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A138947 Square array T[i+1,j] = prime(T[i,j]), T[1,j] = j-th nonprime = A018252(j); read by upward antidiagonals.

%I A138947 #17 Jan 03 2020 19:46:19
%S A138947 1,4,2,6,7,3,8,13,17,5,9,19,41,59,11,10,23,67,179,277,31,12,29,83,331,
%T A138947 1063,1787,127,14,37,109,431,2221,8527,15299,709,15,43,157,599,3001,
%U A138947 19577,87803
%N A138947 Square array T[i+1,j] = prime(T[i,j]), T[1,j] = j-th nonprime = A018252(j); read by upward antidiagonals.
%C A138947 For i>1, T[i,j] = A018252(j)-th number among those not occurring in rows < i.
%C A138947 A permutation of the integers > 0.
%C A138947 Transpose of A114537. See that sequence and the link for more information and references.
%D A138947 Alexandrov, Lubomir. "On the nonasymptotic prime number distribution." arXiv preprint math/9811096 (1998). (See Appendix.)
%H A138947 N. Fernandez, <a href="http://www.borve.org/primeness/FOP.html">An order of primeness, F(p)</a>.
%H A138947 N. Fernandez, <a href="/A006450/a006450.html">An order of primeness</a> [cached copy, included with permission of the author]
%F A138947 T[i,j] = j-th number for which A078442 equals i-1.
%e A138947 The first row (1,4,6,8,9,10...) of the array gives the nonprime numbers A018252.
%e A138947 The 2nd row (2,7,13,19,23,29,37,...) of the array gives the primes with nonprime index, A000040(A018252(j)) = A007821(j).
%e A138947 The i-th row is { A000040(k) | A049076(k)=i-1 } = A078442^{-1}(i-1).
%e A138947 Column j is the sequence b(n+1)=prime(b(n)) starting with b(j)=A018252(j): A007097, A057450, A057451, A057452, A057453, A057456, A057457, ...
%t A138947 t[1, 1] = 1; t[1, 2] = 4; t[1, k_] := (p = t[1, k-1]; If[ PrimeQ[p+1], p+2, p+1]); t[n_ /; n > 1, k_] := Prime[t[n-1, k]]; Flatten[ Table[ t[n, k-n+1], {k, 1, 9}, {n, 1, k}]] (* _Jean-François Alcover_, Oct 03 2011 *)
%o A138947 (PARI) p=c=0; T=matrix( 10,10, i,j, if( i==1, while( isprime(c++),); p=c, p=prime(p))); A138947=concat( vector( vecmin( matsize( T )),i, vector( i,j, T[ j,i+1-j ])))
%Y A138947 Cf. A018252, A007821, A006450, A049076, A007097.
%Y A138947 If the antidiagonals are read in the opposite direction we get A114537.
%K A138947 nonn,tabl
%O A138947 1,2
%A A138947 _M. F. Hasler_, Apr 28 2008