A378176 Sum over all multiplicative partitions mu of n (with factors > 1) of the encoding as Product_{j in mu} prime(j).

1, 3, 5, 16, 11, 28, 17, 67, 48, 62, 31, 156, 41, 94, 102, 303, 59, 270, 67, 334, 158, 172, 83, 743, 218, 224, 343, 508, 109, 707, 127, 1173, 292, 316, 336, 1651, 157, 364, 372, 1587, 179, 1091, 191, 926, 960, 448, 211, 3468, 516, 1202, 528, 1198, 241, 2209

Offset: 1

Author

Alois P. Heinz, Nov 18 2024

Keywords

• nonn

Examples

			The multiplicative partitions of n=8 are {[8], [4,2], [2,2,2]}, encodings give {prime(8), prime(4)*prime(2), prime(2)^3} = {19, 7*3, 3^3} = {19, 21, 27}; the sum gives a(8) = 67.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Row sums of A378175.

Cf. A000040, A001055, A006450, A145519, A377853.

Programs

Maple

b:= proc(n) option remember; `if`(n=1, {1}, {seq(map(x-> x*
      ithprime(d), b(n/d))[], d=numtheory[divisors](n) minus {1})})
    end:
a:= n-> add(i, i=b(n)):
seq(a(n), n=1..54);

Formula

a(prime(n)) = a(A000040(n)) = A006450(n).