A373365 -id:A373365 - cp's OEIS frontend

A373369 a(n) = gcd(A001414(n), A059975(n)), where A001414 and A059975 are fully additive with a(p) = p and a(p) = p-1, respectively.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 2, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 6, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3

Offset: 1

Antti Karttunen, Jun 05 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A059975, A345452 (positions of even terms).

Cf. also A082299, A306709, A373362, A373363, A373364, A373365.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
A059975(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1)); };
A373369(n) = gcd(A001414(n), A059975(n));

A373366 a(n) = gcd(A064097(n), A083345(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 5, 7, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 7, 1, 1, 8, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 1, 4, 1, 1, 1, 2, 9, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jun 02 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A064097, A083345.

Cf. also A373363, A373365.

A064097(n) = if(1==n,0,1+A064097(n-(n/vecmin(factor(n)[,1]))));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A373366(n) = gcd(A064097(n), A083345(n));

A373370 a(n) = gcd(bigomega(n), A064097(n)), where bigomega is number of prime factors with repetition, and A064097 is a quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 1, 3, 1, 5, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 4, 1, 4, 1, 3, 1, 2, 1, 1, 1, 3, 3, 4, 1, 1, 1, 4, 3

Offset: 1

Antti Karttunen, Jun 05 2024

nonn

As A001222 and A064097 are both fully additive sequences, all sequences that give the positions of multiples of some natural number k in this sequence are closed under multiplication, i.e., are multiplicative semigroups.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001222, A064097.

Cf. also A373365, A373366.

A064097(n) = if(1==n,0,1+A064097(n-(n/vecmin(factor(n)[,1]))));
A373370(n) = gcd(bigomega(n), A064097(n));

cp's OEIS Frontend

A373369 a(n) = gcd(A001414(n), A059975(n)), where A001414 and A059975 are fully additive with a(p) = p and a(p) = p-1, respectively.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A373366 a(n) = gcd(A064097(n), A083345(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A373370 a(n) = gcd(bigomega(n), A064097(n)), where bigomega is number of prime factors with repetition, and A064097 is a quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI