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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A144313 Prime numbers p such that p - 1 is the fourth a-figurate number, seventh b-figurate number and possibly tenth c-figurate number for some a, b and c and not a d-figurate number for any nontrivial d.

Original entry on oeis.org

29, 71, 113, 239, 281, 449, 491, 659, 701, 827, 911, 953, 1373, 1499, 1583, 1667, 1709, 1877, 2003, 2087, 2129, 2213, 2339, 2423, 2549, 2591, 2633, 2801, 2843, 2969, 3221, 3347, 3389, 3557, 3767, 3851, 4229, 4271, 4397, 4481, 4649, 4691, 4733, 5153, 5279
Offset: 1

Views

Author

Reikku Kulon, Sep 17 2008

Keywords

Comments

Appears to be necessarily a subset of A007528.
The 46th Mersenne prime exponent (Mpe, A000043) 43112609 is a member: 43112608 is the fourth 7185436-figurate number and the seventh 2052983-figurate number and is not a k-figurate number for any other k except 43112608 (trivially). Several other Mersenne prime exponents are members of this sequence.
It is conjectured:
- that this sequence is infinite;
- that there is a unique set {4, 7, 10, 16, ...} (A138694?) giving the possible orders in k-figurate numbers for the set S of all Mpe for which Mpe - 1 is a (4, 7) or (4, 10) k-figurate number;
- that the ratio of Mpe in this sequence to those not approaches a nonzero value;
- that a characteristic function f(n) exists which equals 1 iff n is in S.
Contribution from Reikku Kulon, Sep 18 2008: (Start)
Subset of the integers n such that n is congruent to 29 modulo 42. The case where p - 1 is a tenth c-figurate number occurs when p is also congruent to 281 modulo 630.
The first three primes where c is defined are 281, 911 and 2801, with c = 8, 22, 64; c is congruent to 8 modulo 14. All such primes are necessarily congruent to 1 modulo 10.
The first invalid values of c are 36 and 50, which correspond to the semiprimes 1541 = 23 * 67 and 2171 = 13 * 167. Both of these are members of A071331 and A098237. The next invalid value of c, 78, corresponds to 3431 = 47 * 73, once again a member of both sequences.
The first primes where a, b, c and d are all defined (which therefore excludes them from this sequence) are the consecutive 6581, 7211 and 7841, all members of A140856, A140732, A142076, A142317 and A142905. (End)

Crossrefs

Cf. A000040, A007528, A138694, A131500, A096676, A000043, A000668
Contribution from Reikku Kulon, Sep 18 2008: (Start)
Cf. A071331, A098237 (semiprimes)
Cf. A140856, A140732, A142076, A142317, A142905 (a, b, c and d all defined) (End)

A130788 Solar planets' average orbit velocity (inverse ratio relative to one of Mercury), multiplied by 3 and rounded to the nearest integer.

Original entry on oeis.org

3, 4, 5, 6, 8, 11, 15, 21, 26, 30
Offset: 1

Views

Author

Alexander R. Povolotsky, Aug 19 2007

Keywords

Comments

a(5) refers to the asteroid belt and a(10) to Pluto, neither of which are presently considered planets (and neither of which appear in Malisoff's table). - Charles R Greathouse IV, Jan 16 2016

Examples

			A..B.......C..............D...............E................F.........G.....H
N.Planet.RltOrbDst.AvgOrbtVlct.SqrInvRltAvgOrbVlct.InvRltAvgOrbVlct.3*F...intg
...........#.......km/sec......compare with E......sqrt(E).............RndOff(G)
1.Mercury.1.......47.958.(47.95830/47.95830)^2=1.....1..............3.......3
2.Venus ..1.868...35.020.(47.95830/35.020)^2.=1.875..1.3694546..4.1083638...4
3.Earth...2.584...29.77..(47.95830/29.77)^2..=2.595..1.6109607..4.8328821...5
4.Mars....3.938...24.2...(47.95830/24.2)^2...=3.927..1.98174793.5.94524379..6.
5.AstrBlt.7.158.17.926*..(47.95830/17.9258)^2=7.158..2.67537338.8.02612014..8
6.Jupiter.13.444.13.1....(47.95830/13.1)^2...=13.402.3.66093893.10.9828168..11
7.Saturn..24.649.9.638...(47.95830/9.638)^2..=24.760.4.97595974.14.9278792..15
8.Uranus..49.561.6.83....(47.95830/6.83)^2...=49.304.7.02171303.21.0651391..21
9.Neptune.77.674.5.45....(47.95830/5.45)^2...=77.435.8.79968807.26.3990642..26
10.Pluto.101.912.4.75....(47.95830/4.75)^2...=101.939.10.0964842.30.2894526.30

References

  • William M. Malisoff, Some new laws for the solar system, Science, 70: 328-329, 1929.

Links

Crossrefs

Cf. A131500.

Extensions

Name edited by Charles R Greathouse IV, Jan 16 2016

A278658 Value of Margot's planetary discriminant Pi for n-th planet from the Sun.

Original entry on oeis.org

130, 950, 810, 54, 40000, 6100, 420, 300
Offset: 1

Views

Author

Felix Fröhlich, Nov 25 2016

Keywords

Comments

A value >= 1 indicates that the planet has sufficient mass to clear its orbit.
A planet is a solar system body that meets the definition of "planet" passed on August 24, 2006 under Resolution 5A of the 26th General Assembly of the International Astronomical Union.

Links

Crossrefs

Cf. A116448, A118652, A130788, A131500.
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