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.

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A277710 Square array A(r,c), where each row r lists all numbers k for which A264977(k) = r, read by downwards antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 5, 2, 13, 10, 3, 29, 26, 39, 4, 41, 58, 75, 20, 9, 61, 82, 147, 52, 21, 6, 85, 122, 207, 116, 45, 78, 7, 125, 170, 291, 164, 93, 150, 11, 8, 173, 250, 411, 244, 189, 294, 19, 40, 81, 209, 346, 579, 340, 381, 414, 35, 104, 105, 18, 253, 418, 819, 500, 657, 582, 67, 232, 165, 42, 23, 281, 506, 927, 692, 765, 822, 131, 328, 213, 90, 43, 12
Offset: 1

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Author

Antti Karttunen, Oct 29 2016

Keywords

Comments

Alternative description: Each row r lists the positions of A019565(r) in A277330.
Odd terms occur only on rows with odd index, and even terms only on rows with even index. Specifically: all terms k on row r are equal to r modulo 4, thus the first differences of each row are all multiples of 4.
All the terms on any particular row are either all multiples of two (or respectively: three, or six), or none of them are.

Examples

			The top left 12 x 12 corner of the array:
   1,   5,  13,  29,  41,   61,   85,  125,  173,  209,  253,  281
   2,  10,  26,  58,  82,  122,  170,  250,  346,  418,  506,  562
   3,  39,  75, 147, 207,  291,  411,  579,  819,  927, 1155, 1635
   4,  20,  52, 116, 164,  244,  340,  500,  692,  836, 1012, 1124
   9,  21,  45,  93, 189,  381,  657,  765,  873, 1317, 1533, 1749
   6,  78, 150, 294, 414,  582,  822, 1158, 1638, 1854, 2310, 3270
   7,  11,  19,  35,  67,  131,  259,  311,  359,  515,  619,  655
   8,  40, 104, 232, 328,  488,  680, 1000, 1384, 1672, 2024, 2248
  81, 105, 165, 213, 333,  429,  669,  861, 1341, 1725, 2685, 2721
  18,  42,  90, 186, 378,  762, 1314, 1530, 1746, 2634, 3066, 3498
  23,  43,  79,  83, 103,  155,  163,  203,  307,  323,  403,  611
  12, 156, 300, 588, 828, 1164, 1644, 2316, 3276, 3708, 4620, 6540

Links

Crossrefs

Transpose: A277709.
Column 1: A277711, sorted into ascending order: A277817.
Row 1: A277701, Row 2: A277712 (= 2*A277701), Row 3: A277713, Row 4: 4*A277701, Row 5: A277715, Row 6: 2*A277713. Row 8: 8*A277701, Row 10: 2*A277715.
Cf. A277824 (the index of the column where n is located in this array).
Cf. A019565, A264977, A277330, A277816 and permutation pair A277695 & A277696.

Formula

A(r,1) = A277711(r); for c > 1, A(r,c) = A277816(A(r,c-1)).
Other identities. For all r>=1, c>=1:
A(2*r,c) = 2*A(r,c).
A(r,c) modulo 4 = r modulo 4.

Extensions

The dispersion-style formula added by Antti Karttunen, Nov 06 2016

A277696 Permutation of natural numbers: a(1) = 1; a(2n) = A277817(1+a(n)), a(2n+1) = A277816(a(n)).

Original entry on oeis.org

1, 2, 5, 3, 10, 7, 13, 4, 39, 15, 26, 9, 11, 18, 29, 6, 20, 92, 75, 24, 27, 49, 58, 14, 21, 16, 19, 31, 42, 62, 41, 8, 78, 33, 52, 270, 172, 196, 147, 47, 312, 56, 51, 126, 101, 143, 82, 23, 22, 34, 45, 28, 80, 32, 35, 64, 59, 96, 90, 153, 118, 95, 61, 12, 40, 224, 150, 66, 57, 129, 116, 1134, 534, 606, 316, 752, 404, 520, 207, 120, 55, 1400, 600
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2016

Keywords

Comments

This sequence can be represented as a binary tree. Each left hand child is produced as A277817(1+n), and each right hand child as A277816(n), when the parent node contains n:
1
................../ \..................
2 5
3......../ \........10 7......../ \........13
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
4 39 15 26 9 11 18 29
6 20 92 75 24 27 49 58 14 21 16 19 31 42 62 41
etc.

Links

Crossrefs

Inverse: A277695.
Cf. A277710, A277816, A277817.
Cf. A277701 (the rightmost edge of the tree).

Formula

a(1) = 1; and then after, a(2n) = A277817(1+a(n)), a(2n+1) = A277816(a(n)).

A277815 a(n) = the largest k < n for which A264977(k) = A264977(n), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 7, 0, 5, 0, 0, 0, 0, 0, 11, 4, 9, 14, 0, 0, 17, 10, 15, 0, 13, 0, 0, 0, 0, 0, 19, 0, 25, 22, 3, 8, 29, 18, 23, 28, 21, 0, 0, 0, 0, 34, 27, 20, 37, 30, 47, 0, 33, 26, 31, 0, 41, 0, 0, 0, 0, 0, 35, 0, 57, 38, 55, 0, 0, 50, 39, 44, 53, 6, 43, 16, 0, 58, 79, 36, 61, 46, 0, 56, 73, 42, 71, 0, 45, 0, 0, 0, 0, 0, 51
Offset: 0

Views

Author

Antti Karttunen, Nov 06 2016

Keywords

Links

Crossrefs

Cf. A264977, A277695, A277814, A277817 (the positions of zeros).
Left inverse of A277816.

Programs

  • Scheme
    (define (A277815 n) (if (zero? n) n (let ((v (A264977 n))) (let loop ((k (- n 1))) (cond ((zero? k) 0) ((= v (A264977 k)) k) (else (loop (- k 1))))))))

Formula

For all n >= 0, a(A277816(n)) = n.

A285111 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(A005117(1+n)) = 2*a(n), a(A065642(n)) = 1 + 2*a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 7, 5, 12, 16, 13, 14, 10, 24, 15, 32, 27, 26, 25, 28, 20, 48, 55, 9, 30, 11, 21, 64, 54, 52, 31, 50, 56, 40, 111, 96, 110, 18, 51, 60, 22, 42, 41, 49, 128, 108, 223, 17, 103, 104, 61, 62, 447, 100, 43, 112, 80, 222, 109, 192, 220, 57, 63, 36, 102, 120, 113, 44, 84, 82, 895, 98, 256, 99, 221, 216, 446, 34, 207, 23
Offset: 1

Views

Author

Antti Karttunen, Apr 17 2017

Keywords

Comments

Note the indexing: the domain starts from 1, while the range includes also zero.

Links

Crossrefs

Inverse: A285112.
Cf. A005117, A008683, A013928, A065642, A285328.
Similar or related permutations: A243343, A243345, A277695, A284571.

Programs

  • Python
    from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
from functools import reduce
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a285328(n):
    if core(n) == n: return 1
    k=n - 1
    while k>0:
        if a007947(k) == a007947(n): return k
        else: k-=1
def a013928(n): return sum([1 for i in range(1, n) if core(i) == i])
def a(n):
    if n<3: return n - 1
    if core(n)==n: return 2*a(a013928(n))
    else: return 1 + 2*a(a285328(n))
print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Apr 17 2017

Formula

a(1) = 0, a(2) = 1, and for n > 2, if A008683(n) <> 0 [when n is squarefree], a(n) = 2*a(A013928(n)), otherwise a(n) = 1 + 2*a(A285328(n)).

A284571 Permutation of natural numbers: a(1) = 1, a(A005117(1+n)) = 2*a(n), a(A065642(1+n)) = 1 + 2*a(n).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 9, 5, 12, 32, 17, 18, 10, 24, 33, 64, 65, 34, 11, 36, 20, 48, 129, 7, 66, 19, 37, 128, 130, 68, 49, 22, 72, 40, 97, 96, 258, 14, 69, 132, 38, 74, 73, 21, 256, 260, 81, 13, 29, 136, 15, 98, 521, 44, 39, 144, 80, 194, 257, 192, 516, 23, 137, 28, 138, 264, 45, 76, 148, 146, 197, 42, 512, 147, 193, 520, 162, 26, 27
Offset: 1

Views

Author

Antti Karttunen, Apr 17 2017

Keywords

Links

Crossrefs

Inverse: A284572.
Cf. A005117, A008683, A013928, A065642, A285328.
Similar or related permutations: A243343, A243345, A277695, A285111.

Programs

  • Python
    from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a285328(n):
    if core(n) == n: return 1
    k=n - 1
    while k>0:
        if a007947(k) == a007947(n): return k
        else: k-=1
def a013928(n): return sum(1 for i in range(1, n) if core(i) == i)
def a(n):
    if n==1: return 1
    if core(n)==n: return 2*a(a013928(n))
    else: return 1 + 2*a(a285328(n) - 1)
[a(n) for n in range(1, 121)] # Indranil Ghosh, Apr 17 2017

Formula

a(1) = 1, for n > 1, if A008683(n) <> 0 [when n is squarefree], a(n) = 2*a(A013928(n)), otherwise a(n) = 1 + 2*a(A285328(n)-1).
Showing 1-5 of 5 results.

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