cp's OEIS Frontend

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.

Showing 1-7 of 7 results.

A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 21, 35, 49, 11, 12, 27, 55, 77, 121, 13, 14, 33, 65, 91, 143, 169, 17, 16, 39, 85, 119, 187, 221, 289, 19, 18, 45, 95, 133, 209, 247, 323, 361, 23, 20, 51, 115, 161, 253, 299, 391, 437, 529, 29, 22, 57, 125, 203, 319, 377, 493, 551, 667
Offset: 2

Views

Author

Yasutoshi Kohmoto, Jun 05 2003

Keywords

Comments

This is permutation of natural numbers larger than 1.
From Antti Karttunen, Dec 19 2014: (Start)
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.
For navigating in this array:
A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.
A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.
First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).
(End)
The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015

Examples

			The top left corner of the array:
   2,   4,   6,    8,   10,   12,   14,   16,   18,   20,   22,   24,   26
   3,   9,  15,   21,   27,   33,   39,   45,   51,   57,   63,   69,   75
   5,  25,  35,   55,   65,   85,   95,  115,  125,  145,  155,  175,  185
   7,  49,  77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329
  11, 121, 143,  187,  209,  253,  319,  341,  407,  451,  473,  517,  583
  13, 169, 221,  247,  299,  377,  403,  481,  533,  559,  611,  689,  767
  17, 289, 323,  391,  493,  527,  629,  697,  731,  799,  901, 1003, 1037
  19, 361, 437,  551,  589,  703,  779,  817,  893, 1007, 1121, 1159, 1273
  23, 529, 667,  713,  851,  943,  989, 1081, 1219, 1357, 1403, 1541, 1633
  29, 841, 899, 1073, 1189, 1247, 1363, 1537, 1711, 1769, 1943, 2059, 2117
  ...

Links

Crossrefs

Transpose of A083140.
One more than A249741.
Inverse permutation: A252460.
Column 1: A000040, Column 2: A001248.
Row 1: A005843, Row 2: A016945, Row 3: A084967, Row 4: A084968, Row 5: A084969, Row 6: A084970.
Main diagonal: A083141.
First semiprime in each column occurs at A251717; A251718 & A251719 with additional criteria. A251724 gives the corresponding semiprimes for the latter. See also A251728.
Permutations based on mapping numbers between this array and A246278: A249817, A249818, A250244, A250245, A250247, A250249. See also: A249811, A249814, A249815.
Also used in the definition of the following arrays of permutations: A249821, A251721, A251722.
Cf. A002260, A004736, A004280, A020639, A038179, A055396, A078898, A138511, A249820, A249730, A249735, A249744, A250469, A250470, A250472, A250474.

Programs

  • Mathematica
    lim = 11; a = Table[Take[Prime[n] Select[Range[lim^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], lim], {n, lim}]; Flatten[Table[a[[i, n - i + 1]], {n, lim}, {i, n}]] (* Michael De Vlieger, Jan 04 2016, after Yasutoshi Kohmoto at A083140 *)

Extensions

More terms from Hugo Pfoertner, Jun 13 2003

A250470 a(n) = A249817(A064989(A249818(n))).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 8, 7, 19, 2, 9, 11, 10, 5, 23, 6, 29, 1, 12, 13, 15, 4, 31, 17, 14, 3, 37, 10, 41, 7, 16, 19, 43, 2, 25, 9, 18, 11, 47, 8, 21, 5, 20, 23, 53, 6, 59, 29, 22, 1, 27, 14, 61, 13, 24, 15, 67, 4, 71, 31, 26, 17, 35, 22, 73, 3, 28, 37, 79, 10, 33, 41, 30, 7, 83, 12, 55, 19, 32, 43, 39, 2, 89, 25, 34, 9, 97, 26, 101
Offset: 1

Views

Author

Antti Karttunen, Dec 06 2014

Keywords

Comments

Odd bisection, A250472, is a permutation of natural numbers. A250479 gives the even bisection.
For odd numbers n >= 3, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1. In other words, 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 n.

Links

Crossrefs

Cf. A055396, A064989, A078898, A083140, A083221, A249744, A249810, A249820, A249817, A249818, A250469.
Odd bisection: A250472.
Even bisection: A250479.
Differs from A064989 for the first time at n=21, where a(21) = 8, while
A064989(21) = 10.

Programs

Formula

a(n) = A249817(A064989(A249818(n))).
Other identities. For all n >= 1:
a(A250469(n)) = n. [This is an inverse function for injection A250469.]
For all odd numbers n >= 3: A055396(a(n)) = A055396(n)-1.

A266403 Self-inverse permutation of natural numbers: a(n) = A250470(A263273(A250469(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 8, 17, 6, 13, 10, 11, 20, 9, 14, 71, 22, 7, 26, 19, 12, 23, 16, 21, 24, 41, 18, 53, 28, 31, 56, 29, 38, 107, 58, 67, 74, 61, 32, 197, 40, 25, 68, 59, 50, 137, 64, 73, 62, 49, 44, 227, 76, 27, 80, 55, 30, 89, 34, 43, 66, 37, 48, 91, 46, 69, 60, 35, 42, 65, 70, 15, 78, 47, 36, 119, 52
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2016

Keywords

Links

Crossrefs

Cf. A000035, A250469, A250470, A263273.
Cf. A265369, A265904, A266190, A266401 (other conjugates or similar derivations of A263273).
Cf. also A250471, A250472, A264996, A266404, A266415, A266416, A266645, A266646.

Programs

Formula

a(n) = A250470(A263273(A250469(n))).
As a composition of related permutations:
a(n) = A266415(A266645(n)) = A266646(A266416(n)).
a(n) = A250472(A264996(A250471(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A253554 a(1) = 1, a(2n) = n, a(2n+1) = A250470(2n+1).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 4, 5, 7, 6, 11, 7, 6, 8, 13, 9, 17, 10, 8, 11, 19, 12, 9, 13, 10, 14, 23, 15, 29, 16, 12, 17, 15, 18, 31, 19, 14, 20, 37, 21, 41, 22, 16, 23, 43, 24, 25, 25, 18, 26, 47, 27, 21, 28, 20, 29, 53, 30, 59, 31, 22, 32, 27, 33, 61, 34, 24, 35, 67, 36, 71, 37, 26, 38, 35, 39, 73, 40, 28, 41, 79, 42, 33, 43, 30
Offset: 1

Views

Author

Antti Karttunen, Jan 12 2015

Keywords

Comments

Divide the even numbers by two, and for odd numbers n >= 3, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1.
For any number n >= 2 in binary trees A252753 and A252755, a(n) gives the number which is the parent of n.

Links

Crossrefs

Bisections: A000027 and A250472.
Cf. A253555 (the number of iterations needed to reach 1 from n).
Cf. A055396, A078898, A250470, A252753, A252755.
Differs from A252463 for the first time at n=21, where a(21) = 8, while A252463(21) = 10.

Programs

  • Scheme
    (define (A253554 n) (cond ((<= n 1) n) ((even? n) (/ n 2)) (else (A250470 n))))

Formula

a(1) = 1, a(2n) = n, a(2n+1) = A250470(2n+1).

A250471 Permutation of natural numbers: a(n) = (A250469(n) + 1) / 2.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 6, 11, 13, 14, 7, 17, 9, 20, 18, 23, 10, 26, 12, 29, 28, 32, 15, 35, 25, 38, 33, 41, 16, 44, 19, 47, 43, 50, 39, 53, 21, 56, 48, 59, 22, 62, 24, 65, 58, 68, 27, 71, 61, 74, 63, 77, 30, 80, 46, 83, 73, 86, 31, 89, 34, 92, 78, 95, 60, 98, 36, 101, 88, 104, 37, 107, 40, 110, 93, 113, 72, 116, 42, 119, 103, 122, 45, 125, 67, 128, 108, 131, 49
Offset: 1

Views

Author

Antti Karttunen, Dec 06 2014

Keywords

Links

Crossrefs

Inverse: A250472.
Cf. A048673, A250469.

Programs

Formula

a(n) = (A250469(n) + 1) / 2.

A266415 Permutation of natural numbers: a(n) = A250470(A263273(A003961(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 8, 17, 10, 13, 6, 11, 22, 9, 20, 71, 28, 7, 18, 19, 16, 23, 14, 21, 64, 41, 26, 227, 58, 31, 74, 29, 82, 53, 12, 67, 52, 61, 24, 107, 46, 25, 30, 59, 40, 65, 56, 73, 190, 49, 44, 197, 76, 27, 230, 55, 172, 137, 38, 43, 220, 37, 32, 571, 244, 69, 60, 35, 34, 89, 72, 15, 154, 47, 68, 479, 70
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2016

Keywords

Links

Crossrefs

Inverse: A266416.
Cf. A000035, A003961, A250470, A263273.
Related permutations: A048673, A250472, A264985, A264996, A266403, A266646.

Programs

Formula

a(n) = A250470(A263273(A003961(n))).
As a composition of related permutations:
a(n) = A266403(A266646(n)).
a(n) = A250472(A264996(A048673(n))) = A250472(1+A264985(-1+A048673(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A250479 Even bisection of A250470: a(n) = A250470(2n).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 10, 7, 19, 2, 9, 11, 8, 5, 23, 6, 29, 1, 14, 13, 15, 4, 31, 17, 22, 3, 37, 10, 41, 7, 12, 19, 43, 2, 25, 9, 26, 11, 47, 8, 27, 5, 34, 23, 53, 6, 59, 29, 20, 1, 39, 14, 61, 13, 38, 15, 67, 4, 71, 31, 18, 17, 35, 22, 73, 3, 16, 37, 79, 10, 63, 41, 46, 7, 83, 12, 65, 19, 58, 43, 75, 2, 89, 25, 28, 9, 97, 26, 101
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2014

Keywords

Links

Crossrefs

Cf. A250470, A250472 (the other bisection).
Differs from A064989 for the first time at n=55, where a(55) = 27, while A064989(55) = 21.

Programs

Formula

a(n) = A250470(2*n).
Showing 1-7 of 7 results.

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