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A007956 Product of proper divisors of n. 30
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 (list; graph; refs; listen; history; text; internal format)
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

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

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

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

CROSSREFS

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

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

Cf. A001065 (sums of proper divisors).

Sequence in context: A324370 A324193 A264859 * A107754 A181569 A274085

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

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Last modified July 16 04:09 EDT 2019. Contains 325064 sequences. (Running on oeis4.)