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A308503
Square array T(n, k), n, k > 0, read by antidiagonals upwards: T(n, k) = f(g(n) | g(k)), where f is defined over the set of finite sequences of nonnegative integers with no trailing zero as f(e) = Sum_{k = 1..#e} prime(k)^e_k, g is the inverse of f, and | denotes concatenation.
1
1, 2, 2, 3, 6, 3, 4, 15, 10, 4, 5, 12, 21, 18, 5, 6, 35, 20, 75, 14, 6, 7, 30, 55, 36, 33, 30, 7, 8, 77, 42, 245, 28, 105, 22, 8, 9, 24, 91, 150, 65, 60, 39, 54, 9, 10, 45, 40, 847, 66, 385, 44, 375, 50, 10, 11, 70, 63, 72, 119, 210, 85, 108, 147, 42, 11, 12
OFFSET
1,2
COMMENTS
The function f establishes a natural bijection from the set of finite sequences of nonnegative integers with no trailing zero to the set of natural numbers based on prime factorization.
If we consider the set of finite sequences of signed integers with no trailing zero, then we obtain a bijection to the set of positive rational numbers.
The function g is defined by:
- g(1) = () (the empty sequence),
- g(n) = the n-th row of A067255 for any n > 1.
FORMULA
For any m, n, k > 0:
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 1) = T(1, n) = n (1 is a neutral element),
- T(2, k) = 2*A003961(k),
- h(T(n, k)) = h(n) + h(k) for h = A001221, A001222, A061395,
- the function i -> T(n, i)/n is completely multiplicative and equals the A061395(n)-th iterate of A003961.
EXAMPLE
Array T(n, k) begins:
n\k| 1 2 3 4 5 6 7 8 9 10
---+----------------------------------------------------
1| 1 2 3 4 5 6 7 8 9 10
2| 2 6 10 18 14 30 22 54 50 42
3| 3 15 21 75 33 105 39 375 147 165
4| 4 12 20 36 28 60 44 108 100 84
5| 5 35 55 245 65 385 85 1715 605 455
6| 6 30 42 150 66 210 78 750 294 330
7| 7 77 91 847 119 1001 133 9317 1183 1309
8| 8 24 40 72 56 120 88 216 200 168
9| 9 45 63 225 99 315 117 1125 441 495
10| 10 70 110 490 130 770 170 3430 1210 910
PROG
(PARI) T(n, k) = { my (e=concat(apply(m -> my (f=factor(m), w=#f~, v=vector(if (w, primepi(f[w, 1]), 0))); for (j=1, w, v[primepi(f[j, 1])]=f[j, 2]); v, [n, k]))); prod (i=1, #e, if (e[i], prime(i)^e[i], 1)) }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Jun 02 2019
STATUS
approved