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A064553
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a(1) = 1, a(prime(i)) = i + 1 for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0; prime = A000040.
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16
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1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16, 8, 18, 9, 16, 15, 12, 10, 24, 16, 14, 27, 20, 11, 24, 12, 32, 18, 16, 20, 36, 13, 18, 21, 32, 14, 30, 15, 24, 36, 20, 16, 48, 25, 32, 24, 28, 17, 54, 24, 40, 27, 22, 18, 48, 19, 24, 45, 64, 28, 36, 20, 32, 30, 40, 21, 72, 22, 26
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n) <= n for all n and a(x) = x iff x = 2^i * 3^j for i, j >= 0: a(A003586(n)) = A003586(n) for n > 0. By definition a is completely multiplicative and also surjective. a(p) < a(q) for primes p < q.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..8000
T. D. Noe, Plot of A064553
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FORMULA
| Let the prime factorization of n be p1^e1...pk^ek, then a(n) = (pi(p1)+1)^e1...(pi(pk)+1)^ek, where pi(p) is the index of prime p. - T. D. Noe (noe(AT)sspectra.com), Dec 12 2004
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EXAMPLE
| a(5) = a(prime(3)) = 3 + 1 = 4; a(14) = a(2*7) = a(prime(1)* prime(4)) = (1+1)*(4+1) = 10.
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MATHEMATICA
| nn=100; a=Table[0, {nn}]; a[[1]]=1; Do[If[PrimeQ[i], a[[i]]=PrimePi[i]+1, p=FactorInteger[i][[1, 1]]; a[[i]] = a[[p]]*a[[i/p]]], {i, 2, nn}]; a (* T. D. Noe, Dec 12 2004, revised Sep 27 2011 *)
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PROG
| Haskell:
a064553 :: Integer -> Integer
a064553 n = r n 1
where r 1 y = y
r x y = r (x `div` spf) (y * (a049084 spf + 1))
where spf = a020639 x
-- eop.
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CROSSREFS
| Cf. A000040, A049084, A020639, A064554, A064555, A001055, A003586, A064557, A064558, A003963.
Sequence in context: A199424 A184163 A184160 * A126012 A096908 A162684
Adjacent sequences: A064550 A064551 A064552 * A064554 A064555 A064556
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KEYWORD
| mult,nice,nonn
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AUTHOR
| Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 21 2001
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