

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
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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 kmultiplicatively perfect if M(n) = n^k. All such numbers are of the form p q^(k1) or p^(2k1). This statement is in Sandor's paper. Therefore all 2multiplicatively perfect numbers are semiprime p*q or cubes p^3.  Walter Kehowski, Sep 13 2005
All 2multiplicatively perfect numbers except 1 have 4 divisors (as implied by Kehowski) and the converse is also true that all numbers with 4 divisors are 2multiplicatively 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. SpringerVerlag, 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. 3132.
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 !! (n1)
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: A036455 A291127 A211337 * A030513 A161918 A294729
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



