A278510 -id:A278510 - cp's OEIS frontend

A258851 The pi-based arithmetic derivative of n: a(p) = pi(p) for p prime, a(uv) = a(u)v + u*a(v), where pi = A000720.

0, 0, 1, 2, 4, 3, 7, 4, 12, 12, 11, 5, 20, 6, 15, 19, 32, 7, 33, 8, 32, 26, 21, 9, 52, 30, 25, 54, 44, 10, 53, 11, 80, 37, 31, 41, 84, 12, 35, 44, 84, 13, 73, 14, 64, 87, 41, 15, 128, 56, 85, 55, 76, 16, 135, 58, 116, 62, 49, 17, 136, 18, 53, 120, 192, 69, 107

Offset: 0

Alois P. Heinz, Jun 12 2015

The pi-based variant of the arithmetic derivative of n (A003415).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Arithmetic derivative
Index entries for sequences computed from indices in prime factorization

Column k=1 of A258850, A258997.
First differences give A258863.
Partial sums give A258864.
Cf. A000720 (pi), A003415, A020639, A032742, A055396, A056239, A258861, A258862, A258995, A278510.

with(numtheory):
a:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..100);

a[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; a[0] = 0; Array[a, 100, 0] (* Jean-François Alcover, Apr 24 2016 *)

A258851(n)=n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i]) \\ M. F. Hasler, Jul 13 2015

(define (A258851 n) (if (<= n 1) 0 (+ (* (A055396 n) (A032742 n)) (* (A020639 n) (A258851 (A032742 n)))))) ;; Antti Karttunen, Mar 07 2017

a(n) = n * Sum e_j*pi(p_j)/p_j for n = Product p_j^e_j with pi = A000720.
a(A258861(n)) = n; A258861 = pi-based antiderivative of n.
a(a(A258862(n))) = n; A258862 = second pi-based antiderivative of n.
a(a(a(A258995(n)))) = n; A258995 = third pi-based antiderivative of n.
a(0) = a(0*p) = a(0)*p + 0*a(p) = a(0)*p for all p => a(0) = 0.
a(p) = a(1*p) = a(1)*p + 1*a(p) = a(1)*p + a(p) for all p => a(1) = 0.
a(u^v) = v * u^(v-1) * a(u).

A354871 a(n) = gcd(A056239(n), A258851(n)).

Original entry on oeis.org

0, 1, 2, 2, 3, 1, 4, 3, 4, 1, 5, 4, 6, 5, 1, 4, 7, 1, 8, 1, 2, 3, 9, 1, 6, 1, 6, 2, 10, 1, 11, 5, 1, 1, 1, 6, 12, 1, 4, 6, 13, 1, 14, 1, 1, 1, 15, 2, 8, 1, 1, 4, 16, 1, 2, 1, 2, 1, 17, 1, 18, 1, 8, 6, 3, 1, 19, 3, 1, 1, 20, 1, 21, 1, 4, 2, 1, 1, 22, 1, 8, 1, 23, 4, 2, 1, 4, 4, 24, 1, 2, 1, 1, 1, 1, 1, 25, 1, 3, 4

Offset: 1

Antti Karttunen, Jun 11 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A056239, A258851, A278510, A354872, A354873.

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A354871(n) = gcd(A056239(n), A258851(n));

a(n) = gcd(A056239(n), A258851(n)).
a(n) = gcd(A056239(n), A278510(n)) = gcd(A258851(n), A278510(n)).
For n > 1, a(n) = A056239(n) / A354872(n) = A258851(n) / A354873(n).

A354872 a(n) = A056239(n) / gcd(A056239(n), A258851(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 1, 5, 1, 5, 3, 2, 1, 5, 1, 7, 1, 3, 1, 6, 1, 1, 7, 8, 7, 1, 1, 9, 2, 1, 1, 7, 1, 7, 7, 10, 1, 3, 1, 7, 9, 2, 1, 7, 4, 7, 5, 11, 1, 7, 1, 12, 1, 1, 3, 8, 1, 3, 11, 8, 1, 7, 1, 13, 2, 5, 9, 9, 1, 7, 1, 14, 1, 2, 5, 15, 3, 2, 1, 8, 5, 11, 13, 16, 11, 7, 1, 9, 3, 2, 1, 10

Offset: 2

Antti Karttunen, Jun 11 2022

Numerator of fraction A056239(n) / A258851(n).

Antti Karttunen, Table of n, a(n) for n = 2..65537
Index entries for sequences computed from indices in prime factorization

Cf. A056239, A258851, A278510, A354871, A354873 (denominators).

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A354872(n) = { my(u=A056239(n)); (u/gcd(u, A258851(n))); };
\\ Or:
A354872(n) = numerator(A056239(n)/A258851(n));

a(n) = A056239(n) / A354871(n) = A056239(n) / gcd(A056239(n), A258851(n)).

A354873 a(n) = A258851(n) / gcd(A056239(n), A258851(n)).

Original entry on oeis.org

1, 1, 2, 1, 7, 1, 4, 3, 11, 1, 5, 1, 3, 19, 8, 1, 33, 1, 32, 13, 7, 1, 52, 5, 25, 9, 22, 1, 53, 1, 16, 37, 31, 41, 14, 1, 35, 11, 14, 1, 73, 1, 64, 87, 41, 1, 64, 7, 85, 55, 19, 1, 135, 29, 116, 31, 49, 1, 136, 1, 53, 15, 32, 23, 107, 1, 32, 73, 117, 1, 204, 1, 61, 35, 54, 79, 127, 1, 208, 27, 67, 1, 47, 43, 71, 22

Offset: 2

Antti Karttunen, Jun 11 2022

Denominator of fraction A056239(n) / A258851(n).

Antti Karttunen, Table of n, a(n) for n = 2..16384
Index entries for sequences computed from indices in prime factorization

Cf. A056239, A258851, A278510, A354871, A354872 (numerators).

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A354873(n) = { my(u=A258851(n)); (u/gcd(u, A056239(n))); };
\\ Or:
A354873(n) = denominator(A056239(n)/A258851(n));

a(n) = A258851(n) / A354871(n) = A258851(n) / gcd(A056239(n), A258851(n)).

A353509 a(n) = A353379(n) - A001222(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 2, 2, 0, 0, 4, 0, 4, 2, 2, 0, 7, 0, 2, 0, 4, 0, 9, 0, 0, 2, 2, 2, 8, 0, 2, 2, 7, 0, 9, 0, 4, 4, 2, 0, 10, 0, 4, 2, 4, 0, 7, 2, 7, 2, 2, 0, 16, 0, 2, 4, 0, 2, 9, 0, 4, 2, 9, 0, 14, 0, 2, 4, 4, 2, 9, 0, 10, 0, 2, 0, 16, 2, 2, 2, 7, 0, 16, 2, 4, 2, 2, 2, 15, 0, 4, 4, 8, 0, 9

Offset: 1

Antti Karttunen, Apr 29 2022

nonn

The difference of A258851 (primepi-based arithmetic derivative) and A056239 (sum of prime indices with multiplicity) applied to A181819, the prime shadow of n.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A001222, A056239, A181819, A258851, A278510, A353379.

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A353509(n) = (A258851(A181819(n))-bigomega(n));

a(n) = A278510(A181819(n)) = A353379(n) - A001222(n).

cp's OEIS Frontend

A258851 The pi-based arithmetic derivative of n: a(p) = pi(p) for p prime, a(u*v) = a(u)*v + u*a(v), where pi = A000720.

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A354871 a(n) = gcd(A056239(n), A258851(n)).

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A354872 a(n) = A056239(n) / gcd(A056239(n), A258851(n)).

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A354873 a(n) = A258851(n) / gcd(A056239(n), A258851(n)).

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A353509 a(n) = A353379(n) - A001222(n).

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A258851 The pi-based arithmetic derivative of n: a(p) = pi(p) for p prime, a(uv) = a(u)v + u*a(v), where pi = A000720.