005(n), A003973(n)).

a(p) = A048673(p) for all primes p.

a(p^3) = 2*A048673(p)^3 - 2*A048673(p)^2 + A048673(p). [The denominator A336839(p^3) = 1 for all p]

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336838(n) = numerator(sigma(A003961(n))/numdiv(n));

Crossrefs

Cf. A000005, A003961, A003973, A048673, A057020, A336840, A336918.

Cf. A336839 (denominators).

Sequence in context: A345040 A196378 A336848 * A051298 A069870 A067180

Adjacent sequences: A336835 A336836 A336837 * A336839 A336840 A336841

Keyword

nonn,frac

Author

Antti Karttunen, Aug 07 2020

Status

approved