A363473 Triangle read by rows: T(n, k) = k * prime(n - k + A061395(k)) for 1 < k <= n, and T(n, 1) = A008578(n).

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, 30, 49, 17, 34, 51, 44, 65, 42, 77, 16, 19, 38, 57, 52, 85, 66, 91, 24, 27, 23, 46, 69, 68, 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

Offset: 1

Author

Werner Schulte, Jan 05 2024

Keywords

• nonn
• easy
• tabl

Comments

Conjecture: this is a permutation of the natural numbers.

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.

Examples

			Triangle begins:
n\k :   1    2    3    4    5    6    7    8    9   10   11   12   13
=====================================================================
 1  :   1
 2  :   2    4
 3  :   3    6    9
 4  :   5   10   15    8
 5  :   7   14   21   12   25
 6  :  11   22   33   20   35   18
 7  :  13   26   39   28   55   30   49
 8  :  17   34   51   44   65   42   77   16
 9  :  19   38   57   52   85   66   91   24   27
10  :  23   46   69   68   95   78  119   40   45   50
11  :  29   58   87   76  115  102  133   56   63   70  121
12  :  31   62   93   92  145  114  161   88   99  110  143   36
13  :  37   74  111  116  155  138  203  104  117  130  187   60  169
etc.
		

Crossrefs

Cf. A000040, A001747, A001749, A001750, A008578, A112773, A138636, A253560, A272470, A061395.

Programs

PARI

T(n, k) = { if(k==1, if(n==1, 1, prime(n-1)), i=floor((k+1)/2);
            while(k % prime(i) != 0, i=i-1); k*prime(n-k+i)) }

SageMath

def prime(n): return sloane.A000040(n)
def A061395(n): return prime_pi(factor(n)[-1][0]) if n > 1 else 0
def T(n, k):
     if k == 1: return prime(n - 1) if n > 1 else 1
     return k * prime(n - k + A061395(k))
for n in range(1, 11): print([T(n,k) for k in range(1, n+1)])
# Peter Luschny, Jan 07 2024

Formula

T(n, n) = A253560(n) for n > 0.

T(n, 1) = A008578(n) for n > 0.

T(n, 2) = A001747(n) for n > 1.

T(n, 3) = A112773(n) for n > 2.

T(n, 4) = A001749(n-3) for n > 3.

T(n, 5) = A001750(n-2) for n > 4.

T(n, 6) = A138636(n-4) for n > 5.

T(n, 7) = A272470(n-3) for n > 6.