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A165558
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Integers which are half of their arithmetic derivative.
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1
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0, 16, 108, 729, 12500, 84375, 3294172, 9765625, 22235661, 2573571875, 678223072849, 1141246682444, 7703415106497, 891598970659375, 1211500426369012, 8177627877990831, 234966429149994773, 946484708100790625
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| All integers of the form p^p*q^q, with q and p two distinct primes, are in the sequence. [R. J. Mathar, Sep 26 2009]
6*10^8 < a(10) <= 2573571875. a(11) <= 678223072849. [From Donovan Johnson (donovan.johnson(AT)yahoo.com), Nov 03 2010]
By a result of Ufnarovski and Ahlander, an integer is in this sequence if and only if it has the form p^(2p) or p^p*q^q, with p and q distinct primes. See comments from A072873. [From Nathaniel Johnston (nathaniel(AT)nathanieljohnston.com), Nov 22 2010]
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FORMULA
| {n: A003415(n) = 2*n}.
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EXAMPLE
| For k =84375 = 3^3*5^5, so A003415(k)/2 = 84375*(3/3+5/5)/2 = 84375 = k, which adds k=84375 to the sequence.
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MAPLE
| P:= proc(p) local a, b, m, n, i, ok, t1, t2, t3; a:=0; for n from 0 by 1 to p do
b:=1000000000039; ok:=0; if n<=1 then a:=0; ok:=1; fi; if isprime(n) then a:=1; ok:=1; fi; if ok=0 then t1:=ifactor(b*n); m:=nops(t1); t2:=0; for i from 1 to m do t3:=op(i, t1); if nops(t3)=1 then t2:=t2+1/op(t3); else t2:=t2+op(2, t3)/op(op(1, t3)); fi; od;
t2:=t2-1/b; a:=n*t2; fi; if a=2*n then lprint(n, a); fi; od; end: P(100000);
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CROSSREFS
| Cf. A003415, A072873
Sequence in context: A083469 A056001 A163725 * A155871 A120668 A053526
Adjacent sequences: A165555 A165556 A165557 * A165559 A165560 A165561
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KEYWORD
| nonn
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AUTHOR
| Paolo P. Lava & Giorgio Balzarotti (paoloplava(AT)gmail.com), Sep 22 2009
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EXTENSIONS
| Entries checked by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 26 2009
a(7)-a(9) from Donovan Johnson (donovan.johnson(AT)yahoo.com), Nov 03 2010
a(10)-a(18) and general form from Nathaniel Johnston (nathaniel(AT)nathanieljohnston.com), Nov 22 2010
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