A141810 -id:A141810 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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

Cf. A000027, A001221, A001222 (with repetition), A027746, A141809, A141810.
a(A013939(A000040(n))+1) = A000040(n).
Cf. A020639, A027750.
A284411 gives column medians.

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

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

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

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

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

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

A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 3, 7, 8, 9, 2, 5, 11, 4, 3, 13, 2, 7, 3, 5, 16, 17, 2, 9, 19, 4, 5, 3, 7, 2, 11, 23, 8, 3, 25, 2, 13, 27, 4, 7, 29, 2, 3, 5, 31, 32, 3, 11, 2, 17, 5, 7, 4, 9, 37, 2, 19, 3, 13, 8, 5, 41, 2, 3, 7, 43, 4, 11, 9, 5, 2, 23, 47, 16, 3, 49, 2, 25, 3, 17, 4, 13, 53, 2, 27, 5, 11, 8, 7, 3

Offset: 1

Leroy Quet, Jul 07 2008

In other words, except for row 1, row n contains the unitary prime power divisors of n, sorted by the prime. - Franklin T. Adams-Watters, May 05 2011

A034684(n) = smallest term of n-th row; A028233(n) = T(n,1); A053585(n) = T(n,A001221(n)); A008475(n) = sum of n-th row for n > 1. - Reinhard Zumkeller, Jan 29 2013

			60 has the prime factorization 2^2 * 3^1 * 5^1, so row 60 is (4,3,5).
From _M. F. Hasler_, Oct 12 2018: (Start)
The table starts:
    n : largest prime powers dividing n
    1 :  1
    2 :  2
    3 :  3
    4 :  4
    5 :  5
    6 :  2, 3
    7 :  7
    8 :  8
    9 :  9
   10 :  2, 5
   11 : 11
   12 :  4, 3
   etc. (End)

Reinhard Zumkeller, Rows n=1..10000 of triangle, flattened
Eric Weisstein's World of Mathematics, Prime Factorization

A027748, A124010 are used in a formula defining this sequence.
Cf. A001221 (row lengths), A008475 (row sums), A028233 (column 1), A034684 (row minima), A053585 (right edge).
Cf. A060175, A141810, A213925.

a141809 n k = a141809_row n !! (k-1)
a141809_row 1 = [1]
a141809_row n = zipWith (^) (a027748_row n) (a124010_row n)
a141809_tabf = map a141809_row [1..]
-- Reinhard Zumkeller, Mar 18 2012

f[{x_, y_}] := x^y; Table[Map[f, FactorInteger[n]], {n, 1, 50}] // Grid (* Geoffrey Critzer, Apr 03 2015 *)

A141809_row(n)=if(n>1, [f[1]^f[2]|f<-factor(n)~], [1]) \\ M. F. Hasler, Oct 12 2018, updated Aug 19 2022

T(n,k) = A027748(n,k)^A124010(n,k) for n > 1, k = 1..A001221(n). - Reinhard Zumkeller, Mar 15 2012

A140831 Numbers in whose canonical prime factorization the powers of the primes do not form an increasing sequence.

Original entry on oeis.org

12, 24, 40, 45, 48, 56, 60, 63, 80, 84, 90, 96, 112, 120, 126, 132, 135, 144, 156, 160, 168, 175, 176, 180, 189, 192, 204, 208, 224, 228, 240, 252, 264, 270, 275, 276, 280, 288, 297, 300, 312, 315, 320, 325, 336, 348, 350, 351, 352, 360, 372, 378, 384, 405

Offset: 1

Leroy Quet, Jul 18 2008

Previous name was: Let p^b(n,p) be the largest power of the prime p that divides n. The integer n is included if the list of p^b(n,p)'s, where each p is a distinct prime divisor of n, arranged by size of each p^b(n,p) is not in the same order as the list of p^b(n,p)'s arranged by size of each prime p.
This sequence contains no squarefree integers.
90 is the smallest integer in this sequence but not in sequence A126855.
The number of terms < 10^n: 0, 12, 151, 1575, 16154, 161630, 1617052, ..., . - Robert G. Wilson v, Aug 31 2008
If k is in the sequence, then all powers of k are in the sequence. - Mike Jones, Jun 16 2022
If k is in the sequence then k*A020639(k)^m is in the sequence for m >= 0. - David A. Corneth, Jun 16 2022
Conjecture: There are infinitely many terms k such that k+1 is also a term. - Mike Jones, Jun 18 2022

			The prime factorization of 90 is, when arranged by size of the distinct primes, 2^1 * 3^2 * 5^1. Since 3^2 is > 5^1, even though 5 > 3, 90 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Complement of A383397.

Cf. A020639, A141809, A141810, A126855.

fQ[n_] := Block[{f = First@# ^ Last@# & /@ FactorInteger@n}, f != Sort@f]; Select[ Range@ 407, fQ@# &] (* Robert G. Wilson v, Aug 31 2008 *)

is(n) = { my(f = factor(n)); for(i = 1, #f~-1, if(f[i,1]^f[i,2] > f[i+1,1]^f[i+1,2], return(1) ) ); 0 } \\ David A. Corneth, Jun 16 2022

More terms from Robert G. Wilson v, Aug 31 2008

Simpler name from Mike Jones, Jun 15 2022

A141404 Irregular array: For any prime p that divides n, if the highest power of the prime p that divides n is p^b(n,p), then p^b(n,p) = Sum_{k=1..m} a(n,k), where m is the order of the prime-power p^b(n,p) among the prime-powers (each being the highest power of each prime q that divides n, where q divides n) when they are ordered by size. Row 1 = (1).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 1, 7, 8, 9, 2, 3, 11, 3, 1, 13, 2, 5, 3, 2, 16, 17, 2, 7, 19, 4, 1, 3, 4, 2, 9, 23, 3, 5, 25, 2, 11, 27, 4, 3, 29, 2, 1, 2, 31, 32, 3, 8, 2, 15, 5, 2, 4, 5, 37, 2, 17, 3, 10, 5, 3, 41, 2, 1, 4, 43, 4, 7, 5, 4, 2, 21, 47, 3, 13, 49, 2, 23, 3, 14, 4, 9, 53, 2, 25, 5, 6, 7, 1, 3, 16, 2

Offset: 1

Leroy Quet, Aug 03 2008

Row n contains A001221(n) terms.

			The prime factorization of 300 is 2^2 *3^1 *5^2. So the prime powers ordered by size are 3, 4, 25. Therefore row 300 is (3,1,21), because 3=3, 3+1 = 4, 3+1+21 = 25.

Cf. A141810, A001221.

{ A141404row(n) = my(f); if(n==1,return([1])); f=factorint(n); f=vecsort(vector(matsize(f)[1],i,f[i,1]^f[i,2])); vector(#f,i,f[i]-if(i>1,f[i-1])); } \\ Max Alekseyev, May 07 2009

Extended by Max Alekseyev, May 07 2009

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

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A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

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A140831 Numbers in whose canonical prime factorization the powers of the primes do not form an increasing sequence.

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