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A008474 If n = Product (p_j^k_j) then a(n) = Sum (p_j + k_j). 17
0, 3, 4, 4, 6, 7, 8, 5, 5, 9, 12, 8, 14, 11, 10, 6, 18, 8, 20, 10, 12, 15, 24, 9, 7, 17, 6, 12, 30, 13, 32, 7, 16, 21, 14, 9, 38, 23, 18, 11, 42, 15, 44, 16, 11, 27, 48, 10, 9, 10, 22, 18, 54, 9, 18, 13, 24, 33, 60, 14, 62, 35, 13, 8, 20, 19, 68, 22, 28, 17, 72, 10, 74, 41, 11, 24, 20, 21 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(1) = 0 by convention, but could equally be taken to be 1 or 2.

Since only the primes p_j with nonzero exponents k_j in the factorization of n are considered in Sum (p_j + k_j), to the empty product (1) should correspond the empty sum (0). a(1) = 0 is thus the most natural choice. - Daniel Forgues, Apr 06 2010

Conjecture: for m > 4, by iterating the map m -> A008474(m) one always reaches 5 [tested up to m = 320000]. - Ivan N. Ianakiev, Nov 10 2014

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime Factorization

FORMULA

Additive with a(p^e) = p + e.

MAPLE

A008474 := proc(n) local e, j; e := ifactors(n)[2]:

add(e[j][1]+e[j][2], j=1..nops(e)) end:

seq (A008474(n), n=1..60);

# Peter Luschny, Jan 17 2011

MATHEMATICA

A008474[n_]:=Plus@@Flatten[FactorInteger[n]]; Table[A008474[n], {n, 200}] (* Zak Seidov, May 23 2005 *)

PROG

(Haskell)

a008474 n = sum $ zipWith (+) (a027748_row n) (a124010_row n)

-- Reinhard Zumkeller, Feb 11 2012, Aug 27 2011

(PARI) {for(k=1, 79,

M=factor(k); smt =0;

for(i=1, matsize(M)[1], for(j=1, matsize(M)[2], smt=smt+M[i, j]));

print1(smt, ", "))} \\\ Douglas Latimer, Apr 27 2012

(PARI) a(n)=my(f=factor(n)); sum(i=1, #f~, f[i, 1]+f[i, 2]) \\ Charles R Greathouse IV, Jun 03 2015

CROSSREFS

Cf. A107737, A107738, A000026, A027748, A124010, A250030.

Sequence in context: A164326 A317645 A117571 * A111611 A100478 A112376

Adjacent sequences:  A008471 A008472 A008473 * A008475 A008476 A008477

KEYWORD

nonn

AUTHOR

Olivier Gérard

EXTENSIONS

More terms from Zak Seidov, May 23 2005

STATUS

approved

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Last modified March 29 21:32 EDT 2020. Contains 333117 sequences. (Running on oeis4.)