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A007422 Multiplicatively perfect numbers n: product of divisors of n is n^2.
(Formerly M4068)
26
1, 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Or, numbers n such that product of proper divisors of n is n.

A084110(a(n)) = 1, see also A084116. - Reinhard Zumkeller, May 12 2003

If M(n) denotes the product of the divisors of n, then n is said to be k-multiplicatively perfect if M(n) = n^k. All such numbers are of the form p q^(k-1) or p^(2k-1). This statement is in Sandor's paper. Therefore all 2-multiplicatively perfect numbers are semiprime p*q or cubes p^3. - Walter Kehowski, Sep 13 2005

All 2-multiplicatively perfect numbers except 1 have 4 divisors (as implied by Kehowski) and the converse is also true that all numbers with 4 divisors are 2-multiplicatively perfect. - Howard Berman (howard_berman(AT)hotmail.com), Oct 24 2008

Also 1 followed by numbers n such that A000005(n) = 4. - Nathaniel Johnston, May 03 2011

Fixed points of A007956. - Reinhard Zumkeller, Jan 26 2014

REFERENCES

Kenneth Ireland and Michael Ira Rosen, A Classical Introduction to Modern Number Theory. Springer-Verlag, NY, 1982, p. 19.

E. Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen ueber Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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

Jozsef Sandor, Multiplicatively perfect numbers, J. Ineq. Pure Appl. Math. 2(2001), no. 1, article 3, 6 pp.

Eric Weisstein's World of Mathematics, Divisor Product.

Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

EXAMPLE

The divisors of 10 are 1, 2, 5, 10 and 1 * 2 * 5 * 10 = 100 = 10^2.

MAPLE

k:=2: MPL:=[]: for z from 1 to 1 do for n from 1 to 5000 do if convert(divisors(n), `*`) = n^k then MPL:=[op(MPL), n] fi od; od; MPL; # Walter Kehowski, Sep 13 2005

MATHEMATICA

Select[Range[200], Times@@Divisors[#] == #^2 &]  (* Harvey P. Dale, Mar 27 2011 *)

PROG

(MAGMA) IsA007422:=func< n | &*Divisors(n) eq n^2 >; [ n: n in [1..200] | IsA007422(n) ]; // Klaus Brockhaus, May 04 2011

(Haskell)

a007422 n = a007422_list !! (n-1)

a007422_list = [x | x <- [1..], a007956 x == x]

-- Reinhard Zumkeller, Jan 26 2014

(PARI) is(n)=n==1 || numdiv(n) == 4 \\ Charles R Greathouse IV, Oct 15 2015

CROSSREFS

Cf. A030513 (same as this sequence but without the 1), A027751, A006881 (subsequence), A030078 (subsequence), A236473.

Sequence in context: A291127 A211337 * A030513 A161918 A294729 A242270

Adjacent sequences:  A007419 A007420 A007421 * A007423 A007424 A007425

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Some numbers were omitted - thanks to Erich Friedman for pointing this out.

STATUS

approved

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Last modified February 25 03:06 EST 2018. Contains 299630 sequences. (Running on oeis4.)