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%I #14 Apr 08 2021 10:48:35
%S 1,1,1,1,2,1,1,3,3,1,1,2,5,2,1,1,5,3,3,5,1,1,6,7,4,7,6,1,1,7,15,5,5,
%T 15,7,1,1,2,11,6,11,6,11,2,1,1,3,3,7,35,35,7,3,3,1,1,10,5,4,13,30,13,
%U 4,5,10,1,1,11,21,9,5,77,77,5,9,21,11,1
%N Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime exponents of numbers (see Comments section for precise definition).
%C To compute T(n, k):
%C - write the prime exponents of n and of k on two lines, right aligned (these lines correspond to rows of A067255 in reversed order),
%C - to "multiply" two prime numbers: take the smallest,
%C - to "add" two prime numbers: take the largest,
%C - for example, for T(12, 14):
%C (11 7 5 3 2)
%C 12 --> 1 2
%C 14 --> x 1 0 0 1
%C ---------
%C 1 1
%C 0 0
%C 0 0
%C + 1 1
%C -----------
%C 1 1 0 1 1 --> 462 = T(12, 14)
%C This sequence is closely related to lunar multiplication (A087062):
%C - for any b > 1, let S_b be the set of nonnegative integers m such that A051903(m)< b,
%C - there is a natural bijection f from S_b to the set of nonnegative integers:
%C f(Product_{k >= 0} prime(k)^d(k)) = Sum_{k >= 0} d(k) * b^k,
%C - let g be the inverse of f,
%C - then for any numbers n and k in S_b, we have:
%C T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base b,
%C - the corresponding addition table is A003990.
%H Rémy Sigrist, <a href="/A343033/b343033.txt">Table of n, a(n) for n = 1..10011</a>
%H <a href="/index/Di#dismal">Index entries for sequences related to dismal (or lunar) arithmetic</a>
%F T(n, k) = T(k, n).
%F T(n, 1) = 1.
%F T(n, 2) = A007947(n).
%F T(n, 3) = A328915(n).
%F T(n, 4) = A007948(n).
%F T(n, n) = A343035(n).
%F A051903(T(n, k)) = min(A051903(n), A051903(k)).
%e Array T(n, k) begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e ---- - -- -- -- -- --- --- -- -- --- --- --- --- ---
%e 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 2| 1 2 3 2 5 6 7 2 3 10 11 6 13 14 --> A007947
%e 3| 1 3 5 3 7 15 11 3 5 21 13 15 17 33 --> A328915
%e 4| 1 2 3 4 5 6 7 4 9 10 11 12 13 14 --> A007948
%e 5| 1 5 7 5 11 35 13 5 7 55 17 35 19 65
%e 6| 1 6 15 6 35 30 77 6 15 210 143 30 221 462
%e 7| 1 7 11 7 13 77 17 7 11 91 19 77 23 119
%e 8| 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e 9| 1 3 5 9 7 15 11 9 25 21 13 45 17 33
%e 10| 1 10 21 10 55 210 91 10 21 110 187 210 247 910
%e 11| 1 11 13 11 17 143 19 11 13 187 23 143 29 209
%e 12| 1 6 15 12 35 30 77 12 45 210 143 60 221 462
%e 13| 1 13 17 13 19 221 23 13 17 247 29 221 31 299
%e 14| 1 14 33 14 65 462 119 14 33 910 209 462 299 238
%o (PARI) T(n,k) = { my (r=1, pp=factor(n)[,1]~, qq=factor(k)[,1]~); for (i=1, #pp, for (j=1, #qq, my (p=prime(primepi(pp[i])+primepi(qq[j])-1), v=valuation(r, p), w=min(valuation(n, pp[i]), valuation(k, qq[j]))); if (w>v, r*=p^(w-v)))); r }
%Y Cf. A003990, A007947, A007948, A051903, A067255, A087062, A328915, A342767, A343035.
%K nonn,tabl
%O 1,5
%A _Rémy Sigrist_ and _N. J. A. Sloane_, Apr 03 2021
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