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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.
3
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
1,1
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.
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).
EXAMPLE
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.
PROG
(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)); };
CROSSREFS
KEYWORD
nonn,easy,tabl
AUTHOR
Antti Karttunen, Jun 21 2024
STATUS
approved