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 1-10 of 10 results.

A381230 Index of first occurrence of n in A000319, or -1 if n never appears there.

Original entry on oeis.org

6, 0, 17, 45, 13, 42, 938, 920, 57403, 7865, 15862, 313445, 54988, 53976, 57573, 61372, 83445, 14998, 360213, 57322, 59858, 2070497, 2021631, 14918, 58001, 1394543, 79664, 2056395, 85989, 11, 8, 71662, 5495575, 5652770, 360110, 2070250, 5947332, 5514279, 5514772, 2015129, 78877, 305403, 1031, 1869352, 5652707, 4823241
Offset: 0

Views

Author

N. J. A. Sloane, Feb 28 2025

Keywords

Comments

If a(57) is not -1, then a(57) > 14333074. - Tim Peters, Mar 13 2025

Examples

			0 is the 6th term of A000319, so a(0) = 6. 1 is the 0th term, so a(1) = 0.

Links

Crossrefs

Cf. A000319, A381231.

Extensions

a(8)-a(10) from Hugo Pfoertner, Feb 28 2025
a(11)-a(31) from Tim Peters, Mar 01 2025
a(32) from Tim Peters, Mar 02 2025
a(33)-a(45) from Tim Peters, Mar 04 2025

A381231 Index of first occurrence of -n in A000319, or -1 if -n never appears there.

Original entry on oeis.org

6, 3, 4, 5, 48, 15642, 14, 87, 924, 23, 1074, 1066, 14524, 7051, 15000, 71709, 57604, 57554, 2056626, 2036049, 16068, 86, 934, 37142, 57439, 72635, 57342, 1394559, 358329, 2112076, 16941, 2015018, 57124, 27572, 1837444, 29, 2058540, 54694, 2075246, 359870, 76579, 7844, 61424, 55065, 61434, 2016279, 71877, 2271483, 305269, 57405, 1842679
Offset: 0

Views

Author

N. J. A. Sloane, Feb 28 2025

Keywords

Comments

If a(54) is not -1, then a(54) > 14333074. - Tim Peters, Mar 13 2025

Links

Crossrefs

Cf. A000319, A381230.

Extensions

a(5)-a(17) from Hugo Pfoertner, Feb 28 2025
a(18)-a(50) from Tim Peters, Mar 01 2025

A381234 RUNS transform of A000319.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 7, 1, 1, 2, 1, 1, 2, 1, 2, 1, 8, 1, 1, 1, 1, 18, 1, 1, 1, 1, 1, 1, 1, 1, 1, 825, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 76, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1144, 1, 1, 188, 1, 1, 4571, 1, 1, 62
Offset: 1

Views

Author

N. J. A. Sloane, Mar 16 2025

Keywords

Comments

Based on Tim Peters's data file in A000319.

Examples

			A000319 begins 1, 1, 74, -1, -2, -3, 0, 1, 30, -2, -2, 29, 1, 4, -6, 0, 1, 2, -1, -1, -1, -1, -2,  ...
so the RUNS transform is 2, 1, 1, 1, 1, 1, 1, 2, 1, ...
and A381235 starts 1, 74, -1, -2, -3, 0, 1, 30, -2, 29, ...

Links

Crossrefs

Cf. A000319, A381235.

A381235 a(n) = the term of A000319 that is repeated in the n-th run of equal consecutive terms.

Original entry on oeis.org

1, 74, -1, -2, -3, 0, 1, 30, -2, 29, 1, 4, -6, 0, 1, 2, -1, -2, -9, 0, 1, 2, -2, -35, -1, -2, -3, 0, 1, 5, -2, 3, 1, -4, -1, -2, -3, 1, 2, -1, -2, -3, 0, 1, 2, -1, -2, -21, -7, 0, 1, -4, -1, -2, 7, 2, -3, 1, -8, -3, 1, -3, 1, 3, 0, 1, -22, 0, 1, 6, -1, -2, -3, 1, 6, 0, 1, 2, -1, -2, 5, -3, 1, 42, -2, 2, -1, -2, -3, 1, -11, -1, -2, -4, -1
Offset: 1

Views

Author

N. J. A. Sloane, Mar 16 2025

Keywords

Comments

Or, replace each run of consecutive equal terms in A000319 by a single term.
Based on Tim Peters's data file in A000319.

Examples

			See A381234.

Links

Crossrefs

Cf. A000319, A381234.

A053873 Numbers n such that OEIS sequence A_n contains n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 14, 16, 19, 26, 27, 36, 37, 52, 59, 62, 69, 72, 115, 119, 120, 121, 134, 161, 164, 174, 177, 188, 189, 190, 193, 194, 195, 196, 209, 224, 265, 267, 277
Offset: 1

Views

Author

Jens Voß, Mar 30 2000

Keywords

Comments

A number n is in this sequence iff n appears anywhere in the terms of A_n, not just in the terms that are visible in the entry.
Is 53873 in this sequence? (A rhetorical question!) - Tanya Khovanova, Aug 09 2007
Is 53169 in this sequence? (A rhetorical question!). - Raymond Wang, Oct 07 2008
I skipped 241 since it appears that A000241(14) > 241, but as the 13th and further terms are not known this is not certain. The next term in the sequence is almost surely 319, but finding the least k for which A000319(k) = 319 requires calculating a chaotic sequence to high precision. - Charles R Greathouse IV, Jul 20 2007
241 is not in this sequence, since A000241(13) <= 225 and A000241(14) >= 0.8594*315 (see comments in A000241). - Danny Rorabaugh, Mar 13 2015

Examples

			4 is not in A000004, so 4 is not in this sequence.
60 is not in A000060, so 60 is not in this sequence.
86 is not in A000086, so 86 is not in this sequence.

Links

Crossrefs

Complement of A053169.

Extensions

More terms from N. J. A. Sloane, Aug 24 2006
a(23)-a(25) from Charles R Greathouse IV, Aug 30 2006
a(26)-a(40) from Charles R Greathouse IV, Jul 20 2007
Typo in one entry corrected by Olaf Voß, Feb 25 2008

A381226 a(n) is the number of distinct positive integers that can be obtained by starting with n!, and optionally applying the operations square root, floor, and ceiling, in any order.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 8, 9, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18
Offset: 1

Views

Author

N. J. A. Sloane, Feb 24 2025

Keywords

Comments

This sequence, A381227, and A381228 arose in connection with the problem of showing that every positive integer can be represented using a single 4. Hans Havermann has pointed out that A139004 is related to this question and has many references. - N. J. A. Sloane, Feb 25 2025

Examples

			For n = 8, 8! = 40320; sqrt(40320) = 200.798..., floor and ceiling give 200 and 201. Sqrt(200) = 14.142..., and floor and ceiling give 14 and 15. From 14 we get 3 and 4; from 3 we get 1 and 2. 15 and 4 give nothing more. In all, we get a(8) = 9 different numbers: 40320, 200, 201, 14, 15, 3, 4, 1, 2.
Note that at each step, we must consider three "parents": if x was a term at the previous step, we get floor(sqrt(x)), sqrt(x), and ceiling(sqrt(x)) as potential parents at the next step.

Crossrefs

Cf. A139004, A381227-A381229.
Motivated by trying to understand A000319.

Programs

  • PARI
    f(n) = my(t); if(n<4, [1..n], t=sqrtint(n); if(issquare(n), concat(f(t), n), Set(concat([f(t), f(t+1), [n]]))));
a(n) = #f(n!); \\ Jinyuan Wang, Feb 25 2025

Extensions

More terms from Jinyuan Wang, Feb 25 2025

A000329 Nearest integer to b(n), where b(n) = tan(b(n-1)), b(0) = 1.

Original entry on oeis.org

1, 2, 75, -1, -1, -2, 1, 2, 31, -1, -2, 29, 1, 5, -6, 1, 1, 3, -1, -1, -1, -1, -1, -9, 1, 1, 1, 2, -2, -35, 0, 0, -1, -1, -1, -1, -1, -1, -2, 1, 1, 1, 5, -1, -2, 4, 1, 2, -4, 0, 0, 0, -1, -1, -1, -1, -1, -1, -2, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

We have a(11764189) = 329, from b(11764189) ~ 328.86367. This value was found using interval arithmetic with MPFR's tangent function (rounding the results downward and upward at each step), starting at an initial precision of 70000 bits. - Matthew House, Nov 17 2024

Links

Crossrefs

Cf. A000319.

Programs

  • Mathematica
    Round[NestList[Tan, 1, 100]] (* Matthew House, Nov 17 2024 *)

Extensions

Terms a(71) and beyond from Peter J. Taylor, Nov 23 2017

A381238 a(n) = floor(b(n)), where b(n) = sec^2(b(n-1)), b(0)=1.

Original entry on oeis.org

1, 3, 1, 4, 66, 1, 11, 4, 8, 3, 1, 6, 1, 143, 2, 1, 481, 10, 14, 6, 1, 4, 11, 2, 1, 140, 3, 1, 3, 1, 7, 14, 2, 2, 1, 3, 1, 39, 3, 1, 2159, 3, 1, 4, 3, 1, 3, 1, 14, 10, 1, 11, 6, 1, 3, 1, 4, 185, 1, 3, 1, 4, 78, 1, 3, 1, 6, 1, 4, 3, 2, 2, 1, 11, 17, 8, 15, 1, 16, 5, 1, 14, 4, 8, 7, 1, 20, 7, 96, 14, 62, 1, 3, 2, 2, 1, 91, 1, 10, 97, 1
Offset: 0

Views

Author

N. J. A. Sloane, Mar 17 2025

Keywords

Comments

a(0)-a(100) were computed using Maple with 10000 digits of precision.
A variant of A000319 (note that sec(x)^2 is the derivative of tan(x)).
Conjecture: Every nonnegative number appears in this sequence. (See A382148.)

Links

Crossrefs

Cf. A000319, A005699, A382148.

Programs

  • Mathematica
    Floor[NestList[Sec[#]^2 &, 1, 100]] (* Paolo Xausa, Mar 18 2025 *)

A058634 a(n) = floor(b(n)), where b(n) = 1/cos(b(n-1)), b(0) = 1.

Original entry on oeis.org

1, 1, -4, -2, 2, -2, 9, -2, 1, -4, -2, 1, -4, -2, 1, -4, -2, 1, -3, -2, 1, -3, -2, 1, -4, -2, 1, -4, -2, 2, -2, 12, 1, 1, -4, -2, 1, -3, -2, 2, -2, 8, -3, -2, -6, 1, 16, -3, -2, 7, -12, 1, -4, -2, 2, -3, -3, -3, -2, -6, 1, 2, -2, 2, -3, -2, -6, 1, 4, -3, -2, 9, -2, 2, -3, -2, -4, -2, 1, -3, -2, 1, -3, -2, 2, -3, -2, -7, 1, 4, -2, 70, 4, -2, 27
Offset: 0

Views

Author

Avi Peretz (njk(AT)netvision.net.il), Dec 26 2000

Keywords

Examples

			a(3) = -2 because: b(0) = 1, b(1) = 1/cos(1) = 1.850815..., b(2) = 1/cos(1.850815) = -3.618291..., b(3) = 1/cos(-3.618291) = -1.125468... and a(3) = floor(b(3)) = -2

Crossrefs

Cf. A000319.

Programs

  • PARI
    default(realprecision,1000); b=1; for(n=0, 94, print1(floor(b)", "); b=1/cos(b))

Extensions

More terms from Ralf Stephan, Mar 23 2003

A250221 Least k such that A_n(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 11, -1, 16, 12, -1, 9, -1, 11, -1, -1, -1, 11, -1, 8, -1, -1, 126, -1, -1, -1, -1, -1, -1, 26, 27, -1, -1, -1, -1, -1, -1, -1, -1, 29, 31, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 33, -1, -1, -1, -1, -1, -1, 19, -1, -1, 45, -1, -1, -1, -1, -1, -1, 35, -1, -1, 8
Offset: 1

Views

Author

Eric Chen, Dec 24 2014

Keywords

Comments

a(A053169(n)) = -1, but what is a(53169)?
a(319) is the first unknown term. (See A000319)
a(241) should be -1. (See A000241)

Crossrefs

Cf. A053873, A053169.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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