cp's OEIS Frontend

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

%I A007422 M4068 #71 Feb 16 2025 08:32:31
%S A007422 1,6,8,10,14,15,21,22,26,27,33,34,35,38,39,46,51,55,57,58,62,65,69,74,
%T A007422 77,82,85,86,87,91,93,94,95,106,111,115,118,119,122,123,125,129,133,
%U A007422 134,141,142,143,145,146,155,158,159,161,166,177,178,183,185,187
%N A007422 Multiplicatively perfect numbers j: product of divisors of j is j^2.
%C A007422 Or, numbers j such that product of proper divisors of j is j.
%C A007422 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
%C A007422 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
%C A007422 Also 1 followed by numbers j such that A000005(j) = 4. - _Nathaniel Johnston_, May 03 2011
%C A007422 Fixed points of A007956. - _Reinhard Zumkeller_, Jan 26 2014
%D A007422 Kenneth Ireland and Michael Ira Rosen, A Classical Introduction to Modern Number Theory. Springer-Verlag, NY, 1982, p. 19.
%D A007422 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.
%D A007422 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007422 T. D. Noe, <a href="/A007422/b007422.txt">Table of n, a(n) for n = 1..1000</a>
%H A007422 William Chau, <a href="https://www.jstor.org/stable/24340619">The tau, sigma, rho functions, and some related numbers</a>, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.11.No10.pdf">entire issue</a>.
%H A007422 József Sándor, <a href="http://www.emis.de/journals/JIPAM/article119.html?sid=119">Multiplicatively perfect numbers</a>, J. Ineq. Pure Appl. Math., Vol. 2, No. 1 (2001), Article 3, 6 pp.
%H A007422 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorProduct.html">Divisor Product</a>.
%H A007422 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePerfectNumber.html">Multiplicative Perfect Number</a>.
%F A007422 A084110(a(n)) = 1, see also A084116. - _Reinhard Zumkeller_, May 12 2003
%F A007422 The number of terms not exceeding x is N(x) ~ x * log(log(x))/log(x) (Chau, 2004). - _Amiram Eldar_, Jun 29 2022
%e A007422 The divisors of 10 are 1, 2, 5, 10 and 1 * 2 * 5 * 10 = 100 = 10^2.
%p A007422 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
%p A007422 # second Maple program:
%p A007422 q:= n-> n=1 or numtheory[tau](n)=4:
%p A007422 select(q, [$1..200])[];  # _Alois P. Heinz_, Dec 17 2021
%t A007422 Select[Range[200], Times@@Divisors[#] == #^2 &]  (* _Harvey P. Dale_, Mar 27 2011 *)
%o A007422 (Magma) IsA007422:=func< n | &*Divisors(n) eq n^2 >; [ n: n in [1..200] | IsA007422(n) ]; // _Klaus Brockhaus_, May 04 2011
%o A007422 (Haskell)
%o A007422 a007422 n = a007422_list !! (n-1)
%o A007422 a007422_list = [x | x <- [1..], a007956 x == x]
%o A007422 -- _Reinhard Zumkeller_, Jan 26 2014
%o A007422 (PARI) is(n)=n==1 || numdiv(n) == 4 \\ _Charles R Greathouse IV_, Oct 15 2015
%o A007422 (Python)
%o A007422 from math import isqrt
%o A007422 from sympy import primepi, integer_nthroot, primerange
%o A007422 def A007422(n):
%o A007422     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)))
%o A007422     m, k = n, f(n)
%o A007422     while m != k:
%o A007422         m, k = k, f(k)
%o A007422     return m # _Chai Wah Wu_, Aug 16 2024
%Y A007422 Cf. A030513 (same as this sequence but without the 1), A027751, A006881 (subsequence), A030078 (subsequence), A084110, A084116, A236473.
%K A007422 nonn,nice,easy
%O A007422 1,2
%A A007422 _N. J. A. Sloane_
%E A007422 Some numbers were omitted - thanks to _Erich Friedman_ for pointing this out.