A030628 1 together with numbers of the form p*q^4 and p^9, where p and q are primes.

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

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

Index to sequences related to prime signature

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 *)
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

Cf. A030515, A030627, A030629.

Sequence in context: A362969 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