A342916 -id:A342916 - cp's OEIS frontend

A342915 a(n) = gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

1, 3, 4, 1, 6, 1, 8, 3, 2, 1, 12, 1, 14, 3, 8, 1, 18, 1, 20, 3, 2, 1, 24, 1, 2, 3, 4, 1, 30, 1, 32, 3, 2, 1, 12, 1, 38, 3, 8, 1, 42, 1, 44, 9, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 60, 1, 62, 3, 32, 1, 6, 1, 68, 3, 2, 1, 72, 1, 74, 3, 4, 1, 6, 1, 80, 9, 2, 1, 84, 1, 2, 3, 8, 1, 90, 1, 4, 3, 2, 1, 24, 1, 98, 3, 4, 1, 102

Offset: 1

Antti Karttunen, Mar 29 2021

nonn

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

Cf. A001615, A342916, A342917.
Cf. also A049559, A342458.
After n=1 differs from A143771 for the first time at n=44, where a(44) = 9, while A143771(44) = 3.

psi[n_] := If[n==1, 1, Times @@ ((#1+1)*#1^(#2-1)& @@@ FactorInteger[n])];
a[n_] := GCD[n+1, psi[n]];
Array[a, 105] (* Jean-François Alcover, Dec 22 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A342915(n) = gcd(1+n,A001615(n));

a(n) = gcd(1+n, A001615(n)).

a(n) = (1+n) / A342916(n) = A001615(n) / A342917(n).

A342917 a(n) = A001615(n) / gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 1, 1, 6, 1, 12, 1, 4, 6, 18, 1, 24, 1, 8, 3, 24, 1, 36, 1, 12, 16, 36, 1, 48, 15, 14, 9, 48, 1, 72, 1, 16, 24, 54, 4, 72, 1, 20, 7, 72, 1, 96, 1, 8, 36, 72, 1, 96, 28, 30, 18, 84, 1, 108, 9, 32, 40, 90, 1, 144, 1, 32, 3, 96, 14, 144, 1, 36, 48, 144, 1, 144, 1, 38, 30, 120, 16, 168, 1, 16, 54, 126, 1, 192, 54, 44, 15, 144

Offset: 1

Antti Karttunen, Mar 29 2021

The scatter plot shows two distinct "fans" separated by a gap. Why?

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

Cf. A001615, A342915, A342916.

Cf. also A160595.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A342917(n) = { my(u=A001615(n)); (u/gcd(1+n,u)); };

a(n) = A001615(n) / A342915(n) = A001615(n) / gcd(1+n, A001615(n)).

A342918 a(n) = (1+n) / A143771(n).

Original entry on oeis.org

1, 1, 1, 5, 1, 7, 1, 3, 5, 11, 1, 13, 1, 5, 2, 17, 1, 19, 1, 7, 11, 23, 1, 25, 13, 9, 7, 29, 1, 31, 1, 11, 17, 35, 3, 37, 1, 13, 5, 41, 1, 43, 1, 15, 23, 47, 1, 49, 25, 17, 13, 53, 1, 55, 7, 19, 29, 59, 1, 61, 1, 21, 8, 65, 11, 67, 1, 23, 35, 71, 1, 73, 1, 25, 19, 77, 13, 79, 1, 27, 41, 83, 1, 85, 43, 29, 11, 89, 1, 91, 23

Offset: 1

Antti Karttunen, Mar 29 2021

nonn

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

Cf. A143771, A339873.

After n=1 differs from A342916 for the first time at n=44, where a(44) = 15, while A342916(44) = 5.

A143771(n) = my(d = divisors(n)); gcd(vector(#d, k, d[k]+n/d[k])); \\ From A143771.
A342918(n) = ((1+n)/A143771(n));

a(n) = (1+n) / A143771(n).

cp's OEIS Frontend

A342915 a(n) = gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

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A342917 a(n) = A001615(n) / gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

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A342918 a(n) = (1+n) / A143771(n).

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