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A114537 Dispersion of the primes (an array read by downward antidiagonals).

%I A114537 #51 Jan 03 2020 19:47:15
%S A114537 1,2,4,3,7,6,5,17,13,8,11,59,41,19,9,31,277,179,67,23,10,127,1787,
%T A114537 1063,331,83,29,12,709,15299,8527,2221,431,109,37,14,5381,167449,
%U A114537 87803,19577,3001,599,157,43,15,52711,2269733,1128889,219613,27457,4397,919,191,47
%N A114537 Dispersion of the primes (an array read by downward antidiagonals).
%C A114537 A number is prime if and only if it does not lie in Column 1. As a sequence, a permutation of the natural numbers. The fractal sequence of this dispersion is A022447 and the transposition sequence is A114538.
%C A114537 The dispersion of the composite numbers is given at A114577.
%D A114537 Alexandrov, Lubomir. "On the nonasymptotic prime number distribution." arXiv preprint math/9811096 (1998). (See Appendix.)
%D A114537 Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria, 45 (1997) 157-168.
%H A114537 Robert G. Wilson v, <a href="/A114537/b114537.txt">Table of n, a(n) for n = 1..172</a>
%H A114537 Neil Fernandez, <a href="http://www.borve.org/primeness/FOP.html">An order of primeness, F(p)</a>
%H A114537 Neil Fernandez, <a href="/A006450/a006450.html">An order of primeness</a> [cached copy, included with permission of the author]
%H A114537 Neil Fernandez, <a href="http://www.borve.org/primeness/intro.html">The Exploring Primeness Project</a>
%H A114537 Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions and Dispersions</a>.
%H A114537 Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
%H A114537 Robert G. Wilson v, <a href="/A114537/a114537_2.txt">The Northwest Corner of the Primeness Array (24 x 24).</a>
%F A114537 T(r,1) = A018252(r). T(r,c) = prime(T(r,c-1)), c>1. [_R. J. Mathar_, Oct 22 2010]
%e A114537 Northwest corner of the Primeness array:
%e A114537 1   2   3    5    11     31     127       709       5381       52711        648391
%e A114537 4   7  17   59   277   1787   15299    167449    2269733    37139213     718064159
%e A114537 6  13  41  179  1063   8527   87803   1128889   17624813   326851121    7069067389
%e A114537 8  19  67  331  2221  19577  219613   3042161   50728129   997525853   22742734291
%e A114537 9  23  83  431  3001  27457  318211   4535189   77557187  1559861749   36294260117
%e A114537 10  29 109  599  4397  42043  506683   7474967  131807699  2724711961   64988430769
%e A114537 12  37 157  919  7193  72727  919913  14161729  259336153  5545806481  136395369829
%e A114537 14  43 191 1153  9319  96797 1254739  19734581  368345293  8012791231  200147986693
%e A114537 15  47 211 1297 10631 112129 1471343  23391799  440817757  9672485827  243504973489
%e A114537 16  53 241 1523 12763 137077 1828669  29499439  563167303 12501968177  318083817907
%e A114537 18  61 283 1847 15823 173867 2364361  38790341  751783477 16917026909  435748987787
%e A114537 20  71 353 2381 21179 239489 3338989  56011909 1107276647 25366202179  664090238153
%e A114537 21  73 367 2477 22093 250751 3509299  59053067 1170710369 26887732891  705555301183
%e A114537 22  79 401 2749 24859 285191 4030889  68425619 1367161723 31621854169  835122557939
%e A114537 24  89 461 3259 30133 352007 5054303  87019979 1760768239 41192432219 1099216100167
%e A114537 25  97 509 3637 33967 401519 5823667 101146501 2062666783 48596930311 1305164025929
%e A114537 26 101 547 3943 37217 443419 6478961 113256643 2323114841 55022031709 1484830174901
%e A114537 27 103 563 4091 38833 464939 6816631 119535373 2458721501 58379844161 1579041544637
%p A114537 A114537 := proc(r,c) option remember; if c = 1 then A018252(r) ; else ithprime(procname(r,c-1)) ; end if; end proc: # _R. J. Mathar_, Oct 22 2010
%t A114537 NonPrime[n_] := FixedPoint[n + PrimePi@# + 1 &, n]; t[n_, k_] := Nest[Prime, NonPrime[n], k]; Table[ t[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten
%t A114537 (* or to view the table *) Table[t[n, k], {n, 0, 6}, {k, 0, 10}] // TableForm (* _Robert G. Wilson v_, Dec 26 2005 *)
%Y A114537 Columns 1-13: A018252, A007821, A049078, A049079, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046.
%Y A114537 Rows 1-7: A007097, A057450, A057451, A057452, A057453, A057456, A057457.
%Y A114537 Diagonal: A181441.
%Y A114537 Cf. A000040, A007821, A114538, A006450.
%Y A114537 If the antidiagonals are read in the opposite direction we get A138947.
%K A114537 nonn,tabl,nice
%O A114537 1,2
%A A114537 _Clark Kimberling_, Dec 07 2005