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.

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

Offset: 1

Author

Franklin T. Adams-Watters, Nov 01 2006

Keywords

• easy
• nonn
• tabf

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

Examples

			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]
...
		

Links

• Reinhard Zumkeller, Rows n = 1..10000 of triangle, flattened
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

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

Sorted rows: A118914, A212171.

Programs

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

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
PrimeSignature := proc(n) local F, e, k; F := ifactors(n)[2]; [seq(e, e = seq(F[k][2], k = 1..nops(F)))] end:
ListTools:-Flatten([[0], seq(PrimeSignature(n), n = 1..73)]); # Peter Luschny, Jun 15 2025

Mathematica

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

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

Formula

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

Extensions

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