OFFSET
1,2
COMMENTS
Also 1 together with numbers with 10 divisors. Also numbers n such that product of all proper divisors of n equals n^4.
If M(n) denotes the product of all 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). The sequence A030628 is therefore 5-multiplicatively perfect. See the Links for A007422. - Walter Kehowski, Sep 13 2005
REFERENCES
D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976. p. 119.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry for 48, page 106, 1997.
LINKS
R. J. Mathar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Divisor Product
FORMULA
MAPLE
with(numtheory): k:=5: 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
Join[{1}, Select[Range[6000], DivisorSigma[0, #]==10&]] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)
Select[Range[2500], Times@@Most[Divisors[#]]==#^4&] (* Harvey P. Dale, Nov 04 2024 *)
PROG
(PARI) {v=[]; for(n=1, 500, v=concat(v, if(numdiv(n)==10, n, ", ")); ); v} \\ Jason Earls, Jun 18 2001
(PARI) list(lim)=my(v=List([1]), t); forprime(p=2, (lim\2+.5)^(1/4), t=p^4; forprime(q=2, lim\t, if(p==q, next); listput(v, t*q))); forprime(p=2, (lim+.5)^(1/9), listput(v, p^9)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Apr 26 2012
CROSSREFS
KEYWORD
nonn,easy,nice,changed
AUTHOR
EXTENSIONS
Better description from Sharon Sela (sharonsela(AT)hotmail.com), Dec 23 2001
More terms from Walter Kehowski, Sep 13 2005
STATUS
approved