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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, 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: A036455 A291127 A211337 * A030513 A161918 A294729 Adjacent sequences:  A007419 A007420 A007421 * A007423 A007424 A007425 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS Some numbers were omitted - thanks to Erich Friedman for pointing this out. STATUS approved

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Last modified October 15 09:22 EDT 2019. Contains 328026 sequences. (Running on oeis4.)