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

1,2

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

Links

T. D. Noe, Rows n=1..2048 of triangle, flattened

Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Example

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

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 *)

Prog

(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
(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

Crossrefs

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

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

Cf. A020639, A027750.

A284411 gives column medians.

Sequence in context: A086418 A100761 A336964 * A361650 A328852 A000705

Adjacent sequences: A027745 A027746 A027747 * A027749 A027750 A027751

Keyword

nonn,easy,tabf,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved