A007422 Multiplicatively perfect numbers j: product of divisors of j is j^2. (Formerly M4068)

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

Offset

1,2

Comments

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

If M(j) denotes the product of the divisors of j, then j is said to be k-multiplicatively perfect if M(j) = j^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 j such that A000005(j) = 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.

Edmund 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

William Chau, The tau, sigma, rho functions, and some related numbers, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; entire issue.

József Sándor, Multiplicatively perfect numbers, J. Ineq. Pure Appl. Math., Vol. 2, No. 1 (2001), Article 3, 6 pp.

Eric Weisstein's World of Mathematics, Divisor Product.

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

Formula

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

The number of terms not exceeding x is N(x) ~ x * log(log(x))/log(x) (Chau, 2004). - Amiram Eldar, Jun 29 2022

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
# second Maple program:
q:= n-> n=1 or numtheory[tau](n)=4:
select(q, [$1..200])[];  # Alois P. Heinz, Dec 17 2021

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
(Python)
from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A007422(n):
    def f(x): return int(n-1+x-primepi(integer_nthroot(x, 3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 16 2024

Crossrefs

Cf. A030513 (same as this sequence but without the 1), A027751, A006881 (subsequence), A030078 (subsequence), A084110, A084116, 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