A351827 Sum of the numbers <= n that are either prime, a divisor of n, or both.

1, 3, 6, 10, 11, 17, 18, 30, 27, 28, 29, 51, 42, 56, 57, 70, 59, 92, 78, 112, 99, 100, 101, 155, 126, 127, 137, 147, 130, 191, 161, 221, 194, 195, 196, 246, 198, 236, 237, 280, 239, 322, 282, 352, 351, 328, 329, 447, 378, 414, 380, 411, 382, 496, 437, 492, 439, 440, 441, 598

Offset

1,2

Links

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

Formula

a(n) = Sum_{k=1..n} k * (u(k) + v(n/k) - u(k)*v(n/k)), where u(n) is the prime characteristic (A010051) and v(n) = 1 - ceiling(n) + floor(n).

a(n) = sigma(n) - sopf(n) + Sum_{p<=n, p prime} p. - Wesley Ivan Hurt, Dec 31 2023

Maple

N:= 200: # for a(1)..a(N)
P:= select(isprime, [2, seq(i, i=3..N, 2)]):
f:= proc(n) convert(NumberTheory:-Divisors(n) union convert(P[1..ListTools:-BinaryPlace(P, n+1)], set), `+`) end proc:
map(f, [$1..N]); # Robert Israel, Mar 02 2026

Mathematica

{1}~Join~Table[DivisorSigma[1, n] + Total[Prime@ Range[PrimePi[n]]] - Total@ FactorInteger[n][[All, 1]], {n, 2, 60}] (* Michael De Vlieger, Mar 02 2026 *)

Crossrefs

Cf. A000203 (sigma), A008472 (sopf), A010051, A034387, A351519.

Sequence in context: A105359 A105355 A183545 * A351828 A158975 A282876

Adjacent sequences: A351824 A351825 A351826 * A351828 A351829 A351830

Keyword

nonn

Author

Wesley Ivan Hurt, Feb 20 2022

Status

approved