A373844 Triangle read by rows: T(n,k) = A276086(1 + A002110(n) + A002110(k)), 1 <= k <= n, where A276086 is the primorial base exp-function.

18, 30, 50, 42, 70, 98, 66, 110, 154, 242, 78, 130, 182, 286, 338, 102, 170, 238, 374, 442, 578, 114, 190, 266, 418, 494, 646, 722, 138, 230, 322, 506, 598, 782, 874, 1058, 174, 290, 406, 638, 754, 986, 1102, 1334, 1682, 186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922, 222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738

Offset: 1

Author

Antti Karttunen, Jun 21 2024

Keywords

• nonn
• easy
• tabl

Comments

Triangle giving all products of three primes, of which one is even (2) and two are odd (not necessarily distinct), so that the product is of the form 4m+2.

The only terms such that T(n, k) > A373845(n, k) > 1 are 30, 42, 110 at positions T(2,1), T(3,1), T(4,2), and the corresponding terms in A373845 are 6, 14, 38.

Examples

			Triangle begins as:
   18,
   30,  50,
   42,  70,  98,
   66, 110, 154, 242,
   78, 130, 182, 286, 338,
  102, 170, 238, 374, 442,  578,
  114, 190, 266, 418, 494,  646,  722,
  138, 230, 322, 506, 598,  782,  874, 1058,
  174, 290, 406, 638, 754,  986, 1102, 1334, 1682,
  186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922,
  222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738,
etc.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of the triangle

Crossrefs

Cf. A002110, A276086, A370121, A373845.

Cf. also A087112, A369979, A373848.

Programs

PARI

A002110(n) = prod(i=1,n,prime(i));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A373844(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A276086(1+(A002110(1+c)+x)); };

Formula

For n, k >= 1, T(n, k) = A276086(1+A370121(n, k)).

For n, k >= 1, T(n, k) = 2*A087112(n+1, k+1).