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A030628
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1 together with numbers of the form p*q^4 and p^9, where p and q are primes.
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15
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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)
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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MAPLE
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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
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MATHEMATICA
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Better description from Sharon Sela (sharonsela(AT)hotmail.com), Dec 23 2001
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STATUS
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approved
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