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%I A030628 #47 Feb 21 2025 19:39:51
%S A030628 1,48,80,112,162,176,208,272,304,368,405,464,496,512,567,592,656,688,
%T A030628 752,848,891,944,976,1053,1072,1136,1168,1250,1264,1328,1377,1424,
%U A030628 1539,1552,1616,1648,1712,1744,1808,1863,1875,2032,2096,2192,2224,2349,2384
%N A030628 1 together with numbers of the form p*q^4 and p^9, where p and q are distinct primes.
%C A030628 Also 1 together with numbers with 10 divisors. Also numbers n such that product of all proper divisors of n equals n^4.
%C A030628 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
%D A030628 D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976. p. 119.
%D A030628 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry for 48, page 106, 1997.
%H A030628 R. J. Mathar, <a href="/A030628/b030628.txt">Table of n, a(n) for n = 1..1000</a>
%H A030628 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorProduct.html">Divisor Product</a>
%H A030628 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A030628 Union A178739 U A179665 {1}. - _R. J. Mathar_, Apr 03 2011
%p A030628 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
%t A030628 Join[{1},Select[Range[6000],DivisorSigma[0,#]==10&]] (* _Vladimir Joseph Stephan Orlovsky_, May 05 2011 *)
%t A030628 Select[Range[2500],Times@@Most[Divisors[#]]==#^4&] (* _Harvey P. Dale_, Nov 04 2024 *)
%o A030628 (PARI) {v=[]; for(n=1,500,v=concat(v, if(numdiv(n)==10,n,",")); ); v} \\ _Jason Earls_, Jun 18 2001
%o A030628 (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
%o A030628 (Python)
%o A030628 from sympy import primepi, primerange, integer_nthroot
%o A030628 def A030628(n):
%o A030628     def bisection(f,kmin=0,kmax=1):
%o A030628         while f(kmax) > kmax: kmax <<= 1
%o A030628         kmin = kmax >> 1
%o A030628         while kmax-kmin > 1:
%o A030628             kmid = kmax+kmin>>1
%o A030628             if f(kmid) <= kmid:
%o A030628                 kmax = kmid
%o A030628             else:
%o A030628                 kmin = kmid
%o A030628         return kmax
%o A030628     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])
%o A030628     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A030628 Cf. A030515, A030627, A030629.
%K A030628 nonn,easy,nice,changed
%O A030628 1,2
%A A030628 _Jeff Burch_
%E A030628 Better description from Sharon Sela (sharonsela(AT)hotmail.com), Dec 23 2001
%E A030628 More terms from _Walter Kehowski_, Sep 13 2005

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