A387279 Inverse Möbius transform of A006530.

1, 3, 4, 5, 6, 9, 8, 7, 7, 13, 12, 14, 14, 17, 14, 9, 18, 15, 20, 20, 18, 25, 24, 19, 11, 29, 10, 26, 30, 29, 32, 11, 26, 37, 20, 23, 38, 41, 30, 27, 42, 37, 44, 38, 22, 49, 48, 24, 15, 23, 38, 44, 54, 21, 28, 35, 42, 61, 60, 44, 62, 65, 28, 13, 32, 53, 68, 56

Offset

1,2

Comments

Sum of the greatest prime divisor of all the divisors of n.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Aloe Poliszuk)

Formula

a(n) = Sum_{d|n} A006530(d).

Order the prime divisors of n as p_{1} > p_{2} > ... > p_{m}, and let e_{j} be the greatest exponent such that p_{j}^e_{j} divides n. Then, a(n) = 1 + Sum_{i=1..m} p_{i} * e_{i} * Product_{j=1..i} (e_{j} + 1).

Maple

a:= proc(n) uses numtheory;
      add(max(factorset(d)[], 1), d=divisors(n))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 24 2025

Prog

(PARI)
Gpf(n) = if(n==1, 1, vecmax(factorint(n)[, 1]));
a(n) = sumdiv(n, d, Gpf(d));
(PARI) a(n) = my(fac = factorint(n)); 1 + sum(X=1, #fac[, 1], fac[X, 1] * fac[X, 2] * prod(Y=1, X-1, fac[Y, 2]+1));

Crossrefs

Cf. A006530, A000040, A020639.

Sequence in context: A037348 A277898 A212640 * A333694 A364064 A338321

Adjacent sequences: A387276 A387277 A387278 * A387280 A387281 A387282

Keyword

nonn

Author

Aloe Poliszuk, Sep 24 2025

Status

approved