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.

0, 0, 1, 2, 4, 6, 7, 30, 12, 12, 17, 210, 20, 2310, 67, 28, 32, 30030, 33, 510510, 44, 104, 431, 9699690, 52, 60, 4633, 54, 148, 223092870, 71, 6469693230, 80, 652, 60077, 192, 84, 200560490130, 1021039, 6956, 108, 7420738134810, 229, 304250263527210, 884, 114, 19399403, 13082761331670030, 128, 420, 145, 90124, 9292, 614889782588491410, 135, 1116, 324

Offset

0,4

Links

Antti Karttunen, Table of n, a(n) for n = 0..1001

Index entries for sequences related to primorial numbers

Formula

a(n) = n * Sum e_j * A276085(p_j)/p_j for n = Product p_j^e_j, where for primes p, A276085(p) = A002110(A000720(p)-1).

a(n) = n * 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).

Prog

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

Crossrefs

Cf. A002110, A108951, A276085, A276150, A328771, 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))].

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

Sequence in context: A061937 A065804 A374123 * A275693 A135033 A317647

Adjacent sequences: A328765 A328766 A328767 * A328769 A328770 A328771

Keyword

nonn

Author

Antti Karttunen, Oct 28 2019

Status

approved