A356299 -id:A356299 - cp's OEIS frontend

A327858 Greatest common divisor of the arithmetic derivative and the primorial base exp-function: a(n) = gcd(A003415(n), A276086(n)).

1, 2, 1, 1, 1, 1, 5, 1, 3, 6, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 10, 1, 1, 1, 10, 15, 3, 1, 1, 1, 1, 1, 14, 1, 6, 5, 1, 21, 2, 1, 1, 1, 1, 3, 3, 25, 1, 7, 14, 15, 10, 7, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 3, 3, 18, 1, 1, 3, 2, 1, 1, 1, 1, 3, 5, 5, 18, 1, 1, 1, 6, 1, 1, 1, 2, 15, 2, 35, 1, 1, 2, 3, 2, 49, 6, 1, 1, 7, 15, 35, 1, 7, 1, 1, 1

Offset: 0

Antti Karttunen, Sep 30 2019

Sequence contains only terms of A048103.

Proof that A046337 gives the positions of even terms: see Charlie Neder's Feb 25 2019 comment in A235992 and recall that A276086 is never a multiple of 4, as it is a permutation of A048103, and furthermore it toggles the parity. See also comment in A327860. - Antti Karttunen, May 01 2022

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A048103, A235992, A276086, A327859, A328382, A351234, A354348, A356299, A358669, A359423, A359589 (Dirichlet inverse of a(n)-1), A369971, A373849.
Cf. A046337 (positions of even terms), A356311 (positions of 1's), A356310 (their characteristic function).
Cf. also A085731, A324198, A328572 [= gcd(A276086(n), A327860(n))], A345000, A373145, A373843.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12], f, g}, f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; g[n_] := Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[GCD[f@ #, g@ #] &, 105]] (* Michael De Vlieger, Sep 30 2019 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n),A276086(n));

a(n) = gcd(A003415(n), A276086(n)).
a(p) = 1 for all primes p.
a(n) = A276086(A351234(n)). - Antti Karttunen, May 01 2022
From Antti Karttunen, Dec 05 2022: (Start)
For n >= 2, a(n) = gcd(A003415(n), A328382(n)).
(End)
For n >= 2, a(n) = A358669(n) / A359423(n). For n >= 1, A356299(n) | a(n). - Antti Karttunen, Jan 09 2023
a(n) = gcd(A003415(n), A373849(n)) = gcd(A276086(n), A369971(n)) = A373843(A276086(n)). - Antti Karttunen, Jun 21 & 23 2024

Verbal description added to the definition by Antti Karttunen, May 01 2022

A046337 Odd numbers with an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 9, 15, 21, 25, 33, 35, 39, 49, 51, 55, 57, 65, 69, 77, 81, 85, 87, 91, 93, 95, 111, 115, 119, 121, 123, 129, 133, 135, 141, 143, 145, 155, 159, 161, 169, 177, 183, 185, 187, 189, 201, 203, 205, 209, 213, 215, 217, 219, 221, 225, 235, 237, 247, 249, 253, 259

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A028260.
Setwise difference A005408 \ A067019.
Setwise difference A028260 \ A063745.
Union of A359161 and A359163.
Union of A327862 and A360110.
Subsequence of A345452, of A356312 and of A359371.
Positions of positive terms in A166698, positions of even terms in A327858 and A356299.
Subsequences: A002557, A046315 (odd semiprimes), A056913, A359596, A359607, A359608 (without its term 2).
Cf. A000035, A008836, A046338, A046470, A353557 (characteristic function), A358777.
Cf. also A036349, A297845.

Select[Range[1,301,2],EvenQ[PrimeOmega[#]]&] (* Harvey P. Dale, Jul 25 2011 *)

lista(nn) = {forstep(n=1, nn, 2, if (bigomega(n) % 2 == 0, print1(n, ", ")));} \\ Michel Marcus, Jul 04 2015

{k | A000035(k) > 0 and A008836(k) > 0}. - Antti Karttunen, Jan 13 2023

cp's OEIS Frontend

A327858 Greatest common divisor of the arithmetic derivative and the primorial base exp-function: a(n) = gcd(A003415(n), A276086(n)).

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