A265668 Table read by rows: prime factors of squarefree numbers; a(1) = 1 by convention.

1, 2, 3, 5, 2, 3, 7, 2, 5, 11, 13, 2, 7, 3, 5, 17, 19, 3, 7, 2, 11, 23, 2, 13, 29, 2, 3, 5, 31, 3, 11, 2, 17, 5, 7, 37, 2, 19, 3, 13, 41, 2, 3, 7, 43, 2, 23, 47, 3, 17, 53, 5, 11, 3, 19, 2, 29, 59, 61, 2, 31, 5, 13, 2, 3, 11, 67, 3, 23, 2, 5, 7, 71, 73, 2

Offset

1,2

Comments

For n > 1: A072047(n) = length of row n;

T(n,1) = A073481(n); T(n,A001221(n)) = A073482(n);

for n > 1: A111060(n) = sum of row n;

A005117(n) = product of row n.

Links

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Example

.   n | T(n,*)  A5117(n)    n | T(n,*)  A5117(n)    n | T(n,*)   A5117(n)
. ----+---------+------   ----+---------+------   ----+----------+------
.   1 | [1]     |  1       21 | [3,11]  | 33       41 | [2,3,11] | 66
.   2 | [2]     |  2       22 | [2,17]  | 34       42 | [67]     | 67
.   3 | [3]     |  3       23 | [5,7]   | 35       43 | [3,23]   | 69
.   4 | [5]     |  5       24 | [37]    | 37       44 | [2,5,7]  | 70
.   5 | [2,3]   |  6       25 | [2,19]  | 38       45 | [71]     | 71
.   6 | [7]     |  7       26 | [3,13]  | 39       46 | [73]     | 73
.   7 | [2,5]   | 10       27 | [41]    | 41       47 | [2,37]   | 74
.   8 | [11]    | 11       28 | [2,3,7] | 42       48 | [7,11]   | 77
.   9 | [13]    | 13       29 | [43]    | 43       49 | [2,3,13] | 78
.  10 | [2,7]   | 14       30 | [2,23]  | 46       50 | [79]     | 79
.  11 | [3,5]   | 15       31 | [47]    | 47       51 | [2,41]   | 82
.  12 | [17]    | 17       32 | [3,17]  | 51       52 | [83]     | 83
.  13 | [19]    | 19       33 | [53]    | 53       53 | [5,17]   | 85
.  14 | [3,7]   | 21       34 | [5,11]  | 55       54 | [2,43]   | 86
.  15 | [2,11]  | 22       35 | [3,19]  | 57       55 | [3,29]   | 87
.  16 | [23]    | 23       36 | [2,29]  | 58       56 | [89]     | 89
.  17 | [2,13]  | 26       37 | [59]    | 59       57 | [7,13]   | 91
.  18 | [29]    | 29       38 | [61]    | 61       58 | [3,31]   | 93
.  19 | [2,3,5] | 30       39 | [2,31]  | 62       59 | [2,47]   | 94
.  20 | [31]    | 31       40 | [5,13]  | 65       60 | [5,19]   | 95  .

Mathematica

FactorInteger[#][[All, 1]]&/@Select[Range[100], SquareFreeQ]//Flatten (* Harvey P. Dale, Apr 27 2018 *)

Prog

(Haskell)
import Math.NumberTheory.Primes.Factorisation (factorise)
import Data.Maybe (mapMaybe)
a265668 n k = a265668_tabf !! (n-1) !! (k-1)
a265668_row n = a265668_tabf !! (n-1)
a265668_tabf = [1] : mapMaybe f [2..] where
   f x = if all (== 1) es then Just ps else Nothing
         where (ps, es) = unzip $ factorise x

Crossrefs

Cf. A005117, A027748, A124010, A072047 (row lengths), A073481, A073482, A001221, A111060, A049200, A062822.

Sequence in context: A232931 A060084 A361503 * A273087 A236434 A138182

Adjacent sequences: A265665 A265666 A265667 * A265669 A265670 A265671

Keyword

nonn,tabf

Author

Reinhard Zumkeller, Dec 13 2015

Status

approved