The On-Line Encyclopedia of Integer Sequences (OEIS)

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A328768

The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.
(history; published version)

#20 by Michael De Vlieger at Mon Jul 08 13:59:28 EDT 2024

STATUS

proposed

approved

#19 by Antti Karttunen at Mon Jul 08 13:34:45 EDT 2024

STATUS

editing

proposed

#18 by Antti Karttunen at Mon Jul 08 13:19:20 EDT 2024

CROSSREFS

Cf. A002110, A108951, A276085, A276150, A328771, A373982, A374211.

Cf. also A003415, A258851, A328769, A328845, A328846, A373982, A374211.

For variants of the same formula, see A003415, A258851, A328769, A328845, A328846, A371192.

#17 by Antti Karttunen at Mon Jul 08 13:13:00 EDT 2024

CROSSREFS

Cf. also A003415, A258851, A328769, A328845, A328846, A373982, A374211.

Cf. A042965 (indices of even terms), A016825 (of odd terms), A152822 (antiparity of terms), A373992 (indices of multiples of 3), A374212 (2-adic valuation), A374213 (3-adic valuation), A374123 [a(n) mod 360].

Cf. A374031 [gcd(a(n), A276085(n))], A374116 [gcd(a(n), A328845(n))].

STATUS

approved

editing

#16 by Susanna Cuyler at Mon Oct 28 20:01:01 EDT 2019

STATUS

proposed

approved

#15 by Antti Karttunen at Mon Oct 28 17:32:22 EDT 2019

STATUS

editing

proposed

#14 by Antti Karttunen at Mon Oct 28 16:58:30 EDT 2019

FORMULA

A276150(a(n)) = A328771n * Sum e_j * (p_j)#/(p_j^2) for n = Product p_j^e_j with (p_j)# = A034386(p_j).

For all n >= 0, A276150(a(n)) = A328771(n).

#13 by Antti Karttunen at Mon Oct 28 16:56:49 EDT 2019

PROG

(PARI) A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i, 1]))/(f[i, 1]^2)));

#12 by Antti Karttunen at Mon Oct 28 16:52:34 EDT 2019

NAME

The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1) for , where prime p (i) = A000040(i), a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

#11 by Antti Karttunen at Mon Oct 28 16:51:27 EDT 2019

PROG

(PARI)

\\ Alternatively as:

A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };

A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A276085(f[i, 1])/f[i, 1]));