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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A246278 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)).

Original entry on oeis.org

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 27, 35, 49, 11, 12, 21, 125, 77, 121, 13, 14, 45, 55, 343, 143, 169, 17, 16, 33, 175, 91, 1331, 221, 289, 19, 18, 81, 65, 539, 187, 2197, 323, 361, 23, 20, 75, 625, 119, 1573, 247, 4913, 437, 529, 29, 22, 63, 245, 2401, 209, 2873, 391, 6859, 667, 841, 31
Offset: 2

Views

Author

Antti Karttunen, Aug 21 2014

Keywords

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.
This array can be obtained by taking every second column from array A242378, starting from its column 2.
Permutation of natural numbers larger than 1.
The terms on row n are all divisible by n-th prime, A000040(n).
Each column is strictly growing, and the terms in the same column have the same prime signature.
A055396(n) gives the row number of row where n occurs,
and A246277(n) gives its column number, both starting from 1.
From Antti Karttunen, Jan 03 2015: (Start)
A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.
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.
(End)

Examples

			The top left corner of the array:
   2,     4,     6,     8,    10,    12,    14,    16,    18, ...
   3,     9,    15,    27,    21,    45,    33,    81,    75, ...
   5,    25,    35,   125,    55,   175,    65,   625,   245, ...
   7,    49,    77,   343,    91,   539,   119,  2401,   847, ...
  11,   121,   143,  1331,   187,  1573,   209, 14641,  1859, ...
  13,   169,   221,  2197,   247,  2873,   299, 28561,  3757, ...

Links

Crossrefs

First row: A005843 (the even numbers), from 2 onward.
Row 2: A249734, Row 3: A249827.
Column 1: A000040 (primes), Column 2: A001248 (squares of primes), Column 3: A006094 (products of two successive primes), Column 4: A030078 (cubes of primes).
Transpose: A246279.
Inverse permutation: A252752.
One more than A246275.
Cf. A005940, A242378, A246259, A000040, A002260, A004736, A003961, A055396, A083221, A114537, A246277 (terms of A348717 halved), A246675, A246684, A249818, A252759, A253515.
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), A372562 (A252748), A372563 (A286385), A378979 (* deficiency, A033879), A379008 (* (probably), A294898), A379010 (* A000010, Euler phi), A379011 (* A083254).
Cf. A329050 (subtable).

Programs

  • Mathematica
    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 *)
  • Scheme
    (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.
(define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))

Formula

A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).
As a composition of other similar sequences:
a(n) = A122111(A253561(n)).
a(n) = A249818(A083221(n)).
For all n >= 1, a(n+1) = A005940(1+A253551(n)).
A(n, k) = A341606(n, k) * A355925(n, k). - Antti Karttunen, Jul 22 2022

Extensions

Starting offset of the linear sequence changed from 1 to 2, without affecting the column and row indices by Antti Karttunen, Jan 03 2015

A351722 a(n) is the number of permutations p of {1, 2, ..., 2*n} such that for any k in 1..2*n, k and p(k) do not share a common 1-bit.

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 9, 1, 1, 121, 1089, 729, 729, 1521, 169, 1, 1, 2601, 314721, 1771561, 15944049
Offset: 0

Views

Author

Rémy Sigrist, Apr 06 2022

Keywords

Comments

By the pigeonhole principle, and simply considering parities of k and p(k), there are no such permutation of {1, 2, ..., 2*n+1}.

Examples

			For n = 5:
- we have the following 9 permutations (shown in decimal and in binary):
  p\k  1 2 3 4  5 6 7 8 9 10 |    1   10  11 100  101  110  111 1000 1001 1010
  --- -----------------------+------------------------------------------------
  p1   6 5 4 3 10 9 8 7 2  1 |  110  101 100  11 1010 1001 1000  111   10    1
  p2  10 5 4 3  2 9 8 7 6  1 | 1010  101 100  11   10 1001 1000  111  110    1
  p3   2 5 4 3 10 9 8 7 6  1 |   10  101 100  11 1010 1001 1000  111  110    1
  p4   6 9 4 3 10 1 8 7 2  5 |  110 1001 100  11 1010    1 1000  111   10  101
  p5   6 1 4 3 10 9 8 7 2  5 |  110    1 100  11 1010 1001 1000  111   10  101
  p6  10 9 4 3  2 1 8 7 6  5 | 1010 1001 100  11   10    1 1000  111  110  101
  p7   2 9 4 3 10 1 8 7 6  5 |   10 1001 100  11 1010    1 1000  111  110  101
  p8  10 1 4 3  2 9 8 7 6  5 | 1010    1 100  11   10 1001 1000  111  110  101
  p9   2 1 4 3 10 9 8 7 6  5 |   10    1 100  11 1010 1001 1000  111  110  101
- so a(5) = 9.

Crossrefs

Cf. A000225, A005326, A109468, A253515, A352857.

Programs

  • PARI
    a(n) = matpermanent(matrix(2*n, 2*n, i,j, bitand(i,j)==0))

Formula

a(n) = 1 for any n in A000225 (the only solution is k -> 2*n+1-k).
a(2^k) = 1 for any k >= 0 (the only solution is row 2^k in A253515).
Showing 1-2 of 2 results.

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