A001750 -id:A001750 - cp's OEIS frontend

A163629 Numbers such that the two adjacent integers are a prime and 5 times a prime.

14, 16, 24, 36, 54, 66, 84, 96, 114, 156, 234, 264, 294, 306, 336, 354, 366, 396, 444, 486, 504, 546, 564, 654, 684, 744, 756, 786, 864, 906, 954, 966, 984, 996, 1116, 1164, 1194, 1284, 1554, 1566, 1584, 1656, 1734, 1746, 1866, 1914, 1986, 2004, 2154, 2214

Offset: 1

Juri-Stepan Gerasimov, Aug 02 2009

nonn

			a(1)=14 which lies between 13=A000040(6) and 15=A001750(2).
a(2)=16 which lies between 15=A001750(2) and 17=A000040(7).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A163492.

prQ[n_]:=Module[{a=n-1,b=n+1},(PrimeQ[a]&&PrimeQ[b/5])||(PrimeQ[b] && PrimeQ[a/5])]; Select[Range[3000],prQ] (* Harvey P. Dale, Jun 05 2012 *)

114, 444 added, 1303 removed, 1666 replaced by 1656 - R. J. Mathar, Aug 07 2009

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

Original entry on oeis.org

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

Werner Schulte, Jan 05 2024

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.

			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.

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

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)) }

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

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.

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A163629 Numbers such that the two adjacent integers are a prime and 5 times a prime.

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

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