A246259 Square array: row n contains in ascending order all natural numbers k for which A246271(k)+1 = n, i.e., numbers m for which we need n-1 additional iterations of A003961, starting from A003961(m), before the result is 1 modulo 4.

1, 3, 2, 4, 7, 5, 9, 8, 6, 66, 10, 15, 17, 70, 91, 11, 18, 20, 94, 197, 55, 12, 19, 24, 186, 259, 155, 21, 13, 22, 26, 187, 364, 220, 84, 46, 14, 28, 41, 199, 377, 238, 87, 184, 1362, 16, 29, 45, 237, 413, 467, 189, 414, 1981, 1654, 23, 32, 54, 262, 479, 495, 309, 445, 2378, 3055, 1419

Offset

1,2

Comments

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.

The topmost row, which has row number 1, is the sequence A246261 (numbers n such that A003961(n) is of the form 4k+1), as these are exactly the numbers that do not require any additional iterations of A003961 for the result to be of the form 4k+1 (because it already is of that form).

If a number k occurs in any particular row, then 4k, 9k, 16k, etc. also occur on the same row, because multiplying a number by a perfect square does not affect the result when it is reduced modulo 4.

The array has an infinite number of rows, provided that A246271 is not bounded, because if A246271(n) = k > 0 for some n, then A246271(A003961(n)) = k-1, thus all the preceding rows would also have terms. However, if A246271 really had an absolute maximum m, then we can consider the array to have just m rows. In either case, all the natural numbers occur in it, each just once, thus the sequence forms a permutation of natural numbers.

Applying A003961 to the terms of any row below the first row gives a subsequence of the immediately preceding row.

Links

Antti Karttunen, Table of n, a(n) for n = 1..120; the first 15 antidiagonals of the array

Index entries for sequences that are permutations of the natural numbers

Example

The top left corner of the array:
   1,   3,   4,   9,  10,  11,  12,  13,  14,  16,  23,  25, ...
   2,   7,   8,  15,  18,  19,  22,  28,  29,  32,  43,  50, ...
   5,   6,  17,  20,  24,  26,  41,  45,  54,  57,  61,  68, ...
  66,  70,  94, 186, 187, 199, 237, 262, 264, 278, 280, 286, ...
  91  197, 259, 364, 377, 413, 479, 627, 665, 669, 705, 763, ...
  55, 155, 220, 238, 467, 495, 497, 526, 535, 543, 620, 880, ...
  21,  84,  87, 189, 309, 336, 348, 358, 463, 525, 679, 756, ...
  ...
2 is the least number k such that A003961(k) = 3 modulo 4, but A003961(A003961(k)) = 1 modulo 4 (as indeed A003961(2) = 3, and A003961(3) = 5). Thus A(2,1) = 2.
7 is the second smallest number k satisfying the same property, as A003961(7) = 11 (= 3 mod 4) while A003961(11) = 13 (= 1 mod 4). Thus A(2,2) = 7.
8 is the third smallest number k satisfying the same property, as A003961(8)=27 ( = 3 mod 4) while A003961(27) = 125 = 1 mod 4. Thus A(2,3) = 8.
5 is the least number k such that both A003961(k) and A003961(A003961(k)) = 3 mod 4 but A003961(A003961(A003961(k))) = 1 mod 4. Indeed A003961(5) = 7, A003961(7) = 11 and only at A003961(11) = 13 = 1 mod 4. Thus A(3,1) = 5.

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define (A246259 n) (A246259bi (A002260 n) (A004736 n)))
(define (A246259bi row col) ((rowfun-for-A246259 row) col))
(definec (rowfun-for-A246259 row) (MATCHING-POS 1 1 (lambda (k) (= (- row 1) (A246271 k)))))

Crossrefs

Transpose: A246258.

First row: A246261.

First column: A246280.

Cf. A246278, A246271, A003961, A002260, A004736.

Sequence in context: A362978 A261147 A212952 * A105025 A129594 A214416

Adjacent sequences: A246256 A246257 A246258 * A246260 A246261 A246262

Keyword

nonn,tabl

Author

Antti Karttunen, Aug 22 2014

Status

approved