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A056239
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If n = product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = sum_{k >= 1} k*c_k.
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24
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0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 5, 4, 7, 5, 8, 5, 6, 6, 9, 5, 6, 7, 6, 6, 10, 6, 11, 5, 7, 8, 7, 6, 12, 9, 8, 6, 13, 7, 14, 7, 7, 10, 15, 6, 8, 7, 9, 8, 16, 7, 8, 7, 10, 11, 17, 7, 18, 12, 8, 6, 9, 8, 19, 9, 11, 8, 20, 7, 21, 13, 8, 10, 9, 9, 22, 7, 8, 14, 23, 8, 10, 15, 12, 8, 24, 8, 10
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| A pseudo-logarithmic function in the sense that a(b*c) = a(b)+a(c) and so a(b^c) = c*a(b) and f(n) = k^a(n) is a multiplicative function. Essentially a function from the positive integers onto the partitions of the nonnegative integers (1->0, 2->1, 3->2, 4->1+1, 5->3, 6->1+2 etc.) so each value a(n) appears A000041(a(n)) times, first with the a(n)th prime and last with the a(n)th power of 2. Produces triangular numbers from primorials. - Henry Bottomley (se16(AT)btinternet.com), Nov 22 2001
Michael Nyvang writes (May 08 2006) that the Danish composer Karl Aage Rasmussen discovered this sequence in the 1990's: it has excellent musical properties.
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LINKS
| N. J. A. Sloane, First 10000 terms
Michael Nyvang, Home page
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FORMULA
| Totally additive with a(p) = PrimePi(p), where PrimePi(n) = A000720(n).
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EXAMPLE
| a(12) = 1*2 + 2*1 = 4, since 12 = 2^2 *3^1 = (p_1)^2 *(p_2)^1.
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MAPLE
| # To get 10000 terms. First make prime table: M:=10000; pl:=array(1..M); for i from 1 to M do pl[i]:=0; od: for i from 1 to M do if ithprime(i) > M then break; fi; pl[ithprime(i)]:=i; od:
# Decode Maple's amazing syntax for factoring integers: g:=proc(n) local e, p, t1, t2, t3, i, j, k; global pl; t1:=ifactor(n); t2:=nops(t1); if t2 = 2 and whattype(t1) <> `*` then p:=op(1, op(1, t1)); e:=op(2, t1); t3:=pl[p]*e; else
t3:=0; for i from 1 to t2 do j:=op(i, t1); if nops(j) = 1 then e:=1; p:=op(1, j); else e:=op(2, j); p:=op(1, op(1, j)); fi; t3:=t3+pl[p]*e; od: fi; t3; end; (N. J. A. Sloane, May 10 2006)
A056239 := proc(n) add( numtheory[pi](op(1, p))*op(2, p), p = ifactors(n)[2]) ; end proc: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 20 2010]
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MATHEMATICA
| a[1] = 0; a[2] = 1; a[p_?PrimeQ] := a[p] = PrimePi[p];
a[n_] := a[n] = Total[#[[2]]*a[#[[1]]] & /@ FactorInteger[n]]; a /@ Range[91] (* From Jean-François Alcover, May 19 2011 *)
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CROSSREFS
| Row sums of A112798.
Sequence in context: A117498 A064097 A014701 * A161511 A100197 A057022
Adjacent sequences: A056236 A056237 A056238 * A056240 A056241 A056242
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KEYWORD
| easy,nonn
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AUTHOR
| Leroy Quet Aug 19 2000
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