A344879 -id:A344879 - cp's OEIS frontend

A344875 Multiplicative with a(2^e) = 2^(1+e) - 1, and a(p^e) = p^e - 1 for odd primes p.

1, 3, 2, 7, 4, 6, 6, 15, 8, 12, 10, 14, 12, 18, 8, 31, 16, 24, 18, 28, 12, 30, 22, 30, 24, 36, 26, 42, 28, 24, 30, 63, 20, 48, 24, 56, 36, 54, 24, 60, 40, 36, 42, 70, 32, 66, 46, 62, 48, 72, 32, 84, 52, 78, 40, 90, 36, 84, 58, 56, 60, 90, 48, 127, 48, 60, 66, 112, 44, 72, 70, 120, 72, 108, 48, 126, 60, 72, 78, 124, 80, 120

Offset: 1

Antti Karttunen, Jun 03 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A047994, A065463, A153151, A344876, A344877, A344878, A344879, A344969, A345947.

f[2, e_] := 2^(e + 1) - 1; f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 03 2021 *)

A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };

from math import prod
from sympy import factorint
def A344875(n): return prod((p**(1+e) if p == 2 else p**e)-1 for p, e in factorint(n).items()) # Chai Wah Wu, Jun 01 2022

a(n) = A344878(n) * A344879(n).
Multiplicative with a(p^e) = A153151(p^e). - Antti Karttunen, Jul 01 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (4/5) * A065463 = 0.563553... . - Amiram Eldar, Nov 18 2022

A344878 a(n) is the least common multiple of numbers (2^(1+e2))-1 and those in the set (p_i^e_i)-1, when the odd part of n = Product (p_i^e_i), and e2 is the 2-adic valuation of n.

Original entry on oeis.org

1, 3, 2, 7, 4, 6, 6, 15, 8, 12, 10, 14, 12, 6, 4, 31, 16, 24, 18, 28, 6, 30, 22, 30, 24, 12, 26, 42, 28, 12, 30, 63, 10, 48, 12, 56, 36, 18, 12, 60, 40, 6, 42, 70, 8, 66, 46, 62, 48, 24, 16, 84, 52, 78, 20, 30, 18, 84, 58, 28, 60, 30, 24, 127, 12, 30, 66, 112, 22, 12, 70, 120, 72, 36, 24, 126, 30, 12, 78, 124, 80, 120

Offset: 1

Antti Karttunen, Jun 03 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A000265, A038712, A344875, A344879.

a[n_] := If[n == 1, 1, Module[{p, e}, LCM @@ Table[{p, e} = pe;
     (p^(e + If[p == 2, 1, 0])) - 1, {pe, FactorInteger[n]}]]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

A344878(n) = if(1==n,n, my(f=factor(n)~); lcm(vector(#f, i, (f[1, i]^(f[2, i]+(2==f[1, i]))-1))));

from math import lcm
from sympy import factorint
def A344878(n): return lcm(*(p**(e+int(p==2))-1 for p, e in factorint(n).items())) # Chai Wah Wu, Jun 15 2022

If n = Product (p_i^e_i), then a(n) = LCM of values (p_i^(e_i+[p==2]))-1, where [ ] is the Iverson bracket.
a(n) = lcm(A038712(n), a(A000265(n))).
a(n) = A344875(n) / A344879(n).

A345045 a(n) = A047994(n) / A345044(n), where A047994(n) is multiplicative with a(p^e) = p^e - 1, and A345044(n) gives the least common multiple of the same factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 4, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 2, 1, 1, 1, 1, 1, 6, 4, 1, 2, 1, 1, 4, 6, 1, 2, 1, 2, 1, 1, 1, 2, 3, 1, 2, 1, 1, 4

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

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

Cf. A047994, A345044.

Cf. also A344879, A345047, A353784.

A345045(n) = { my(f=factor(n)~, g=vector(#f, i, (f[1, i]^f[2, i])-1)); factorback(g)/lcm(g); };

a(n) = A047994(n) / A345044(n).

cp's OEIS Frontend

A344875 Multiplicative with a(2^e) = 2^(1+e) - 1, and a(p^e) = p^e - 1 for odd primes p.

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A344878 a(n) is the least common multiple of numbers (2^(1+e2))-1 and those in the set (p_i^e_i)-1, when the odd part of n = Product (p_i^e_i), and e2 is the 2-adic valuation of n.

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Mathematica

PARI

Python

Formula

A345045 a(n) = A047994(n) / A345044(n), where A047994(n) is multiplicative with a(p^e) = p^e - 1, and A345044(n) gives the least common multiple of the same factors.

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