A345947 -id:A345947 - 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

A345937 a(n) = gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 7, 8, 1, 10, 1, 12, 1, 2, 15, 16, 1, 18, 1, 4, 1, 22, 1, 24, 1, 26, 9, 28, 1, 30, 31, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 48, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 63, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 80, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4, 1, 2, 1, 96

Offset: 1

Antti Karttunen, Jun 29 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A047994, A345938, A345939.

Cf. also A049559, A340087, A323409, A345947.

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
A345937(n) = gcd(n-1, A047994(n));

a(n) = gcd(n-1, A047994(n)).
a(n) = A047994(n) / A345938(n).
a(n) = (n-1) / A345939(n), for n > 1.
a(2n-1) = A345947(2n-1), for n >= 1.

A345948 a(n) = A344875(n) / gcd(A153151(n), A344875(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 4, 1, 14, 1, 18, 4, 1, 1, 24, 1, 28, 3, 10, 1, 30, 1, 36, 1, 14, 1, 24, 1, 1, 5, 16, 12, 8, 1, 54, 12, 20, 1, 36, 1, 70, 8, 22, 1, 62, 1, 72, 16, 28, 1, 78, 20, 18, 9, 28, 1, 56, 1, 90, 24, 1, 3, 12, 1, 112, 11, 24, 1, 120, 1, 108, 24, 42, 15, 72, 1, 124, 1, 40, 1, 84, 16, 126, 28, 50, 1, 96, 4, 22

Offset: 1

Antti Karttunen, Jul 01 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A153151, A344875, A345947, A345949.

Cf. also A345938.

{1}~Join~Array[#2/GCD @@ {##} & @@ {Which[# < 2, #, IntegerQ[Log2@ #], 2 # - 1, True, # - 1], Times @@ Map[If[#1 == 2, 2^(#2 + 1) - 1, #1^#2 - 1] & @@ # &, FactorInteger[#]]} &, 91, 2] (* Michael De Vlieger, Jul 06 2021 *)

A153151(n) = if(!n,n,if(!bitand(n,n-1),(n+n-1),(n-1)));
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A345948(n) = { my(u=A344875(n)); (u/gcd(A153151(n), u)); };

a(n) = A344875(n) / A345947(n) = A344875(n) / gcd(A153151(n), A344875(n)).

a(2n-1) = A345938(2n-1), for all n >= 1.

A345949 a(n) = A153151(n) / gcd(A153151(n), A344875(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 5, 7, 1, 23, 1, 25, 1, 9, 1, 29, 1, 1, 8, 11, 17, 5, 1, 37, 19, 13, 1, 41, 1, 43, 11, 15, 1, 47, 1, 49, 25, 17, 1, 53, 27, 11, 14, 19, 1, 59, 1, 61, 31, 1, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 1, 27, 1, 83, 21, 85, 43, 29, 1, 89, 5, 13, 23

Offset: 1

Antti Karttunen, Jul 01 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A153151, A344875, A345947, A345948.

Cf. also A345939.

{1}~Join~Array[#1/GCD @@ {##} & @@ {Which[# < 2, #, IntegerQ[Log2@ #], 2 # - 1, True, # - 1], Times @@ Map[If[#1 == 2, 2^(#2 + 1) - 1, #1^#2 - 1] & @@ # &, FactorInteger[#]]} &, 92, 2] (* Michael De Vlieger, Jul 06 2021 *)

A153151(n) = if(!n,n,if(!bitand(n,n-1),(n+n-1),(n-1)));
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A345949(n) = { my(u=A153151(n)); (u/gcd(u, A344875(n))); };

a(n) = A153151(n) / A345947(n) = A153151(n) / gcd(A153151(n), A344875(n)).

a(2n-1) = A345939(2n-1), for n > 1.

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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A345937 a(n) = gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

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A345948 a(n) = A344875(n) / gcd(A153151(n), A344875(n)).

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A345949 a(n) = A153151(n) / gcd(A153151(n), A344875(n)).

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