A342915 -id:A342915 - cp's OEIS frontend

A344773 Number of divisors d of n for which A342915(d) = A342915(n), where A342915(n) = gcd(1+n, psi(n)).

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 2, 3, 1, 4, 1, 3, 1, 2, 1, 6, 1, 2, 1, 4, 1, 3, 1, 1, 2, 2, 1, 7, 1, 2, 2, 3, 1, 4, 1, 4, 1, 2, 1, 7, 1, 2, 1, 4, 2, 4, 1, 2, 1, 3, 1, 8, 1, 2, 2, 3, 1, 3, 1, 1, 2, 2, 1, 7, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 8, 1, 3, 2, 4, 1, 4, 1, 1, 2

Offset: 1

Antti Karttunen, May 31 2021

nonn

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

Cf. A001615, A342915, A344771.

Cf. also A344774.

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));
A344773(n) = { my(x=A342915(n)); sumdiv(n,d,A342915(d)==x); };

a(n) = Sum_{d|n} [A342915(d) = A342915(n)], where [ ] is the Iverson bracket.

a(n) <= A344771(n).

A344771 Ordinal transform of A342915, where A342915(n) = gcd(1+n, psi(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 5, 1, 3, 2, 6, 1, 7, 1, 4, 2, 8, 1, 9, 3, 5, 2, 10, 1, 11, 1, 6, 4, 12, 2, 13, 1, 7, 3, 14, 1, 15, 1, 1, 5, 16, 1, 17, 6, 8, 3, 18, 1, 19, 4, 9, 7, 20, 1, 21, 1, 10, 2, 22, 2, 23, 1, 11, 8, 24, 1, 25, 1, 12, 4, 26, 3, 27, 1, 2, 9, 28, 1, 29, 10, 13, 5, 30, 1, 31, 5, 14, 11, 32, 2, 33, 1, 15

Offset: 1

Antti Karttunen, May 31 2021

nonn

Number of values of k, 1 <= k <= n, with A342915(k) = A342915(n).

a(p) = 1 for all primes p (and for some other numbers as well).

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

Cf. A001615, A342915, A344773.

Differs from A339914 for the first time at n=44, where a(44) = 1, while A339914(44) = 8.

psi[n_] := If[n= 1, 1, Times@@((#1+1)*#1^(#2-1)& @@@ FactorInteger[n])];
A342915[n_] := GCD[n+1, psi[n]];
b[_] = 0;
a[n_] := a[n] = With[{t = A342915[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 22 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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)));
A342915(n) = gcd(1+n,A001615(n));
v344771 = ordinal_transform(vector(up_to,n,A342915(n)));
A344771(n) = v344771[n];

a(n) <= A344773(n).

A143771 a(n) = gcd(k + n/k), where k is over all divisors of n.

Original entry on oeis.org

2, 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, 3, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 60, 1, 62, 3, 8, 1, 6, 1, 68, 3, 2, 1, 72, 1, 74, 3, 4, 1, 6, 1, 80, 3, 2, 1, 84, 1, 2, 3, 8, 1, 90, 1, 4, 3, 2, 1, 24, 1

Offset: 1

Leroy Quet, Aug 31 2008

nonn

If n is the m-th composite, then a(n) = A143772(m).

If n is prime, then a(n) is defined as n+1, since a(n) = gcd(1+n, n+1).

			a(1) = gcd(1+1) = 2, i.e., the greatest common divisor of a singular set [2].
a(9) = gcd(1+9, 3+3, 9+1) = 2.
a(20) = gcd(1+20, 2+10, 4+5, 5+4, 10+2, 20+1) = 3.
a(44) = gcd(1+44, 2+22, 4+11, 11+4, 22+2, 44+1) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms 2..10000 from Michael De Vlieger)

Cf. A143772, A339873, A339914, A342918 [= (1+n) / a(n)].

After n=1 differs from A342915 for the first time at n=44, where a(44) = 3, while A342915(44) = 9.

A143771 := proc(n) local dvs ; dvs := convert(numtheory[divisors](n),list) ; igcd(seq( op(i,dvs)+n/op(i,dvs), i=1..nops(dvs))) ; end: for n from 2 to 140 do printf("%d,",A143771(n)) ; od: # R. J. Mathar, Sep 05 2008

Table[GCD @@ Map[# + n/# &, Divisors@ n], {n, 2, 96}] (* Michael De Vlieger, Oct 30 2017 *)

a(n) = my(d = divisors(n)); gcd(vector(#d, k, d[k]+n/d[k])); \\ Michel Marcus, Oct 05 2015

Extended by R. J. Mathar, Sep 05 2008

Term a(1) = 2 prepended and Example-section extended by Antti Karttunen, Mar 29 2021

A342458 a(n) = gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 12, 6, 1, 1, 8, 1, 3, 8, 8, 1, 3, 1, 12, 2, 1, 1, 4, 10, 3, 9, 16, 1, 1, 1, 16, 2, 1, 12, 12, 1, 3, 8, 4, 1, 1, 1, 24, 3, 1, 1, 16, 14, 45, 4, 28, 1, 27, 8, 4, 2, 1, 1, 4, 1, 3, 3, 96, 6, 1, 1, 36, 2, 1, 1, 12, 1, 3, 5, 40, 6, 1, 1, 16, 108, 1, 1, 4, 2, 3, 8, 4, 1, 3, 4, 48, 2, 1, 24, 16, 1, 7, 3, 20

Offset: 1

Antti Karttunen, Mar 28 2021

nonn

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

Cf. A001615, A003415, A003557, A342459, A342919.
Cf. A301939 (gives the positions at which a(n) = A001615(n) = A003415(n)).
Cf. also A175732, A342413, A342915.

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
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342458(n) = gcd(A001615(n), A003415(n));

a(n) = gcd(A001615(n), A003415(n)).
a(n) = A003557(n) * A342459(n).
a(n) = A003415(n) / A342919(n).

A342916 a(n) = (1+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

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

Offset: 1

Antti Karttunen, Mar 29 2021

nonn

It is conjectured that a(n) = 1 only when n is a prime, A000040. See Thomas Ordowski's May 21 2017 problem in A001615.

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

Cf. A000040, A001615, A342915, A342917.
Cf. also A160596.
After n=1 differs from A342918 for the first time at n=44, where a(44) = 5, while A342918(44) = 15.

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
A342916(n) = ((1+n)/gcd(1+n,A001615(n)));

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

Incorrect A-number in the formula corrected by Antti Karttunen, May 31 2021

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)).

cp's OEIS Frontend

A344773 Number of divisors d of n for which A342915(d) = A342915(n), where A342915(n) = gcd(1+n, psi(n)).

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A344771 Ordinal transform of A342915, where A342915(n) = gcd(1+n, psi(n)).

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A143771 a(n) = gcd(k + n/k), where k is over all divisors of n.

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A342458 a(n) = gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

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A342916 a(n) = (1+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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