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Product of the proper divisors of n.
43

%I #96 Oct 09 2023 02:48:34

%S 1,1,1,2,1,6,1,8,3,10,1,144,1,14,15,64,1,324,1,400,21,22,1,13824,5,26,

%T 27,784,1,27000,1,1024,33,34,35,279936,1,38,39,64000,1,74088,1,1936,

%U 2025,46,1,5308416,7,2500,51,2704,1,157464,55,175616,57,58,1,777600000,1,62,3969,32768,65

%N Product of the proper divisors of n.

%C From _Bernard Schott_, Feb 01 2019: (Start)

%C a(n) = 1 iff n = 1 or n is prime.

%C a(n) = n when n > 1 iff n has exactly four divisors, equally, iff n is either the cube of a prime or the product of two different primes, so iff n belongs to A030513 (very nice proof in Sierpiński).

%C a(p^3) = 1 * p * p^2 = p^3; a(p*q) = 1 * p * q = p*q.

%C As a(1) = 1, {1} Union A030513 = A007422, fixed points of this sequence. (End)

%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 57.

%D Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

%H Amiram Eldar, <a href="/A007956/b007956.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Zhu Weiyi, <a href="https://vixra.org/abs/1403.0870">On the divisor product sequences</a>, Smarandache Notions J., Vol. 14 (2004), pp. 144-146.

%F a(n) = A007955(n)/n = n^(A000005(n)/2-1) = sqrt(n^(number of factors of n other than 1 and n)).

%F a(n) = Product_{k=1..A000005(n)-1} A027751(n,k). - _Reinhard Zumkeller_, Feb 04 2013

%F a(n) = A240694(n, A000005(n)-1) for n > 1. - _Reinhard Zumkeller_, Apr 10 2014

%F Sum_{k=1..n} 1/a(k) ~ pi(n) + log(log(n))^2 + c_1*log(log(n)) + c_2 + O(log(log(n))/log(n)), where pi(n) = A000720(n) and c_1 and c_2 are constants (Weiyi, 2004; Sandor and Crstici, 2004). - _Amiram Eldar_, Oct 29 2022

%e a(18) = 1 * 2 * 3 * 6 * 9 = 324. - _Bernard Schott_, Jan 31 2019

%p A007956 := n -> mul(i,i=op(numtheory[divisors](n) minus {1,n}));

%p seq(A007956(i), i=1..79); # _Peter Luschny_, Mar 22 2011

%t Table[Times@@Most[Divisors[n]], {n, 65}] (* _Alonso del Arte_, Apr 18 2011 *)

%t a[n_] := n^(DivisorSigma[0, n]/2 - 1); Table[a[n], {n, 1, 65}] (* _Jean-François Alcover_, Oct 07 2013 *)

%o (PARI) A007956(n) = local(a);a=1;fordiv(n,d,a=a*d);a/n \\ _Michael B. Porter_, Dec 01 2009

%o (PARI) a(n)=my(k); if(issquare(n, &k), k^(numdiv(n)-2), n^(numdiv(n)/2-1)) \\ _Charles R Greathouse IV_, Oct 15 2015

%o (Haskell)

%o a007956 = product . a027751_row

%o -- _Reinhard Zumkeller_, Feb 04 2013, Nov 02 2011

%o (Python)

%o from math import isqrt

%o from sympy import divisor_count

%o def A007956(n): return isqrt(n)**(d-2) if (d:=divisor_count(n))&1 else n**((d>>1)-1) # _Chai Wah Wu_, Jun 18 2023

%Y Cf. A000005, A000720, A007955, A048671, A182936, A027751, A066729.

%Y Cf. A007422 (fixed points). A030513 (subsequence).

%Y Cf. A001065 (sums of proper divisors).

%K nonn,easy,nice

%O 1,4

%A R. Muller

%E More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)