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%I #51 Jan 03 2020 19:47:15
%S 1,2,4,3,7,6,5,17,13,8,11,59,41,19,9,31,277,179,67,23,10,127,1787,
%T 1063,331,83,29,12,709,15299,8527,2221,431,109,37,14,5381,167449,
%U 87803,19577,3001,599,157,43,15,52711,2269733,1128889,219613,27457,4397,919,191,47
%N Dispersion of the primes (an array read by downward antidiagonals).
%C 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 The dispersion of the composite numbers is given at A114577.
%D Alexandrov, Lubomir. "On the nonasymptotic prime number distribution." arXiv preprint math/9811096 (1998). (See Appendix.)
%D Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria, 45 (1997) 157-168.
%H Robert G. Wilson v, <a href="/A114537/b114537.txt">Table of n, a(n) for n = 1..172</a>
%H Neil Fernandez, <a href="http://www.borve.org/primeness/FOP.html">An order of primeness, F(p)</a>
%H Neil Fernandez, <a href="/A006450/a006450.html">An order of primeness</a> [cached copy, included with permission of the author]
%H Neil Fernandez, <a href="http://www.borve.org/primeness/intro.html">The Exploring Primeness Project</a>
%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions and Dispersions</a>.
%H 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 Robert G. Wilson v, <a href="/A114537/a114537_2.txt">The Northwest Corner of the Primeness Array (24 x 24).</a>
%F T(r,1) = A018252(r). T(r,c) = prime(T(r,c-1)), c>1. [_R. J. Mathar_, Oct 22 2010]
%e Northwest corner of the Primeness array:
%e 1 2 3 5 11 31 127 709 5381 52711 648391
%e 4 7 17 59 277 1787 15299 167449 2269733 37139213 718064159
%e 6 13 41 179 1063 8527 87803 1128889 17624813 326851121 7069067389
%e 8 19 67 331 2221 19577 219613 3042161 50728129 997525853 22742734291
%e 9 23 83 431 3001 27457 318211 4535189 77557187 1559861749 36294260117
%e 10 29 109 599 4397 42043 506683 7474967 131807699 2724711961 64988430769
%e 12 37 157 919 7193 72727 919913 14161729 259336153 5545806481 136395369829
%e 14 43 191 1153 9319 96797 1254739 19734581 368345293 8012791231 200147986693
%e 15 47 211 1297 10631 112129 1471343 23391799 440817757 9672485827 243504973489
%e 16 53 241 1523 12763 137077 1828669 29499439 563167303 12501968177 318083817907
%e 18 61 283 1847 15823 173867 2364361 38790341 751783477 16917026909 435748987787
%e 20 71 353 2381 21179 239489 3338989 56011909 1107276647 25366202179 664090238153
%e 21 73 367 2477 22093 250751 3509299 59053067 1170710369 26887732891 705555301183
%e 22 79 401 2749 24859 285191 4030889 68425619 1367161723 31621854169 835122557939
%e 24 89 461 3259 30133 352007 5054303 87019979 1760768239 41192432219 1099216100167
%e 25 97 509 3637 33967 401519 5823667 101146501 2062666783 48596930311 1305164025929
%e 26 101 547 3943 37217 443419 6478961 113256643 2323114841 55022031709 1484830174901
%e 27 103 563 4091 38833 464939 6816631 119535373 2458721501 58379844161 1579041544637
%p 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 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 (* or to view the table *) Table[t[n, k], {n, 0, 6}, {k, 0, 10}] // TableForm (* _Robert G. Wilson v_, Dec 26 2005 *)
%Y Columns 1-13: A018252, A007821, A049078, A049079, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046.
%Y Rows 1-7: A007097, A057450, A057451, A057452, A057453, A057456, A057457.
%Y Diagonal: A181441.
%Y Cf. A000040, A007821, A114538, A006450.
%Y If the antidiagonals are read in the opposite direction we get A138947.
%K nonn,tabl,nice
%O 1,2
%A _Clark Kimberling_, Dec 07 2005
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