A027748 Irregular triangle in which first row is 1, n-th row (n > 1) lists distinct prime factors of n.

1, 2, 3, 2, 5, 2, 3, 7, 2, 3, 2, 5, 11, 2, 3, 13, 2, 7, 3, 5, 2, 17, 2, 3, 19, 2, 5, 3, 7, 2, 11, 23, 2, 3, 5, 2, 13, 3, 2, 7, 29, 2, 3, 5, 31, 2, 3, 11, 2, 17, 5, 7, 2, 3, 37, 2, 19, 3, 13, 2, 5, 41, 2, 3, 7, 43, 2, 11, 3, 5, 2, 23, 47, 2, 3, 7, 2, 5, 3, 17, 2, 13, 53, 2, 3, 5, 11, 2, 7, 3, 19, 2, 29, 59, 2, 3, 5, 61, 2, 31

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• tabf
• nice

Comments

Number of terms in n-th row is A001221(n) for n > 1.

From Reinhard Zumkeller, Aug 27 2011: (Start)

A008472(n) = Sum_{k=1..A001221(n)} T(n,k), n>1;

A007947(n) = Product_{k=1..A001221(n)} T(n,k);

A006530(n) = Max_{k=1..A001221(n)} T(n,k).

A020639(n) = Min_{k=1..A001221(n)} T(n,k).

(End)

Subsequence of A027750 that lists the divisors of n. - Michel Marcus, Oct 17 2015

Examples

			Triangle begins:
   1;
   2;
   3;
   2;
   5;
   2, 3;
   7;
   2;
   3;
   2, 5;
  11;
   2, 3;
  13;
   2, 7;
  ...
		

Links

• T. D. Noe, Rows n=1..2048 of triangle, flattened
• Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Crossrefs

Cf. A000027, A001221, A001222 (with repetition), A027746, A141809, A141810.

a(A013939(A000040(n))+1) = A000040(n).

Cf. A020639, A027750.

A284411 gives column medians.

Programs

Haskell

import Data.List (unfoldr)
a027748 n k = a027748_tabl !! (n-1) !! (k-1)
a027748_tabl = map a027748_row [1..]
a027748_row 1 = [1]
a027748_row n = unfoldr fact n where
   fact 1 = Nothing
   fact x = Just (p, until ((> 0) . (`mod` p)) (`div` p) x)
            where p = a020639 x  -- smallest prime factor of x
-- Reinhard Zumkeller, Aug 27 2011

Maple

with(numtheory): [ seq(factorset(n), n=1..100) ];

Mathematica

Flatten[ Table[ FactorInteger[n][[All, 1]], {n, 1, 62}]](* Jean-François Alcover, Oct 10 2011 *)

PARI

print1(1);for(n=2,20,f=factor(n)[,1];for(i=1,#f,print1(", "f[i]))) \\ Charles R Greathouse IV, Mar 20 2013

Python

from sympy import primefactors
for n in range(2, 101):
    print([i for i in primefactors(n)]) # Indranil Ghosh, Mar 31 2017

Extensions

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)