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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. 449
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
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-François 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: A118914 A135063 A370770 * A212171 A337255 A337375
KEYWORD
easy,nonn,tabf
AUTHOR
EXTENSIONS
Name edited by M. F. Hasler, Apr 08 2022
STATUS
approved

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Last modified March 19 09:40 EDT 2024. Contains 370981 sequences. (Running on oeis4.)