A342918 -id:A342918 - 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

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

cp's OEIS Frontend

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

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