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 1-3 of 3 results.

A165571 Lesser prime factor of successively better golden semiprimes.

Original entry on oeis.org

2, 3, 7, 19, 23, 29, 97, 353, 563, 631, 919, 1453, 2207, 15271, 15737, 42797, 49939, 133559, 179317, 287557, 508451, 918011, 1103483, 1981891, 9181097, 16958611, 17351447, 52204391, 66602803, 99641617, 134887397, 487195147, 629449511, 943818943, 1527963169, 2048029369
Offset: 1

Views

Author

Antti Karttunen, Sep 22 2009

Keywords

Comments

See A165569 and A165570 for the definition. Probably a subset of A108541.

Links

Crossrefs

Cf. A108539, A165569, A165570, A165572.

Programs

  • Mathematica
    f[p_] := Module[{x = GoldenRatio * p, p1, p2}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; If[p2 - x > x - p1, p1, p2]]; seq={}; dm = 1; p1 = 1; Do[p1 = NextPrime[p1]; k++; p2 = f[p1]; d = Abs[p2/p1 - GoldenRatio]; If[d < dm, dm = d; AppendTo[seq, p1]], {10^4}]; seq  (* Amiram Eldar, Nov 28 2019 *)

Formula

a(n) = A000040(A165569(n)).
a(n) = A165570(n)/A165572(n).

Extensions

a(16)-a(23) from Donovan Johnson, May 13 2010
a(24)-a(36) from Amiram Eldar, Nov 28 2019

A165572 Greater prime factor of successively better golden semiprimes.

Original entry on oeis.org

3, 5, 11, 31, 37, 47, 157, 571, 911, 1021, 1487, 2351, 3571, 24709, 25463, 69247, 80803, 216103, 290141, 465277, 822691, 1485373, 1785473, 3206767, 14855327, 27439609, 28075231, 84468479, 107765599, 161223523, 218252393, 788298307, 1018470703, 1527131129, 2472296341
Offset: 1

Views

Author

Antti Karttunen, Sep 22 2009

Keywords

Comments

See A165569 and A165570 for the definition. Probably a subset of A108542.

Links

Crossrefs

Cf. A108539, A165569, A165570, A165571.

Programs

  • Mathematica
    f[p_] := Module[{x = GoldenRatio * p, p1, p2}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; If[p2 - x > x - p1, p1, p2]]; seq={}; dm = 1; p1 = 1; Do[p1 = NextPrime[p1]; k++; p2 = f[p1]; d = Abs[p2/p1 - GoldenRatio]; If[d < dm, dm = d; AppendTo[seq, p2]], {10^4}]; seq  (* Amiram Eldar, Nov 28 2019 *)

Formula

a(n) = A108539(A165569(n)).
a(n) = A165570(n)/A165571(n).

Extensions

a(16)-a(23) from Donovan Johnson, May 13 2010
a(24)-a(35) from Amiram Eldar, Nov 28 2019

A165570 Successively better golden semiprimes.

Original entry on oeis.org

6, 15, 77, 589, 851, 1363, 15229, 201563, 512893, 644251, 1366553, 3416003, 7881197, 377331139, 400711231, 2963563859, 4035221017, 28862500577, 52027213697, 133793658289, 418298061641, 1363588753103, 1970239102459, 6355462656397, 136388198153719, 465337655023099
Offset: 1

Views

Author

Antti Karttunen, Sep 22 2009

Keywords

Comments

This is lexicographically earliest sequence of such semiprimes p*q, starting from 6=2*3, that for each successive term p*q, q/p is a better approximant of Golden ratio (1+sqrt(5))/2 than the previous term. See A165569 for the exact procedure.
Can it be proved that this a subset of A108540?
The ratio A165572(n)/A165571(n) converges towards golden ratio = (1+sqrt(5))/2 = 1.618033988749895... as: 1.5, 1.6666666666666667, 1.5714285714285714, 1.631578947368421, 1.608695652173913, 1.6206896551724137, 1.6185567010309279, 1.6175637393767706, 1.6181172291296626, 1.618066561014263, 1.618063112078346, 1.618031658637302, 1.6180335296782964, 1.6180341824372995, 1.6180339327699054, ...

Links

Crossrefs

Cf. A108539, A165569, A165571, A165572.

Programs

  • Mathematica
    f[p_] := Module[{x = GoldenRatio * p, p1, p2}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; If[p2 - x > x - p1, p1, p2]]; seq={}; dm = 1; p1 = 1; Do[p1 = NextPrime[p1]; k++; p2 = f[p1]; d = Abs[p2/p1 - GoldenRatio]; If[d < dm, dm = d; AppendTo[seq, p1*p2]], {10^4}]; seq (* Amiram Eldar, Nov 28 2019 *)

Formula

a(n) = A165571(n)*A165572(n) = A000040(A165569(n))*A108539(A165569(n)).

Extensions

a(16)-a(23) from Donovan Johnson, May 13 2010
a(24)-a(26) from Amiram Eldar, Nov 28 2019
Showing 1-3 of 3 results.

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