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%I #39 Jan 07 2024 14:03:47
%S 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 30,49,17,34,51,44,65,42,77,16,19,38,57,52,85,66,91,24,27,23,46,69,68,
%U 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 Triangle read by rows: T(n, k) = k * prime(n - k + A061395(k)) for 1 < k <= n, and T(n, 1) = A008578(n).
%C Conjecture: this is a permutation of the natural numbers.
%C 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 T(n, n) = A253560(n) for n > 0.
%F T(n, 1) = A008578(n) for n > 0.
%F T(n, 2) = A001747(n) for n > 1.
%F T(n, 3) = A112773(n) for n > 2.
%F T(n, 4) = A001749(n-3) for n > 3.
%F T(n, 5) = A001750(n-2) for n > 4.
%F T(n, 6) = A138636(n-4) for n > 5.
%F T(n, 7) = A272470(n-3) for n > 6.
%e Triangle begins:
%e n\k : 1 2 3 4 5 6 7 8 9 10 11 12 13
%e =====================================================================
%e 1 : 1
%e 2 : 2 4
%e 3 : 3 6 9
%e 4 : 5 10 15 8
%e 5 : 7 14 21 12 25
%e 6 : 11 22 33 20 35 18
%e 7 : 13 26 39 28 55 30 49
%e 8 : 17 34 51 44 65 42 77 16
%e 9 : 19 38 57 52 85 66 91 24 27
%e 10 : 23 46 69 68 95 78 119 40 45 50
%e 11 : 29 58 87 76 115 102 133 56 63 70 121
%e 12 : 31 62 93 92 145 114 161 88 99 110 143 36
%e 13 : 37 74 111 116 155 138 203 104 117 130 187 60 169
%e etc.
%o (PARI)
%o T(n, k) = { if(k==1, if(n==1, 1, prime(n-1)), i=floor((k+1)/2);
%o while(k % prime(i) != 0, i=i-1); k*prime(n-k+i)) }
%o (SageMath)
%o def prime(n): return sloane.A000040(n)
%o def A061395(n): return prime_pi(factor(n)[-1][0]) if n > 1 else 0
%o def T(n, k):
%o if k == 1: return prime(n - 1) if n > 1 else 1
%o return k * prime(n - k + A061395(k))
%o for n in range(1, 11): print([T(n,k) for k in range(1, n+1)])
%o # _Peter Luschny_, Jan 07 2024
%Y Cf. A000040, A001747, A001749, A001750, A008578, A112773, A138636, A253560, A272470, A061395.
%K nonn,easy,tabl
%O 1,2
%A _Werner Schulte_, Jan 05 2024