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%I #50 Sep 08 2022 14:06:22
%S 2,4,3,6,9,5,8,15,25,7,10,27,35,49,11,12,21,125,77,121,13,14,45,55,
%T 343,143,169,17,16,33,175,91,1331,221,289,19,18,81,65,539,187,2197,
%U 323,361,23,20,75,625,119,1573,247,4913,437,529,29,22,63,245,2401,209,2873,391,6859,667,841,31
%N Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).
%C The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
%C This array can be obtained by taking every second column from array A242378, starting from its column 2.
%C Permutation of natural numbers larger than 1.
%C The terms on row n are all divisible by n-th prime, A000040(n).
%C Each column is strictly growing, and the terms in the same column have the same prime signature.
%C A055396(n) gives the row number of row where n occurs,
%C and A246277(n) gives its column number, both starting from 1.
%C From _Antti Karttunen_, Jan 03 2015: (Start)
%C A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.
%C If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276.
%C (End)
%H Antti Karttunen, <a href="/A246278/b246278.txt">Table of n, a(n) for n = 2..1276; the first 50 antidiagonals of the array</a>
%F A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).
%F As a composition of other similar sequences:
%F a(n) = A122111(A253561(n)).
%F a(n) = A249818(A083221(n)).
%F For all n >= 1, a(n+1) = A005940(1+A253551(n)).
%F A(n, k) = A341606(n, k) * A355925(n, k). - _Antti Karttunen_, Jul 22 2022
%e The top left corner of the array:
%e 2, 4, 6, 8, 10, 12, 14, 16, 18, ...
%e 3, 9, 15, 27, 21, 45, 33, 81, 75, ...
%e 5, 25, 35, 125, 55, 175, 65, 625, 245, ...
%e 7, 49, 77, 343, 91, 539, 119, 2401, 847, ...
%e 11, 121, 143, 1331, 187, 1573, 209, 14641, 1859, ...
%e 13, 169, 221, 2197, 247, 2873, 299, 28561, 3757, ...
%t f[p_?PrimeQ] := f[p] = Prime[PrimePi@ p + 1]; f[1] = 1; f[n_] := f[n] = Times @@ (f[First@ #]^Last@ # &) /@ FactorInteger@ n; Block[{lim = 12}, Table[#[[n - k, k]], {n, 2, lim}, {k, n - 1, 1, -1}] &@ NestList[Map[f, #] &, Table[2 k, {k, lim}], lim]] // Flatten (* _Michael De Vlieger_, Jan 04 2016, after _Jean-François Alcover_ at A003961 *)
%o (Scheme)
%o (define (A246278 n) (if (<= n 1) n (A246278bi (A002260 (- n 1)) (A004736 (- n 1))))) ;; Square array starts with offset=2, and we have also tacitly defined a(1) = 1 here.
%o (define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))
%Y First row: A005843 (the even numbers), from 2 onward.
%Y Row 2: A249734, Row 3: A249827.
%Y Column 1: A000040 (primes), Column 2: A001248 (squares of primes), Column 3: A006094 (products of two successive primes), Column 4: A030078 (cubes of primes).
%Y Transpose: A246279.
%Y Inverse permutation: A252752.
%Y One more than A246275.
%Y Cf. A005940, A242378, A246259, A000040, A002260, A004736, A003961, A055396, A083221, A114537, A246277 (terms of A348717 halved), A246675, A246684, A249818, A252759, A253515.
%Y Arrays obtained by applying a particular function (given in parentheses) to the entries of this array. Cases where the columns grow monotonically are indicated with *: A249822 (A078898), A253551 (* A156552), A253561 (* A122111), A341605 (A017665), A341606 (A017666), A341607 (A006530 o A017666), A341608 (A341524), A341626 (A341526), A341627 (A341527), A341628 (A006530 o A341527), A342674 (A341530), A344027 (* A003415, arithmetic derivative), A355924 (A342671), A355925 (A009194), A355926 (A355442), A355927 (* sigma), A356155 (* A258851).
%Y Cf. A329050 (subtable).
%K nonn,tabl
%O 2,1
%A _Antti Karttunen_, Aug 21 2014
%E Starting offset of the linear sequence changed from 1 to 2, without affecting the column and row indices by _Antti Karttunen_, Jan 03 2015
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