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

Author

Jeff Burch

Keywords

• nonn
• easy
• nice

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

Crossrefs

Cf. A030515, A030627, A030629.

Programs

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 *)

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

Python

from sympy import primepi, primerange, integer_nthroot
def A030628(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+x-sum(primepi(x//p**4) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,5)[0])-primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

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

Extensions

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Dec 23 2001

More terms from Walter Kehowski, Sep 13 2005