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A124010 Triangle in which first row is 0, n-th row (n>1) lists the prime signature of n, that is, the exponents of distinct prime factors in factorization of n. 157
0, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

A001222(n)=Sum(T(n,k):1<=k<=A001221(n)); A005361(n)=Product(T(n,k):1<=k<=A001221(n)), n>1; A051903(n)=Max(T(n,k):1<=k<=A001221(n)); A051904(n)=Min(T(n,k):1<=k<=A001221(n)); A067029(n)=T(n,1); A071178(n)=T(n,A001221(n)); A064372(n)=Sum(A064372(T(n,k)):1<=k<=A001221(n)). [Reinhard Zumkeller, Aug 27 2011]

Any finite sequence of natural numbers appears as consecutive terms. [Paul Tek, Apr 27 2013]

For n > 1: n-th row = n-th row of A067255 without zeros. - Reinhard Zumkeller, Jun 11 2013

LINKS

Reinhard Zumkeller, Rows n = 1..10000 of triangle, flattened

Index entries for sequences computed from exponents in factorization of n

FORMULA

n = Product_k A027748(n,k)^a(n,k).

EXAMPLE

Initial values of exponents are:

1, [0]

2, [1]

3, [1]

4, [2]

5, [1]

6, [1, 1]

7, [1]

8, [3]

9, [2]

10, [1, 1]

11, [1]

12, [2, 1]

13, [1]

14, [1, 1]

15, [1, 1]

16, [4]

17, [1]

18, [1, 2]

19, [1]

20, [2, 1]

...

MAPLE

expts:=proc(n) local t1, t2, t3, t4, i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2, t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i, t1); if nops(t4) = 1 then t3:=[op(t3), 1]; else t3:=[op(t3), op(2, t4)]; fi; od; RETURN(t3); end; # N. J. A. Sloane, Dec 20 2007

MATHEMATICA

row[1] = {0}; row[n_] := FactorInteger[n][[All, 2]] // Flatten; Table[row[n], {n, 1, 80}] // Flatten (* Jean-Fran├žois Alcover, Aug 19 2013 *)

PROG

(Haskell)

a124010 n k = a124010_tabf !! (n-1) !! (k-1)

a124010_row 1 = [0]

a124010_row n = f n a000040_list where

   f 1 _      = []

   f u (p:ps) = h u 0 where

     h v e | m == 0 = h v' (e + 1)

           | m /= 0 = if e > 0 then e : f v ps else f v ps

           where (v', m) = divMod v p

a124010_tabf = map a124010_row [1..]

-- Reinhard Zumkeller, Jun 12 2013, Aug 27 2011

(PARI) print1(0); for(n=2, 50, f=factor(n)[, 2]; for(i=1, #f, print1(", "f[i]))) \\ Charles R Greathouse IV, Nov 07 2014

(Python)

from sympy import factorint

def a(n):

    f=factorint(n)

    return [0] if n==1 else [f[i] for i in f]

for n in xrange(1, 21): print a(n) # Indranil Ghosh, May 16 2017

CROSSREFS

Cf. A027748, A001221 (row lengths, n>1), A001222 (row sums), A027746, A020639, A064372.

Sorted rows: A118914, A212171.

Sequence in context: A030358 A118914 A135063 * A212171 A196228 A133364

Adjacent sequences:  A124007 A124008 A124009 * A124011 A124012 A124013

KEYWORD

easy,nonn,tabf

AUTHOR

Franklin T. Adams-Watters, Nov 01 2006

STATUS

approved

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Last modified August 21 13:36 EDT 2017. Contains 290890 sequences.