A125168 a(n) = gcd(n, A032741(n)) where A032741(n) is the number of proper divisors of n.

1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 5, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 4, 1, 1, 3, 1, 1, 7, 1, 1, 5, 1, 1, 3, 1, 5, 3, 1, 1, 1, 1, 7, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 7, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 7

Offset

1,4

Comments

First occurrence of k: 1, 4, 6, 16, 20, 3240000, 42, 256, 162, 18662400, 132, 5308416, 832, 784, 120, 65536, 612, 2985984, 912, 1600, 9240, 98010000, 1380, 1296, 100800, ..., (10^7). - Robert G. Wilson v, Jan 23 2007

Do all values appear? - Robert G. Wilson v, Jan 23 2007

From Bernard Schott, Oct 19 2019: (Start)

a(n) = 1 if n = p^k, p prime, k >= 0 and k <> p or,

n = p*q, p<q primes <> 3 or

n = p*q*r, p<q<r primes <> 7 or,

n = p^2*q, p<q primes <> 5 or

n = p^3*q, p<q primes <> 7.

a(n) = 2 if n = 2^2 or n = 2^(2*p), p prime <> 2,

a(n) = 3 if n = 3*p, p prime <> 3 or n = 3^3,

a(n) = 4 if n = 4*p^2, p prime,

a(n) = 5 if n = 5*p^2, p prime <> 5, or n = 25*p, p prime <> 5, or n = 5^5,

a(n) = 7 if n = 7*p*q with p<q primes <> 7 or n = 7*p^3, p prime <> 7, or n = 7^7,

a(n) = p if n = p^p, p prime. (End)

Links

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

Formula

a(n) = gcd(n, A032741(n)) = gcd(n, A062968(n)).

Example

a(6)=3 because 6 has 3 proper divisors {1,2,3} and gcd(6,3) is 3.

Mathematica

f[n_] := GCD[n, DivisorSigma[0, n] - 1]; Array[f, 105] (* Robert G. Wilson v *).

Prog

(PARI) A125168(n) = gcd(n, numdiv(n)-1); \\ Antti Karttunen, Sep 25 2018

Crossrefs

Cf. A032741, A009191, A062968.

Sequence in context: A320000 A119805 A111957 * A324725 A328392 A051794

Adjacent sequences: A125165 A125166 A125167 * A125169 A125170 A125171

Keyword

easy,nonn

Author

Mitch Cervinka (puritan(AT)toast.net), Jan 12 2007

Extensions

More terms from Robert G. Wilson v, Jan 23 2007

Status

approved