A007422 Multiplicatively perfect numbers j: product of divisors of j is j^2.

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

Examples

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

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.

Crossrefs

Cf. A030513 (same as this sequence but without the 1), A027751, A006881 (subsequence), A030078 (subsequence), A084110, A084116, A236473.

Programs

Haskell

a007422 n = a007422_list !! (n-1)
a007422_list = [x | x <- [1..], a007956 x == x]
-- Reinhard Zumkeller, Jan 26 2014

Magma

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

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 *)

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

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

Extensions

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