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A363331
a(n) is the sum of divisors of n that are both coreful and infinitary.
2
1, 2, 3, 4, 5, 6, 7, 14, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 42, 25, 26, 39, 28, 29, 30, 31, 50, 33, 34, 35, 36, 37, 38, 39, 70, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 78, 55, 98, 57, 58, 59, 60, 61, 62, 63, 84, 65, 66, 67, 68
OFFSET
1,2
COMMENTS
First differs from A363334 at n = 16.
The number of these divisors is A363329(n).
LINKS
FORMULA
Multiplicative with a(p^e) = (Product_{k>=0} ((p^(2^k*(b(k)+1)) - 1)/(p^(2^k) - 1)) - 1, where e = Sum_{k >= 0} b(k) * 2^k is the binary representation of e.
a(n) >= n, with equality if and only if n is in A138302.
a(n) >= A361810(n), with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p) * Sum_{k>=1} a(p^k)/p^(2*k)) = 0.53906337497505398777... .
EXAMPLE
a(8) = 14 since 8 has 3 divisors that are both infinitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.
MATHEMATICA
f[p_, e_] := Times @@ (1 + Flatten[p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], _?(# == 1 &)]))]) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(f[i, 2]); prod(k = 1, #b, if(b[k], f[i, 1]^(2^(#b - k)) + 1, 1)) - 1); }
KEYWORD
nonn,mult
AUTHOR
Amiram Eldar, May 28 2023
STATUS
approved