A342463 a(n) = A342001(A342456(n)); "wild part" of the arithmetic derivative of A342456(n).

1, 1, 1, 2, 1, 2, 12, 8, 1, 2, 6, 4, 50, 24, 16, 16, 1, 2, 6, 4, 126, 62, 46, 26, 1486, 100, 1142, 48, 2056, 32, 342, 10, 1, 2, 6, 4, 94, 24, 72, 18, 242, 120, 1588, 54, 3408, 92, 1740, 22, 6846, 2972, 4340, 766, 5048, 1374, 652, 376, 71156, 22710, 20390, 64, 738580, 4272, 568, 20, 1, 2, 6, 4, 264, 12, 196, 8, 318

Offset: 0

Author

Antti Karttunen, Mar 15 2021

Keywords

• nonn
• look

Comments

Like in A342462, also here the subsequences starting at each n = 2^k seem to be slowly converging towards A329886: 1, 2, 6, 4, 30, 12, 36, 8, 210, 60, ...

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to primorial base

Crossrefs

Cf. A003415, A003557, A005940, A108951, A329886, A342001, A342002, A342456, A342462, A342920.

Programs

Mathematica

Block[{a, f, r = MixedRadix[Reverse@ Prime@ Range@ 24]}, f[n_] := Times @@ MapIndexed[Prime[First[#2]]^#1 &, Reverse@ IntegerDigits[n, r]]; a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ@ n, (Times @@ Map[Prime[PrimePi@ #1 + 1]^#2 & @@ # &, FactorInteger[#]] - Boole[# == 1])*2^IntegerExponent[#, 2] &[a[n/2]], 2 a[(n - 1)/2]]; Array[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &@ f@ a[#] &, 73, 0]] (* Michael De Vlieger, Mar 17 2021 *)

PARI

\\ Needs also code from A342456.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A342463(n) = A342001(A342456(n));

Formula

a(n) = A342001(A342456(n)) = A342002(A329886(n)) = A342920(A005940(1+n)).