A363473 - OEIS

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

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OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..78.

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.

EXAMPLE

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.

PROG

(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

CROSSREFS

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

Sequence in context: A083221 A246278 A359299 * A361748 A377093 A372337

Adjacent sequences: A363470 A363471 A363472 * A363474 A363475 A363476

KEYWORD

nonn,easy,tabl

AUTHOR

Werner Schulte, Jan 05 2024

STATUS

approved