

A030628


1 together with numbers of the form p*q^4 and p^9, where p and q are primes.


11



1, 48, 80, 112, 162, 176, 208, 272, 304, 368, 405, 464, 496, 512, 567, 592, 656, 688, 752, 848, 891, 944, 976, 1053, 1072, 1136, 1168, 1250, 1264, 1328, 1377, 1424, 1539, 1552, 1616, 1648, 1712, 1744, 1808, 1863, 1875, 2032, 2096, 2192, 2224, 2349, 2384
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 kmultiplicatively perfect if M(n)=n^k. All such numbers are of the form p*q^(k1) or p^(2k1). The sequence A030628 is therefore 5multiplicatively 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

Union A178739 U A179665 {1}.  R. J. Mathar, Apr 03 2011


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 *)


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

Cf. A030515, A030627, A030629.
Sequence in context: A261553 A110229 A108608 * A178739 A261548 A065911
Adjacent sequences: A030625 A030626 A030627 * A030629 A030630 A030631


KEYWORD

nonn,easy,nice


AUTHOR

Jeff Burch


EXTENSIONS

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Dec 23, 2001
More terms from Walter Kehowski, Sep 13 2005


STATUS

approved



