A342010 Number of times the term 2 has occurred so far in the range 1..n of A073751.

1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Author

Antti Karttunen, Mar 08 2021

Keywords

• nonn

Comments

Number of prime factors (with multiplicity) in the primorial deflation of the n-th colossally abundant number [A342012(n) = A319626(A004490(n))], provided that the quotient A004490(1+n)/A004490(n) is always a prime.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000120, A001222, A004490, A073751, A319626, A342012, A342013.

Programs

Mathematica

Block[{a = {2}, b, c, f, k, m, n, q = {1}, lim = 105}, f[w_] := Block[{p = w[[1]], i = w[[2]]}, ((Log[(p^(i + 2) - 1)/(p^(i + 1) - 1)])/Log[p]) - 1]; m = {{2, 1}, {3, 0}}; c = 1; b = Array[f[m[[#]]] &, c + 1]; For[n = 2, n <= lim, n++, k = Position[b, Max[b]][[1, 1]]; AppendTo[a, m[[k, 1]]]; AppendTo[q, Boole[m[[k, 1]] == 2]]; m[[k, 2]]++; If[k > c, c++; AppendTo[m, {Prime[k + 1], 0}]; AppendTo[b, f[m[[-1]]]]]; b[[k]] = f[m[[k]]]]; Accumulate@ q] (* Michael De Vlieger, Mar 12 2021, after T. D. Noe at A073751 *)

PARI

v073751 = readvec("b073751_to.txt"); \\ Prepared with gawk '{ print $2 }' < b073751.txt > b073751_to.txt
A073751(n) = v073751[n];
A342010(n) = sum(k=1,n,(2==A073751(k)));

Formula

a(n) = Sum_{k=1..n} [2==A073751(k)], where [ ] is the Iverson bracket.

a(n) = A001222(A342012(n)) = A000120(A342013(n)).