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

%I A125168 #23 Aug 06 2025 01:03:11
%S A125168 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,
%T A125168 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,
%U A125168 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
%N A125168 a(n) = gcd(n, A032741(n)) where A032741(n) is the number of proper divisors of n.
%C A125168 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
%C A125168 Do all values appear? - _Robert G. Wilson v_, Jan 23 2007
%C A125168 From _Bernard Schott_, Oct 19 2019: (Start)
%C A125168 a(n) = 1 if n = p^k, p prime, k >= 0 and k <> p or,
%C A125168             n = p*q, p<q primes <> 3 or
%C A125168             n = p*q*r, p<q<r primes <> 7 or,
%C A125168             n = p^2*q, p<q primes <> 5 or
%C A125168             n = p^3*q, p<q primes <> 7.
%C A125168 a(n) = 2 if n = 2^2 or n = 2^(2*p), p prime <> 2,
%C A125168 a(n) = 3 if n = 3*p, p prime <> 3 or n = 3^3,
%C A125168 a(n) = 4 if n = 4*p^2, p prime,
%C A125168 a(n) = 5 if n = 5*p^2, p prime <> 5, or n = 25*p, p prime <> 5, or n = 5^5,
%C A125168 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,
%C A125168 a(n) = p if n = p^p, p prime. (End)
%H A125168 Antti Karttunen, <a href="/A125168/b125168.txt">Table of n, a(n) for n = 1..65537</a>
%F A125168 a(n) = gcd(n, A032741(n)) = gcd(n, A062968(n)).
%e A125168 a(6)=3 because 6 has 3 proper divisors {1,2,3} and gcd(6,3) is 3.
%t A125168 f[n_] := GCD[n, DivisorSigma[0, n] - 1]; Array[f, 105] (* _Robert G. Wilson v_ *)
%o A125168 (PARI) A125168(n) = gcd(n,numdiv(n)-1); \\ _Antti Karttunen_, Sep 25 2018
%Y A125168 Cf. A032741, A009191, A062968.
%K A125168 easy,nonn
%O A125168 1,4
%A A125168 Mitch Cervinka (puritan(AT)toast.net), Jan 12 2007
%E A125168 More terms from _Robert G. Wilson v_, Jan 23 2007