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Irregular triangle read by rows where row 1 is {1} and row n is the sequence starting with n and repeatedly applying A181819 until a prime number is reached.
48

%I #7 Apr 16 2019 15:27:11

%S 1,2,3,4,3,5,6,4,3,7,8,5,9,3,10,4,3,11,12,6,4,3,13,14,4,3,15,4,3,16,7,

%T 17,18,6,4,3,19,20,6,4,3,21,4,3,22,4,3,23,24,10,4,3,25,3,26,4,3,27,5,

%U 28,6,4,3,29,30,8,5,31,32,11,33,4,3

%N Irregular triangle read by rows where row 1 is {1} and row n is the sequence starting with n and repeatedly applying A181819 until a prime number is reached.

%C The function A181819 maps p^i*...*q^j to prime(i)*...*prime(j) where p through q are distinct primes.

%F T(n,k) = A325239(n,k) for k <= A323014(n).

%F A001222(T(n,k)) = A323023(n,k) for n > 1.

%e Triangle begins:

%e 1 26 4 3 51 4 3 76 6 4 3

%e 2 27 5 52 6 4 3 77 4 3

%e 3 28 6 4 3 53 78 8 5

%e 4 3 29 54 10 4 3 79

%e 5 30 8 5 55 4 3 80 14 4 3

%e 6 4 3 31 56 10 4 3 81 7

%e 7 32 11 57 4 3 82 4 3

%e 8 5 33 4 3 58 4 3 83

%e 9 3 34 4 3 59 84 12 6 4 3

%e 10 4 3 35 4 3 60 12 6 4 3 85 4 3

%e 11 36 9 3 61 86 4 3

%e 12 6 4 3 37 62 4 3 87 4 3

%e 13 38 4 3 63 6 4 3 88 10 4 3

%e 14 4 3 39 4 3 64 13 89

%e 15 4 3 40 10 4 3 65 4 3 90 12 6 4 3

%e 16 7 41 66 8 5 91 4 3

%e 17 42 8 5 67 92 6 4 3

%e 18 6 4 3 43 68 6 4 3 93 4 3

%e 19 44 6 4 3 69 4 3 94 4 3

%e 20 6 4 3 45 6 4 3 70 8 5 95 4 3

%e 21 4 3 46 4 3 71 96 22 4 3

%e 22 4 3 47 72 15 4 3 97

%e 23 48 14 4 3 73 98 6 4 3

%e 24 10 4 3 49 3 74 4 3 99 6 4 3

%e 25 3 50 6 4 3 75 6 4 3 100 9 3

%t red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];

%t Table[NestWhileList[red,n,#>1&&!PrimeQ[#]&],{n,30}]

%Y Row lengths are 1 for n = 1 and A323014(n) for n > 1.

%Y Cf. A001221, A001222, A071625, A118914, A181819, A181821, A182850, A182857, A323022, A323023, A325238, A325239.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, Apr 15 2019