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

1,4

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)

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.

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

Example

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

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 *)

Prog

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

Crossrefs

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

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

Cf. A001065 (sums of proper divisors).

Sequence in context: A324193 A364829 A264859 * A378183 A107754 A181569

Adjacent sequences: A007953 A007954 A007955 * A007957 A007958 A007959

Keyword

nonn,easy,nice

Author

R. Muller

Extensions

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

Status

approved