This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363473 #39 Jan 07 2024 14:03:47
%S A363473 1,2,4,3,6,9,5,10,15,8,7,14,21,12,25,11,22,33,20,35,18,13,26,39,28,55,
%T A363473 30,49,17,34,51,44,65,42,77,16,19,38,57,52,85,66,91,24,27,23,46,69,68,
%U A363473 95,78,119,40,45,50,29,58,87,76,115,102,133,56,63,70,121,31,62,93,92,145,114,161,88,99,110,143,36
%N A363473 Triangle read by rows: T(n, k) = k * prime(n - k + A061395(k)) for 1 < k <= n, and T(n, 1) = A008578(n).
%C A363473 Conjecture: this is a permutation of the natural numbers.
%C A363473 Generalized conjecture: Let T(n, k) = b(k) * prime(n - k + A061395(b(k))) for 1 < k <= n, and T(n, 1) = A008578(n), where b(n), n > 0, is a permutation of the natural numbers with b(1) = 1, then T(n, k), read by rows, is a permutation of the natural numbers.
%F A363473 T(n, n) = A253560(n) for n > 0.
%F A363473 T(n, 1) = A008578(n) for n > 0.
%F A363473 T(n, 2) = A001747(n) for n > 1.
%F A363473 T(n, 3) = A112773(n) for n > 2.
%F A363473 T(n, 4) = A001749(n-3) for n > 3.
%F A363473 T(n, 5) = A001750(n-2) for n > 4.
%F A363473 T(n, 6) = A138636(n-4) for n > 5.
%F A363473 T(n, 7) = A272470(n-3) for n > 6.
%e A363473 Triangle begins:
%e A363473 n\k : 1 2 3 4 5 6 7 8 9 10 11 12 13
%e A363473 =====================================================================
%e A363473 1 : 1
%e A363473 2 : 2 4
%e A363473 3 : 3 6 9
%e A363473 4 : 5 10 15 8
%e A363473 5 : 7 14 21 12 25
%e A363473 6 : 11 22 33 20 35 18
%e A363473 7 : 13 26 39 28 55 30 49
%e A363473 8 : 17 34 51 44 65 42 77 16
%e A363473 9 : 19 38 57 52 85 66 91 24 27
%e A363473 10 : 23 46 69 68 95 78 119 40 45 50
%e A363473 11 : 29 58 87 76 115 102 133 56 63 70 121
%e A363473 12 : 31 62 93 92 145 114 161 88 99 110 143 36
%e A363473 13 : 37 74 111 116 155 138 203 104 117 130 187 60 169
%e A363473 etc.
%o A363473 (PARI)
%o A363473 T(n, k) = { if(k==1, if(n==1, 1, prime(n-1)), i=floor((k+1)/2);
%o A363473 while(k % prime(i) != 0, i=i-1); k*prime(n-k+i)) }
%o A363473 (SageMath)
%o A363473 def prime(n): return sloane.A000040(n)
%o A363473 def A061395(n): return prime_pi(factor(n)[-1][0]) if n > 1 else 0
%o A363473 def T(n, k):
%o A363473 if k == 1: return prime(n - 1) if n > 1 else 1
%o A363473 return k * prime(n - k + A061395(k))
%o A363473 for n in range(1, 11): print([T(n,k) for k in range(1, n+1)])
%o A363473 # _Peter Luschny_, Jan 07 2024
%Y A363473 Cf. A000040, A001747, A001749, A001750, A008578, A112773, A138636, A253560, A272470, A061395.
%K A363473 nonn,easy,tabl
%O A363473 1,2
%A A363473 _Werner Schulte_, Jan 05 2024