A339914 -id:A339914 - cp's OEIS frontend

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

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

A339873 a(n) = 1 + n - A143771(n).

Original entry on oeis.org

0, 0, 4, 0, 6, 0, 6, 8, 10, 0, 12, 0, 12, 8, 16, 0, 18, 0, 18, 20, 22, 0, 24, 24, 24, 24, 28, 0, 30, 0, 30, 32, 34, 24, 36, 0, 36, 32, 40, 0, 42, 0, 42, 44, 46, 0, 48, 48, 48, 48, 52, 0, 54, 48, 54, 56, 58, 0, 60, 0, 60, 56, 64, 60, 66, 0, 66, 68, 70, 0, 72, 0, 72, 72, 76, 72, 78, 0, 78, 80, 82, 0, 84, 84, 84, 80

Offset: 2

Antti Karttunen, Dec 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 2..65539

Cf. A143771, A339914.

A143771(n) = my(d = divisors(n)); gcd(vector(#d, k, d[k]+n/d[k])); \\ From A143771.
A339873(n) = (1+n-A143771(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).

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

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A339873 a(n) = 1 + n - A143771(n).

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

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