A083140 -id:A083140 - cp's OEIS frontend

A114881 Sieve of Eratosthenes minus 1.

1, 2, 3, 4, 8, 5, 6, 24, 14, 7, 10, 48, 34, 20, 9, 12, 120, 76, 54, 26, 11, 16, 168, 142, 90, 64, 32, 13, 18, 288, 220, 186, 118, 84, 38, 15, 22, 360, 322, 246, 208, 132, 94, 44, 17, 28, 528, 436, 390, 298, 252, 160, 114, 50, 19, 30

Offset: 1

Clark Kimberling, Jan 03 2006

A permutation of the natural numbers; row 1 consists of all odds.

			Northwest corner:
1 3 5 7 9
2 8 14 20 26
4 24 34 54 64
6 48 76 90 118

Cf. A083140, A114882, A114883.

A083140(n) - 1

A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 24, 13, 20, 11, 10, 17, 26, 27, 34, 29, 44, 47, 48, 25, 38, 39, 54, 21, 32, 19, 12, 33, 50, 51, 64, 53, 80, 67, 76, 57, 86, 87, 114, 93, 140, 95, 120, 49, 74, 75, 94, 77, 116, 107, 90, 41, 62, 63, 84, 37, 56, 23, 16, 65, 98, 99, 124, 101, 152, 127, 118, 105, 158, 159, 204, 133, 200, 151, 142

Offset: 1

Antti Karttunen, Nov 06 2014

This sequence is a "recursed variant" of A249811.
From Antti Karttunen, Jan 18 2015: (Start)
This can be viewed as an entanglement or encoding permutation where the complementary pairs of sequences to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with another complementary pair: even numbers in the order they appear in A253886 and odd numbers in their usual order: (A253886/A005408).
From the above follows also that this sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent and subtracting one, and each child to the right is obtained by applying A253886 to the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........8                   7......../ \........6
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       14         15       24         13       20         11       10
17 26   27  34     29  44   47  48     25  38   39  54     21  32   19  12
(End)
For listening I recommend some (mostly) percussive MIDI-instrument and the pitch offset set to at least 29 and the tempo (rate) to about 60. - Antti Karttunen, Feb 17 2015

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A249813.
Similar or related permutations: A246684, A249811, A250244, A252755.
Cf. A000079, A000265, A001511, A003602, A006093, A083140, A083221, A249741, A250469, A253886.
Compare also the scatterplot of this sequence to the graphs of A252755 and A246684.
Differs from A246684 for the first time at n=14, where a(14) = 20, while A246684(14) = 26.

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(1) = 1, for n>1: a(n) = A083221(A001511(n), a(A003602(n))) - 1 = A249741(A001511(n), a(A003602(n))).
a(1) = 1, a(2n) = A253886(a(n)), a(2n+1) = (2*a(n+1))-1. - Antti Karttunen, Jan 18 2015
As a composition of other permutations:
a(n) = A250244(A246684(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A000079(n-1)) = A006093(n).

A249820 a(1) = 0 and for n > 1: a(n) = A249810(n) - A078898(n) = A078898(A003961(n)) - A078898(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 2, 0, -1, 0, 6, 0, 4, 0, 1, 0, -4, 0, 11, 0, -4, 4, 3, 0, 3, 0, 25, -1, -7, 0, 20, 0, -7, -1, 12, 0, 7, 0, -2, 4, -8, 0, 44, 0, 0, -2, 0, 0, 36, 0, 22, -2, -13, 0, 23, 0, -12, 8, 90, 0, 0, 0, -5, -2, 4, 0, 77, 0, -16, 4, -3, 0, 4, 0, 55, 28, -19, 0, 41, 0, -19, -4, 15, 0, 43, 0, -2, -3, -20, 0, 155, 0, 12, 5, 24, 0

Offset: 1

Antti Karttunen, Dec 08 2014

sign

a(n) tells how many columns off A003961(n) is from the column where n is in square array A083221 (Cf. A083140, the sieve of Eratosthenes. The column index of n in that table is given by A078898(n)).

			For n = 8 = 2*2*2, A003961(8) = 27 (3*3*3), and while 8 is on row 1 and column 4 of A083221, 27 on the next row is in column 5, thus a(8) = 5 - 4 = 1.
For n = 10 = 2*5, A003961(10) = 21 (3*7), and while 10 is on row 1 and column 5 of A083221, 21 on the next row is in column 4, thus a(10) = 4 - 5 = -1.

Antti Karttunen, Table of n, a(n) for n = 1..3071

Cf. A003961, A078898, A083221, A083140, A246277, A249810, A249817, A249818, A249821, A249822, A251721, A251722.

(define (A249820 n) (if (= 1 n) 0 (- (A249810 n) (A078898 n))))

a(n) = A249810(n) - A078898(n) = A078898(A003961(n)) - A078898(n).

a(k) = 0 when k is a prime or square of prime, among some other numbers.

A249735 Odd bisection of A003961: Replace in 2n-1 each prime factor p(k) with prime p(k+1).

Original entry on oeis.org

1, 5, 7, 11, 25, 13, 17, 35, 19, 23, 55, 29, 49, 125, 31, 37, 65, 77, 41, 85, 43, 47, 175, 53, 121, 95, 59, 91, 115, 61, 67, 275, 119, 71, 145, 73, 79, 245, 143, 83, 625, 89, 133, 155, 97, 187, 185, 161, 101, 325, 103, 107, 385, 109, 113, 205, 127, 203, 425, 209, 169, 215, 343, 131, 235, 137, 253, 875, 139, 149, 265, 221, 217, 605, 151

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

This has the same terms as A007310 (Numbers congruent to 1 or 5 mod 6), but in different order. Apart from 1, they are the numbers that occur below the first two rows of arrays like A246278 and A083221 (A083140).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A249734 (the other bisection of A003961).

Cf. also A007310 (A038179), A249746.

(define (A249735 n) (A003961 (+ n n -1)))

a(n) = A003961(2n - 1).
a(n) = A007310(A249746(n)). [Permutation of A007310, Numbers congruent to 1 or 5 mod 6.]
Other identities. For all n >= 1:
A007310(n) = a(A249745(n)).
A246277(5*a(A048673(n))) = n.
A246277(5*a(n)) = A064216(n).

A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 7, 10, 9, 16, 11, 14, 15, 12, 32, 13, 22, 21, 20, 18, 64, 17, 26, 33, 28, 25, 24, 128, 19, 34, 39, 44, 35, 30, 27, 256, 23, 38, 51, 52, 55, 42, 40, 36, 512, 29, 46, 57, 68, 65, 66, 49, 45, 48, 1024, 31, 58, 69, 76, 85, 78, 77, 56, 50, 54, 2048, 37, 62, 87, 92

Offset: 1

Leroy Quet, Jan 27 2007

This sequence is a permutation of the integers >= 2.
Since the table has been entered by rising instead of falling antidiagonals, the sequence represents the transpose, with columns instead of rows: cf. the "table" link, section "infinite square array". - M. F. Hasler, Oct 22 2019
Start with table headed by primes in the first row, then list beneath each prime(k) the ordered prime(k)-smooth numbers. Read the table by falling antidiagonals to get the terms of this sequence. - David James Sycamore, Jun 23 2024

			Array begins: (rows here appear as columns in the "table" display of the sequence)
   2,  4,  8, 16, 32, 64, 128, 256, 512, ... (A000079)
   3,  6,  9, 12, 18, 24,  27,  36,  48, ... (A065119)
   5, 10, 15, 20, 25, 30,  40,  45,  50, ... (A080193)
   7, 14, 21, 28, 35, 42,  49,  56,  63, ... (A080194)
  11, 22, 33, 44, 55, 66,  77,  88,  99, ... (A080195)
  13, 26, 39, 52, 65, 78,  91, 104, 117, ... (A080196)
The 3rd row, for example, contains the positive integers where the 3rd prime, 5, is the largest prime divisor. That is, each integer in this row is divisible by 5 and may be divisible by 2 or 3 as well, but none of the integers in this row are divisible by primes larger than 5. (So for example, 35 = 5*7 is excluded from the 3rd row.)

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A083140, A006530, A000040 (1st col), A033286 (main diag), A077320.

Cf. A000079, A003586, A051037, A002473, A051038, A080197, A080681.

lpf[n_] := FactorInteger[n][[ -1, 1]];f[n_, m_] := f[n, m] = Block[{k},k = If[m == 1, Prime[n], f[n, m - 1] + 1];While[lpf[k] != Prime[n], k++ ];k];Table[f[ d - m + 1, m], {d, 12}, {m, d}] // Flatten (* Ray Chandler, Feb 09 2007 *)

T=List(); r=c=1; for(n=1,99, #TT[r][1], ); print1(T[r][c]","); r-- && c++ || r=c+c=1) \\ M. F. Hasler, Oct 22 2019

Extended by Ray Chandler, Feb 09 2007

A249811 Permutation of natural numbers: a(n) = A249741(A001511(n), A003602(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 11, 24, 13, 20, 15, 10, 17, 26, 19, 34, 21, 32, 23, 48, 25, 38, 27, 54, 29, 44, 31, 12, 33, 50, 35, 64, 37, 56, 39, 76, 41, 62, 43, 84, 45, 68, 47, 120, 49, 74, 51, 94, 53, 80, 55, 90, 57, 86, 59, 114, 61, 92, 63, 16, 65, 98, 67, 124, 69, 104, 71, 118, 73, 110, 75, 144, 77, 116, 79, 142, 81

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

In the essence, a(n) tells which number in square array A249741 (the sieve of Eratosthenes minus 1) is at the same position where n is in array A135764, which is formed from odd numbers whose binary expansions are shifted successively leftwards on the successive rows. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.

Equally: a(n) tells which number in array A114881 is at the same position where n is in the array A054582, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A249812.
Cf. A000079, A000265, A001511, A003602, A005408, A006093, A007814, A054582, A083140, A083221, A249741.
Similar or related permutations: A249814 ("deep variant"), A246676, A249815, A114881, A209268, A249725, A249741.
Differs from A246676 for the first time at n=14, where a(14)=20, while
A246676(14)=26.

(define (A249811 n) (+ -1 (A083221bi (A001511 n) (A003602 n))))
;; Code for A083221bi given in A083221

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(n) = A083221(A001511(n),A003602(n)) - 1 = A249741(A001511(n),A003602(n)).
As a composition of related permutations:
a(n) = A114881(A209268(n)).
a(n) = A249741(A249725(n)).
a(n) = A249815(A246676(n)).
Other identities. For all n >= 1 the following holds:
a(A000079(n-1)) = A006093(n).

A250472 Permutation of natural numbers: a(n) = A250470(2*n - 1).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 8, 19, 9, 10, 23, 29, 12, 15, 31, 14, 37, 41, 16, 43, 25, 18, 47, 21, 20, 53, 59, 22, 27, 61, 24, 67, 71, 26, 35, 73, 28, 79, 33, 30, 83, 55, 32, 39, 89, 34, 97, 101, 36, 103, 107, 38, 109, 45, 40, 65, 49, 42, 51, 113, 44, 127, 85, 46, 131, 137, 48, 77, 57, 50, 139, 149, 52, 63, 151, 54, 95, 157, 56, 163, 121, 58, 167, 69, 60

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

For n > 1, a(n) tells which number is located immediately above n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains 2n - 1.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Inverse: A250471.
Odd bisection of A250470. The other bisection: A250479.
Cf. A055396, A064216, A078898, A083140, A083221, A114881, A114883.

a(1) = 1, a(n) = A083221(A055396(2*n - 1)-1, A078898(2*n - 1)).

a(n) = A250470(2*n - 1).

A246279 Transpose of array A246278: A(row,1) = 2*row, and for col > 1, A(row,col) = A003961(A(row,col-1)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 21 2014

Permutation of natural numbers larger than 1.

See comments in A246278 (the same array transposed).

			The top-left corner of the array:
   2,     3,     5,     7,    11,    13,    17,    19,    23, ...
   4,     9,    25,    49,   121,   169,   289,   361,   529, ...
   6,    15,    35,    77,   143,   221,   323,   437,   667, ...
   8,    27,   125,   343,  1331,  2197,  4913,  6859, 12167, ...
  10,    21,    55,    91,   187,   247,   391,   551,   713, ...
  12,    45,   175,   539,  1573,  2873,  5491,  8303, 15341, ......

Transpose of A246278.
One more than A246273.
Cf. A003961, A083140.

(define (A246279 n) (A246278bi (A004736 n) (A002260 n)))
(define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))

If col = 1, A(row,col) = 2*row, otherwise A(row,col) = A003961(A(row,col-1)).

A249744 a(n) = 0 if n is 1 or a prime, otherwise, when n = A020639(n) * A032742(n), a(n) = the largest m < n such that A020639(m) = A020639(n), where A020639(n) and A032742(n) are the smallest prime and the largest proper divisor dividing n.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 0, 6, 3, 8, 0, 10, 0, 12, 9, 14, 0, 16, 0, 18, 15, 20, 0, 22, 5, 24, 21, 26, 0, 28, 0, 30, 27, 32, 25, 34, 0, 36, 33, 38, 0, 40, 0, 42, 39, 44, 0, 46, 7, 48, 45, 50, 0, 52, 35, 54, 51, 56, 0, 58, 0, 60, 57, 62, 55, 64, 0, 66, 63, 68, 0, 70, 0, 72, 69, 74, 49, 76, 0, 78, 75, 80, 0, 82, 65, 84, 81, 86, 0, 88, 77, 90, 87, 92, 85, 94, 0, 96, 93, 98, 0, 100

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

For all composite numbers, a(n) tells what is the previous number processed by the sieve of Eratosthenes, i.e., number which is immediately left of n on the same row where n is in arrays like A083140, A083221.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Can be used to compute A078898.

Cf. A000040, A001248, A020639, A083140, A083221, A249738, A250469, A250470.

a[1] = 0; a[?PrimeQ] = 0; a[n] := For[p = FactorInteger[n][[1, 1]]; m = n - p, True, m = m - p, If[FactorInteger[m][[1, 1]] == p, Return[m]]]; Array[a, 102] (* Jean-François Alcover, Mar 08 2016 *)

(define (A249744 n) (* (A020639 n) (A249738 n)))

a(n) = A020639(n) * A249738(n).
Other identities. For all n >= 1 it holds:
a(2n) = 2n-2.
a(A001248(n)) = A000040(n). [I.e., a(p^2) = p for primes p.]

A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).

Original entry on oeis.org

0, 1, 2, 1, 11, 2, 1, 8, 7, 14, 13, 4, 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8

Offset: 0

Jamie Morken, Aug 05 2019

The sequence is Primorial rows of A308121.
Row n has length A005867(n).
Row n > 1 average value = A060753(n)/2.
Row n > 1 has sum = A002110(n-1)*A038110(n)/2.
First value on row(n) = A161527(n-1).
Last value on row(n) = A038110(n) for n > 2.
For n > 1, A060753(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
x + y = A060753(n), where a = A005867(n-1) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
  A038110(n)*A000040(n)-A060753(n).
Related identity:
  A038110(n)/A038111(n)*(Prime(n)^2) - (A038110(n)/A038111(n)*((A038110(n)*Prime(n) - A060753(n))*Prime(n)/A038110(n))) = 1.

			The triangle starts:
row1: 0;
row2: 1;
row3: 2, 1;
row4: 11, 2, 1, 8, 7, 14, 13, 4;
row5: 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8;

Michael De Vlieger, Table of n, a(n) for n = 0..6299 (rows n = 0..6, flattened)
Eric Weisstein's World of Mathematics, Totative

Cf. A058250, A005867, A002110, A038110, A038111, A060753, A286941, A058262, A161527, A083140, A308121.

row[0] = 0; row[n_] := -(v = Numerator[Product[1 - 1/Prime[i], {i, 1, n}] / Prime[n]] * Select[Range[(p = Product[Prime[i], {i, 1, n}])], CoprimeQ[p, #] &]) + Denominator[Product[((pr = Prime[i]) - 1)/pr, {i, 1, n}]] * Range[Length[v]]; Table[row[n], {n, 0, 4}] // Flatten (* Amiram Eldar, Aug 10 2019 *)

cp's OEIS Frontend

A114881 Sieve of Eratosthenes minus 1.

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A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

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A249820 a(1) = 0 and for n > 1: a(n) = A249810(n) - A078898(n) = A078898(A003961(n)) - A078898(n).

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A249735 Odd bisection of A003961: Replace in 2n-1 each prime factor p(k) with prime p(k+1).

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A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

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A249811 Permutation of natural numbers: a(n) = A249741(A001511(n), A003602(n)).

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A250472 Permutation of natural numbers: a(n) = A250470(2*n - 1).

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A246279 Transpose of array A246278: A(row,1) = 2*row, and for col > 1, A(row,col) = A003961(A(row,col-1)).

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A249744 a(n) = 0 if n is 1 or a prime, otherwise, when n = A020639(n) * A032742(n), a(n) = the largest m < n such that A020639(m) = A020639(n), where A020639(n) and A032742(n) are the smallest prime and the largest proper divisor dividing n.

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A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).