A342018 Numbers k such that the arithmetic derivative of A276086(k) is divisible by at least one prime power divisor of the form p^p, where A276086 gives the prime product form of primorial base expansion of its argument.

8, 16, 24, 36, 44, 52, 64, 72, 80, 88, 92, 100, 108, 116, 120, 126, 128, 136, 144, 156, 164, 172, 184, 192, 200, 208, 216, 222, 224, 232, 244, 252, 260, 268, 271, 272, 280, 288, 296, 300, 308, 316, 324, 336, 344, 348, 352, 364, 372, 380, 388, 392, 397, 400, 408, 416, 424, 432, 440, 444, 448, 452, 460, 468, 476, 480, 488, 493, 496

Offset: 1

Author

Antti Karttunen, Mar 04 2021

Keywords

• nonn

Comments

Numbers k for which A342019(k) > 0, or equally, A342007(A327860(k)) = A342017(k) is larger than one, or equally A342007(A342002(k)) > 1, that is, k for which A342023(A342002(k)) = 1.

The first odd term is a(35) = 271.

Examples

			8 is present as A276086(8) = 15, A003415(15) = 8 = 2^3, which is thus divisible by p^p (with p=2 in this case).
271 is present as A276086(271) = 1078, A003415(1078) = 945 = 3^3 * 5 * 7, which is thus divisible by p^p (with p=3 in this case).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to primorial base

Crossrefs

Positions of terms larger than one in A342017, of nonzero terms in A342019.

Cf. A003415, A276086, A327860, A342002, A342007, A342017, A342019, A342023.

Not a subsequence of A342006.

Programs

Mathematica

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Position[#, ?(# > 1 &)][[All, 1]] &@ Array[Function[k, Times @@ Map[#1^Floor[#2/#1] & @@ # &, FactorInteger[#]] &@ If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[#, b] &, 500]] (* _Michael De Vlieger, Mar 12 2021 *)

PARI

isA342018(n) = (A342017(n)>1);

Extensions

Name changed by Antti Karttunen, Mar 12 2021