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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. 305
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; text; internal format)
OFFSET

1,3

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. [Cf. A248692 for example.] 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, 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.

All A000041(a(n)) different n's with the same value a(n) are listed in row a(n) of triangle A215366. - Alois P. Heinz, Aug 09 2012

a(n) is the sum of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(33) = 7 because the partition with Heinz number 33 = 3 * 11 is [2,5]. - Emeric Deutsch, May 19 2015

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization

FORMULA

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

a(n) = Sum_{k=1..A001221(n)} A049084(A027748(k))*A124010(k). - Reinhard Zumkeller, Apr 27 2013

From Antti Karttunen, Oct 11 2014: (Start)

a(n) = n - A178503(n).

a(n) = A161511(A156552(n)).

a(n) = A227183(A243354(n)).

For all n >= 0:

a(A002110(n)) = A000217(n). [Cf. Henry Bottomley's comment above.]

a(A005940(n+1)) = A161511(n).

a(A243353(n)) = A227183(n).

Also, for all n >= 1:

a(A241909(n)) = A243503(n).

a(A122111(n)) = a(n).

a(A242424(n)) = a(n).

A248692(n) = 2^a(n).

(End)

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

A056239 := proc(n) add( numtheory[pi](op(1, p))*op(2, p), p = ifactors(n)[2]) ; end proc: # R. J. Mathar, Apr 20 2010

# alternative:

with(numtheory): a := proc (n) local B: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: add(B(n)[i], i = 1 .. nops(B(n))) end proc: seq(a(n), n = 1 .. 130); # Emeric Deutsch, May 19 2015

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] (* Jean-Fran├žois Alcover, May 19 2011 *)

Table[Total[FactorInteger[n] /. {p_, c_} /; p > 0 :> PrimePi[p] c], {n, 91}] (* Michael De Vlieger, Jul 12 2017 *)

PROG

(Haskell)

a056239 n = sum $ zipWith (*) (map a049084 $ a027748_row n) (a124010_row n)

-- Reinhard Zumkeller, Apr 27 2013

(PARI)

A049084(n) = if(isprime(n), primepi(n), 0); \\ Charles R Greathouse IV

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * A049084(f[i, 1]))); } \\ Antti Karttunen, Oct 26 2014

(Scheme)

(require 'factor) ;; Uses the function factor available in Aubrey Jaffer's SLIB Scheme library.

(define (A056239 n) (apply + (map A049084 (factor n))))

;; Antti Karttunen, Oct 26 2014

CROSSREFS

Row sums of A112798.

Cf. A003963 (gives the corresponding products).

Cf. also A000041, A000217, A000720, A001221, A002110, A005940, A008475, A027748, A049084, A060437, A081401, A082090, A088314, A088318, A088850, A088880, A088881, A088887, A088902, A122111, A124010, A127668, A141128, A153734, A154351, A156552, A163517, A178503, A215366, A215369, A215501, A242422, A242423, A242424, A243070, A243353, A243354, A243503, A248692, A249336, A249337.

Sequence in context: A207034 A226164 A302039 * A161511 A319856 A100197

Adjacent sequences:  A056236 A056237 A056238 * A056240 A056241 A056242

KEYWORD

easy,nonn,hear

AUTHOR

Leroy Quet, Aug 19 2000

STATUS

approved

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Last modified October 17 12:33 EDT 2018. Contains 316280 sequences. (Running on oeis4.)