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A374376
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Array read by downward antidiagonals: T(k,n) is the least number that has k prime factors (counted with multiplicity) and is the concatenation of n primes, or -1 if there is no such number.
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2
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2, 23, -1, 223, 22, -1, 2237, 235, 27, -1, 22273, 2227, 222, 132, -1, 222323, 22223, 2222, 225, 32, -1, 2222273, 222223, 22222, 2223, 252, 729, -1, 22222223, 2222557, 222227, 22225, 2322, 352, 192, -1, 222222227, 22222237, 2222222, 222225, 22232, 2232, 2352, 2112, -1, 2222222377, 222222223
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OFFSET
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1,1
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REFERENCES
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T(k,1) = -1 for k > 1.
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LINKS
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EXAMPLE
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Array starts
2 23 223 2237 22273 ...
-1 22 235 2227 22223 ...
-1 27 222 2222 22222 ...
-1 132 225 2223 22225 ...
-1 32 252 2322 22232 ...
A(4,3) = 225 because 225 = 3^2 * 5^2 is the product of 4 primes (with multiplicity) and is the concatenation of the 3 primes 2, 2 and 5, and is the least number that works.
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MAPLE
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PD[1]:= [2, 3, 5, 7]:
for i from 2 to 7 do PD[i]:= select(isprime, [seq(i, i=10^(i-1)+1..10^i-1, 2)]) od:
dcat:= proc(a, b) 10^(ilog10(b)+1)*a+b end proc:
cp:= proc(m, n) option remember; local d, p, x, R;
if n = 1 then return PD[m] fi;
R:= {};
for d from 1 to m-n+1 do
R:= R union {seq(seq(dcat(p, x), p=PD[d]), x=procname(m-d, n-1))}
od;
R
end proc:
F:= proc(n, N)
local V, count, d, x, v;
if n = 1 then return <2, (-1)$(N-1)> fi;
V:= Vector(N); count:= 0;
for d from n while count < N do
for x in sort(convert(cp(d, n), list)) while count < N do
v:= numtheory:-bigomega(x);
if v <= N and V[v] = 0 then
V[v]:= x; count:= count+1;
fi
od od:
V;
end proc:
N:= 10: M:= Matrix(N, N):
for i from 1 to N do
V:= F(i, N+1-i);
M[i, 1..N+1-i]:= V;
od:
[seq(seq(M[t-i, i], i=1..t-1), t=2..N+1)];
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CROSSREFS
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KEYWORD
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sign,base
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AUTHOR
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STATUS
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approved
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