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A056239 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. +0
23
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; listen)
OFFSET

1,3

COMMENT

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.

LINKS

N. J. A. Sloane, First 10000 terms

Leroy Quet, Home Page (listed in lieu of email address)

Michael Nyvang, Home page

FORMULA

Totally additive with a(p) = PrimePi(p), where PrimePi(n) = A000720(n).

EXAMPLE

a(12) = 1*2 + 2*1 = 4, since 12 = 2^2 *3^1 = (p_1)^2 *(p_2)^1.

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)

CROSSREFS

Row sums of A112798.

Sequence in context: A117498 A064097 A014701 this_sequence A161511 A100197 A057022

Adjacent sequences: A056236 A056237 A056238 this_sequence A056240 A056241 A056242

KEYWORD

easy,nonn

AUTHOR

Leroy Quet Aug 19 2000

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Last modified November 25 08:46 EST 2009. Contains 167481 sequences.


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