A284521 Sum of largest prime power factors of numbers <= n.

1, 3, 6, 10, 15, 18, 25, 33, 42, 47, 58, 62, 75, 82, 87, 103, 120, 129, 148, 153, 160, 171, 194, 202, 227, 240, 267, 274, 303, 308, 339, 371, 382, 399, 406, 415, 452, 471, 484, 492, 533, 540, 583, 594, 603, 626, 673, 689, 738, 763, 780, 793, 846, 873, 884, 892, 911, 940, 999, 1004, 1065, 1096, 1105, 1169, 1182

Offset: 1

Author

Ilya Gutkovskiy, Mar 28 2017

Keywords

• nonn

Comments

Partial sums of A034699.

Examples

			a(1) = 1;
a(2) = 3 because 2 is a prime and 1 + 2 = 3;
a(3) = 6 because 3 is a prime and 3 + 3 = 6;
a(4) = 10 because 4 = 2^2 and 6 + 4 = 10;
a(5) = 15 because 5 is a prime and 10 + 5 = 15;
a(6) = 18 because 12 = 2*3 and 15 + 3 = 18, etc.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A034699, A046670, A073355.

Programs

Maple

g:= n -> max(map(t -> t[1]^t[2], ifactors(n)[2])): g(1):= 1:
ListTools:-PartialSums(map(g, [$1..100])); # Robert Israel, Mar 29 2017

Mathematica

Accumulate[Join[{1}, Table[Last[Select[Divisors[n], PrimePowerQ[#1] & ]], {n, 2, 65}]]]

PARI

a(n) = if (n==1, 1, 1+ sum(k=2, n, f = factor(k); f[#f~,1]^f[#f~,2])); \\ Michel Marcus, Mar 28 2017

Formula

Conjecture: a(n) = O(n^2/log(n)).