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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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A263078 a(n) = greatest k for which A155043(n+k) < A155043(n); a(n) = A263077(n)-n.

Original entry on oeis.org

-1, -2, -1, -2, 1, -4, 5, -2, -3, -4, 1, -6, 5, -2, 3, 2, 5, -6, 11, -2, 9, -4, 11, -2, -3, -4, 15, -6, 19, -8, 29, -2, 27, -4, 37, 12, 47, -4, 45, -6, 55, -8, 65, -2, 51, -4, 61, -6, -1, -2, 69, -4, 79, -6, 77, -8, 83, 2, 81, -12, 79, 10, 77, 76, 75, 6, 73, 16, 71, 14, 69, -12, 67, 22, 65, 20, 73, 18, 77, 16, 27, 26, 37, -12, 35, 34, 45, 20, 51, 18, 49, 40, 47, 26, 45, -12, 43, 42, 41, 40, 39, 30
Offset: 1

Views

Author

Antti Karttunen, Oct 09 2015

Keywords

Examples

			For n=1 we have A049820(1) = 0, thus A155043(1) = 1, and 0 is the only (and thus the largest) number from which zero can be reached with less steps (namely in zero steps, A155043(0) = 0), thus a(1) = 0 - 1 = -1.
For n=7, we have A155043(7) = 4 [as A049820(7) = 5, A049820(5) = 3, A049820(3) = 1, A049820(1) = 0], but there exists x=12 for which we have A049820(12) = 6, A049820(6) = 2, A049820(2) = 0, and this is the largest x such that A155043(x) < A155043(7), thus a(7) = 12 - 7 = 5.

Links

Crossrefs

Cf. A155043, A261089, A262503, A262909, A263077.
Cf. A263079 (indices of the negative terms), A263080 (of the positive terms).

Programs

  • Mathematica
    a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[k = 3 n;
While[a@ k >= a@ n, k--]; k - n, {n, 102}] (* Michael De Vlieger, Oct 13 2015 *)
  • PARI
    A263078 = n -> A263077(n) - n;
for(n=1,124340,write("b263078.txt",n," ",A263078(n)));
\\ Other code as in A263077

Formula

a(n) = A263077(n)-n.

A263079 Numbers n for which there does not exist any x > n such that A155043(x) < A155043(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 38, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 60, 72, 84, 96, 104, 108, 120, 128, 132, 136, 140, 142, 144, 150, 152, 156, 160, 162, 168, 170, 180, 182, 184, 186, 188, 190, 192, 194, 198, 200, 204, 208, 210, 216, 220, 225, 228, 240, 248, 252, 260, 264, 276, 280, 288, 296, 300, 308, 312, 320, 328, 340, 352, 360
Offset: 1

Views

Author

Antti Karttunen, Oct 09 2015

Keywords

Comments

Numbers n for which A263077(n) < n.
Numbers n for which A263078(n) is negative.
Numbers n at which point A155043(n) is the greatest lower bound for the rest of its terms from A155043(n) onward.

Examples

			1 is present because A049820(1) = 0, thus A155043(1) = 1, while all the larger numbers require at least the same number of steps to reach zero.

Links

Crossrefs

Cf. A155043, A261089, A262503, A263077, A263078.
Complement: A263080.

Programs

A263259 a(n) = number of integers k <= n for which A155043(k) = A155043(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 3, 4, 2, 5, 1, 3, 2, 3, 1, 4, 1, 4, 2, 5, 1, 2, 3, 4, 1, 5, 1, 6, 1, 3, 2, 4, 1, 2, 1, 2, 2, 3, 1, 4, 1, 2, 2, 3, 2, 4, 3, 4, 1, 5, 1, 6, 2, 7, 1, 3, 1, 8, 1, 2, 2, 3, 1, 3, 1, 3, 2, 4, 1, 4, 1, 3, 2, 4, 1, 5, 1, 6, 3, 4, 2, 4, 3, 4, 3, 5, 2, 6, 3, 4, 4, 5, 5, 5, 2, 4, 3, 6, 2, 5, 3, 7, 3, 5, 4, 8, 3, 6, 4, 7, 2, 7, 3, 8, 4, 4, 2, 7
Offset: 0

Views

Author

Antti Karttunen, Nov 24 2015

Keywords

Comments

a(n) = one-based index of n in row A155043(n) of table A263265.

Links

Crossrefs

Cf. A000005, A155043, A261089, A262503, A262507, A263089, A263265, A263279.

Formula

Other identities. For all n >= 0:
a(A261089(n)) = 1.
a(A262503(n)) = A262507(n).
A263265(A155043(n), a(n)) = n.

A262502 a(n) = least k such that A261104(k) = n; positions of records in A261104.

Original entry on oeis.org

0, 1, 3, 7, 11, 17, 23, 31, 40, 50, 62, 74, 86, 98, 110, 126, 142, 158, 174, 192, 210, 228, 248, 268, 288, 308, 328, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 738, 768, 798, 828, 860, 892, 924, 956, 988, 1020, 1052, 1084, 1116, 1148, 1180, 1212, 1244, 1280, 1316, 1352, 1388, 1424, 1460, 1496, 1532, 1568, 1604, 1640, 1676, 1716
Offset: 0

Views

Author

Antti Karttunen, Sep 24 2015

Keywords

Comments

a(n+2) should give a safe upper bound for A262503(n), and actually seems to significantly overshoot it when n grows.

Links

Crossrefs

Cf. A261104, A261089, A262503, A262507.
Cf. A262504 (first differences).

Formula

Other identities. For all n >= 0:
A261104(a(n)) = n.

A263080 Numbers n for which there exists x > n such that A155043(x) < A155043(n); numbers n for which A263078(n) is positive.

Original entry on oeis.org

5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121
Offset: 1

Views

Author

Antti Karttunen, Oct 09 2015

Keywords

Examples

			5 is present, because if we start iterating A049820 from it as: A049820(5) = 3, A049820(3) = 1, A049820(1) = 0, we get a path of length 3 to reach zero, thus A155043(5) = 3. On the other hand, if we start from 6, the path is one step shorter: A049820(6) = 2, A049820(2) = 0 [i.e., A155043(6) = 2], thus there exists a number larger than 5 which results a shorter path to zero.

Links

Crossrefs

Cf. A155043, A261089, A262503, A263077, A263078.
Complement: A263079.

Programs

  • Mathematica
    a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Position[Table[k = 3 n; While[a@ k >= a@ n, k--]; k - n, {n, 121}], Integer?Positive] // Flatten (* _Michael De Vlieger, Oct 13 2015 *)
  • PARI
    n=0; i=0; while(i < 10000, n++; if((A263077(n) > n), i++; write("b263080.txt",i," ",n)));
\\ Other code as in A263077.
  • Scheme
    ;; With Antti Karttunen's IntSeq-library.
(define A263080 (MATCHING-POS 1 1 (COMPOSE positive? A263078)))
Previous Showing 11-15 of 15 results.

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