login
A327858
Greatest common divisor of the arithmetic derivative and the primorial base exp-function: a(n) = gcd(A003415(n), A276086(n)).
43
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
0,2
COMMENTS
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
FORMULA
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
MATHEMATICA
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 *)
PROG
(PARI)
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));
CROSSREFS
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.
Sequence in context: A233836 A214719 A356299 * A307372 A259681 A146314
KEYWORD
nonn,base,easy
AUTHOR
Antti Karttunen, Sep 30 2019
EXTENSIONS
Verbal description added to the definition by Antti Karttunen, May 01 2022
STATUS
approved