A370130 -id:A370130 - cp's OEIS frontend

A328242 Numbers k such that A003415(A276086(k)) is a squarefree number, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 17, 30, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77, 210, 211, 212, 213, 215, 217, 218, 219, 220, 221, 223, 225, 226, 227, 241, 242, 243, 245, 246, 247, 248, 249, 250, 251, 254, 255, 256, 257, 270, 273, 274, 275, 276, 277

Offset: 1

Antti Karttunen, Oct 10 2019

nonn

Numbers n such that A276086(n) is either in A328234 or in A000040 (i.e., it is a prime, in which case n itself is a primorial, A002110).

Antti Karttunen, Table of n, a(n) for n = 1..10681
Index entries for sequences related to primorial base

Positions of 1's in A370130. Subsequence of A370132.
Cf. A000040, A002110, A003415, A005117, A008966, A276086, A327860, A328234.
Cf. A328241 (complement).
Cf. A328233, A328240 (subsequences).

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
isA328242(n) = issquarefree(A327860(n));

A369669 The greatest common divisor of the first and the second arithmetic derivative of n.

Original entry on oeis.org

0, 0, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 3, 4, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 27, 16, 1, 1, 1, 16, 1, 1, 4, 4, 1, 1, 16, 4, 1, 1, 1, 16, 1, 5, 1, 16, 1, 3, 4, 4, 1, 27, 16, 4, 1, 1, 1, 4, 1, 1, 1, 64, 3, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 16, 3, 1, 1, 16, 108, 1, 1, 4, 1, 3, 16, 4, 1, 1, 4, 16, 1, 7, 4, 16, 1, 1, 5

Offset: 0

Antti Karttunen, Feb 10 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A003415, A068346, A085731, A370130 [= a(A276086(n))].
Cf. A328393 (positions of 1's), A354874 (their characteristic function).
Cf. A327864 (positions of even terms, also positions of multiples of 4).
Cf. A370119 (positions of multiples of 3).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A369669(n) = { my(d=A003415(n)); gcd(d, A003415(d)); };

a(n) = gcd(A003415(n), A068346(n)).

For n >= 2, a(n) = A085731(A003415(n)).

A370131 a(n) = A068346(A276086(n)), where A068346 is the second arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 0, 0, 1, 5, 10, 0, 1, 12, 1, 16, 44, 7, 39, 16, 42, 608, 391, 55, 135, 365, 455, 1120, 2990, 800, 1100, 1400, 5425, 12575, 21025, 0, 6, 7, 1, 20, 103, 16, 1, 1, 32, 271, 320, 24, 78, 572, 459, 1031, 2887, 635, 1670, 1155, 3335, 19540, 22130, 4225, 7700, 18675, 28100, 68900, 155425, 9, 18, 20, 54, 704, 631, 24, 251

Offset: 0

Antti Karttunen, Feb 10 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2559
Index entries for sequences related to primorial base

Cf. A003415, A068346, A276086, A327860, A370130.

Cf. A002110 (positions of 0's after the initial zero), A328233 (positions of 1's), A328240 (positions of primes), A369651 (= a(A143293(n-1)), for n >= 1).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A370131(n) = A003415(A327860(n));

a(n) = A068346(A276086(n)) = A003415(A327860(n)).

cp's OEIS Frontend

A328242 Numbers k such that A003415(A276086(k)) is a squarefree number, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A369669 The greatest common divisor of the first and the second arithmetic derivative of n.

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A370131 a(n) = A068346(A276086(n)), where A068346 is the second arithmetic derivative, and A276086 is the primorial base exp-function.

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