A346222 -id:A346222 - cp's OEIS frontend

A346623 a(n) = Sum_{ x <= n : x odd and omega(x) = 2 } x.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 15, 15, 15, 15, 15, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 69, 69, 104, 104, 104, 104, 143, 143, 143, 143, 143, 143, 188, 188, 188, 188, 188, 188, 239, 239, 239, 239, 294, 294, 351, 351, 351, 351, 351, 351, 414, 414, 479, 479, 479, 479

Offset: 1

N. J. A. Sloane, Aug 23 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A001221, A007774, A346221, A346222.

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
     `if`(n::odd and nops(ifactors(n)[2])=2, n, 0))
    end:
seq(a(n), n=1..68);  # Alois P. Heinz, Aug 23 2021

a[n_] := a[n] = If[n == 1, 0, a[n-1] + If[OddQ[n] && PrimeNu[n] == 2, n, 0]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 13 2023 *)

from sympy import primefactors
def A346623(n):
    return 0 if n <= 2 else A346623(n-1) + (n if n % 2 and len(primefactors(n)) == 2 else 0) # Chai Wah Wu, Aug 23 2021

A358739 Triangular array read by rows. T(n,k) is the number of n X n matrices A over F_2 such that Sum_{phi} nullity(phi(A)) = k where the sum is over all monic irreducible polynomials in F_2[x] that divide the characteristic polynomial of A, n >= 1, 1 <= k <= n.

Original entry on oeis.org

2, 6, 10, 84, 210, 218, 5040, 19740, 15330, 25426, 1249920, 5780880, 6939660, 7604610, 11979362, 1259919360, 7533267840, 9297061200, 12276675180, 14280964866, 24071588290, 5120312279040, 34082078607360, 48312946523520, 78970351980240, 88215877158444, 112601184828930, 195647202043778

Offset: 1

Geoffrey Critzer, Nov 29 2022

			Triangle begins
        2;
        6,      10;
       84,     210,     218;
     5040,   19740,   15330,   25426;
  1249920, 5780880, 6939660, 7604610, 11979362;
  ...

Cf. A346222 (main diagonal), A083402 (1/2 * column 1), A002416 (row sums).

cp's OEIS Frontend

A346623 a(n) = Sum_{ x <= n : x odd and omega(x) = 2 } x.

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A358739 Triangular array read by rows. T(n,k) is the number of n X n matrices A over F_2 such that Sum_{phi} nullity(phi(A)) = k where the sum is over all monic irreducible polynomials in F_2[x] that divide the characteristic polynomial of A, n >= 1, 1 <= k <= n.

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