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A124010 Triangle in which first row is 0, n-th row (n>1) lists the exponents of distinct prime factors ("ordered prime signature") in the prime factorization of n. 382
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

Most often the prime signature is given as a sorted representative of the multiset of the nonzero exponents, either in increasing order, which yields A118914, or, most commonly, in decreasing order, which yields A212171. - M. F. Hasler, Oct 12 2018

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

(PARI) A124010_row(n)=if(n, factor(n)[, 2]~, [0]) \\ M. F. Hasler, Oct 12 2018

(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 range(1, 21): print(a(n)) # Indranil Ghosh, May 16 2017

CROSSREFS

Cf. A027748, A001221 (row lengths, n>1), A001222 (row sums), A027746, A020639, A064372, A067029 (first column).

Sorted rows: A118914, A212171.

Sequence in context: A030358 A118914 A135063 * A212171 A337255 A337375

Adjacent sequences:  A124007 A124008 A124009 * A124011 A124012 A124013

KEYWORD

easy,nonn,tabf

AUTHOR

Franklin T. Adams-Watters, Nov 01 2006

EXTENSIONS

Name edited by M. F. Hasler, Apr 08 2022

STATUS

approved

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Last modified October 4 15:10 EDT 2022. Contains 357239 sequences. (Running on oeis4.)