cp's OEIS Frontend

A363061 Number of k <= P(n) such that rad(k) | P(n), where rad(n) = A007947(n) and P(n) = A002110(n).

%I A363061 #52 Jul 09 2023 02:43:42
%S A363061 1,2,5,18,68,283,1161,4843,19985,83074,349670,1456458,6107257,
%T A363061 25547835,106115655,440396113,1833079809,7642924612,31705433101,
%U A363061 131711607956,546283729493,2257462298234,9339325821411,38593708318690,159600066415313,661371515924516,2736805917843710
%N A363061 Number of k <= P(n) such that rad(k) | P(n), where rad(n) = A007947(n) and P(n) = A002110(n).
%H A363061 Bert Dobbelaere, <a href="/A363061/a363061.py.txt">Python program</a>
%F A363061 a(n) = A010846(A002110(n)).
%F A363061 a(n) >= 2^n.
%e A363061 a(0) = 1 since P(0) = 1 and 1 | 1.
%e A363061 a(1) = 2 since P(1) = 2 and both 1 | 2 and 2 | 2.
%e A363061 a(2) = 5 since P(2) = 6 and rad(m) | 6 for m = {1, 2, 3, 4, 6}.
%e A363061 a(3) = 18 since P(3) = 30 and rad(m) | 30 for m = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30}, etc.
%e A363061 Regarding a(3), we see that there are 18 terms in the tensor product of prime power ranges of 2, 3, and 5 that do not exceed 30:
%e A363061 5^0X | 2^0 2^1 2^2 2^3 2^4    5^1X | 2^0 2^1 2^2    5^2X | 2^0
%e A363061 --------------------------    ------------------    ----------
%e A363061 3^0  |   1   2   4   8  16    3^0  |   5  10  20    3^0  |  25
%e A363061 3^1  |   3   6  12  24        3^1  |  15  30
%e A363061 3^2  |   9  18
%e A363061 3^3  |  27
%e A363061 Hence, a(3) = 18. This approach proves handy for larger n.
%t A363061 f[1] = 1; f[n_] := Function[w,
%t A363061 ToExpression@ StringJoin["Block[{n = ", ToString@ n,
%t A363061     ", k = 0}, Flatten@ Table[k++, ",
%t A363061     Most@ Flatten@ Map[{#, ", "} &, #], "]; k]"] &@
%t A363061       MapIndexed[
%t A363061         Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[",
%t A363061           ToString@ First@ p, ", n/(",
%t A363061           ToString@ InputForm[Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]],
%t A363061           ")]}"] ]@ w[[First@ #2]] &, w]]@
%t A363061    Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &,
%t A363061      FactorInteger[n][[All, 1]]];
%t A363061    Map[f, FoldList[Times, 1, Prime@ Range@ 9] ]
%Y A363061 Cf. A002110, A007947, A010846.
%K A363061 nonn,hard
%O A363061 0,2
%A A363061 _Michael De Vlieger_, Jun 16 2023
%E A363061 Corrected a(15) and added a(16)-a(23) from _Bert Dobbelaere_, Jun 27 2023
%E A363061 a(24)-a(26) from _Martin Ehrenstein_, Jul 08 2023