A238165 Number of pairs {j, k} with 0 < j < k <= n such that pi(j*n) divides pi(k*n), where pi(.) is given by A000720.

0, 1, 1, 2, 3, 2, 1, 5, 5, 5, 3, 5, 12, 5, 5, 7, 3, 2, 12, 7, 8, 9, 9, 6, 6, 11, 9, 12, 9, 15, 12, 12, 13, 7, 16, 12, 18, 15, 16, 11, 8, 8, 13, 15, 20, 13, 7, 15, 13, 7, 18, 7, 18, 15, 11, 15, 15, 12, 15, 17, 6, 18, 17, 16, 11, 15, 9, 18, 15, 13

Offset

1,4

Comments

Conjecture: (i) a(n) > 0 for all n > 1.

(ii) For any integer n > 4, the sequence pi(k*n)^(1/k) (k = 1, ..., n) is strictly decreasing.

See also A238224 for a refinement of part (i) of this conjecture.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Example

a(5) = 3 since pi(1*5) = 3 divides both pi(3*5) = 6 and pi(5*5) = 9, and pi(2*5) = 4 divides pi(4*5) = 8.
a(7) = 1 since pi(1*7) = 4 divides pi(3*7) = 8.

Mathematica

m[k_, j_]:=Mod[PrimePi[k], PrimePi[j]]==0
a[n_]:=Sum[If[m[k*n, j*n], 1, 0], {k, 2, n}, {j, 1, k-1}]
Do[Print[n, " ", a[n]], {n, 1, 70}]

Crossrefs

Cf. A000720, A237578, A237597, A237598, A237612, A237614, A237615, A238224.

Sequence in context: A055101 A319108 A349633 * A081316 A226362 A375085

Adjacent sequences: A238162 A238163 A238164 * A238166 A238167 A238168

Keyword

nonn

Author

Zhi-Wei Sun, Feb 19 2014

Status

approved