A007956 Product of the proper divisors of n.

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, 27, 784, 1, 27000, 1, 1024, 33, 34, 35, 279936, 1, 38, 39, 64000, 1, 74088, 1, 1936, 2025, 46, 1, 5308416, 7, 2500, 51, 2704, 1, 157464, 55, 175616, 57, 58, 1, 777600000, 1, 62, 3969, 32768, 65

Offset: 1

Author

R. Muller

Keywords

• nonn
• easy
• nice

Comments

From Bernard Schott, Feb 01 2019: (Start)

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

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).

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

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

Examples

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

References

• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 57.
• Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Zhu Weiyi, On the divisor product sequences, Smarandache Notions J., Vol. 14 (2004), pp. 144-146.

Crossrefs

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

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

Cf. A001065 (sums of proper divisors).

Programs

Haskell

a007956 = product . a027751_row
-- Reinhard Zumkeller, Feb 04 2013, Nov 02 2011

Maple

A007956 := n -> mul(i,i=op(numtheory[divisors](n) minus {1,n}));
seq(A007956(i), i=1..79); # Peter Luschny, Mar 22 2011

Mathematica

Table[Times@@Most[Divisors[n]], {n, 65}] (* Alonso del Arte, Apr 18 2011 *)
a[n_] := n^(DivisorSigma[0, n]/2 - 1); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 07 2013 *)

PARI

A007956(n) = local(a);a=1;fordiv(n,d,a=a*d);a/n \\ Michael B. Porter, Dec 01 2009

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

Python

from math import isqrt
from sympy import divisor_count
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

Formula

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

a(n) = Product_{k=1..A000005(n)-1} A027751(n,k). - Reinhard Zumkeller, Feb 04 2013

a(n) = A240694(n, A000005(n)-1) for n > 1. - Reinhard Zumkeller, Apr 10 2014

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

Extensions

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