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 1-9 of 9 results.

A108540 Golden semiprimes: a(n)=p*q and abs(p*phi-q)<1, where phi = golden ratio = (1+sqrt(5))/2.

Original entry on oeis.org

6, 15, 77, 187, 589, 851, 1363, 2183, 2747, 7303, 10033, 15229, 16463, 17201, 18511, 27641, 35909, 42869, 45257, 53033, 60409, 83309, 93749, 118969, 124373, 129331, 156433, 201563, 217631, 232327, 237077, 255271, 270349, 283663, 303533, 326423
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 09 2005; revised Jun 13 2005

Keywords

Examples

			589 = 19*31 and abs(19*phi - 31) = abs(30,7426... - 31) < 1, therefore 589 is a term.

Links

Crossrefs

Cf. A001622, A001358, A050508.

Programs

  • Mathematica
    f[p_] := Module[{x = GoldenRatio * p}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; q = If[x - p1 < p2 - x, p1, p2]; If[Abs[q - x] < 1, q, 0]]; seq = {}; p=1; Do[p = NextPrime[p]; q = f[p]; If[q > 0, AppendTo[seq, p*q]], {100}]; seq (* Amiram Eldar, Nov 28 2019 *)

Formula

a(n) = A108541(n)*A108542(n) = A000040(k)*A108539(k) for some k.

Extensions

Corrected by T. D. Noe, Oct 25 2006

A108542 Greater prime factor of n-th golden semiprime.

Original entry on oeis.org

3, 5, 11, 17, 31, 37, 47, 59, 67, 109, 127, 157, 163, 167, 173, 211, 241, 263, 271, 293, 313, 367, 389, 439, 449, 457, 503, 571, 593, 613, 619, 643, 661, 677, 701, 727, 739, 787, 823, 911, 983, 991, 1021, 1069, 1163, 1187, 1231, 1289, 1381, 1429, 1487, 1523
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 09 2005

Keywords

Comments

abs(phi*A108541(n) - a(n)) < 1, where phi = golden ratio = (1+sqrt(5))/2.

Links

Crossrefs

Cf. A001622, A084127, A108540, A108541, A108543, A108544.
Subsequence of A108539.

Programs

  • Mathematica
    f[p_] := Module[{x = GoldenRatio * p}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; q = If[x - p1 < p2 - x, p1, p2]; If[Abs[q - x] < 1, q, 0]]; seq = {}; p=1; Do[p = NextPrime[p]; q = f[p]; If[q > 0, AppendTo[seq, q]], {200}]; seq (* Amiram Eldar, Nov 28 2019 *)

Formula

a(n) = A108540(n)/A108541(n).

A108541 Lesser prime factor of n-th golden semiprime.

Original entry on oeis.org

2, 3, 7, 11, 19, 23, 29, 37, 41, 67, 79, 97, 101, 103, 107, 131, 149, 163, 167, 181, 193, 227, 241, 271, 277, 283, 311, 353, 367, 379, 383, 397, 409, 419, 433, 449, 457, 487, 509, 563, 607, 613, 631, 661, 719, 733, 761, 797, 853, 883, 919, 941, 971, 997, 1031
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 09 2005

Keywords

Comments

abs(phi*a(n) - A108542(n)) < 1, where phi = golden ratio = (1+sqrt(5))/2.

Links

Crossrefs

Cf. A001622, A084126, A108539, A108540, A108542, A108543, A108544.

Programs

  • Mathematica
    f[p_] := Module[{x = GoldenRatio * p}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; q = If[x - p1 < p2 - x, p1, p2]; If[Abs[q - x] < 1, q, 0]]; seq = {}; p=1; Do[p = NextPrime[p]; If[f[p] > 0, AppendTo[seq, p]], {200}]; seq (* Amiram Eldar, Nov 28 2019 *)

Formula

a(n) = A108540(n)/A108542(n).

A108543 Primes that are factors of golden semiprimes (A108540).

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 41, 47, 59, 67, 79, 97, 101, 103, 107, 109, 127, 131, 149, 157, 163, 167, 173, 181, 193, 211, 227, 241, 263, 271, 277, 283, 293, 311, 313, 353, 367, 379, 383, 389, 397, 409, 419, 433, 439, 449, 457, 487, 503, 509, 563, 571
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 09 2005

Keywords

Links

Crossrefs

Union of A108541 and A108542.
Complement of A108545.
Cf. A108539, A108540, A108544.

Extensions

Corrected by T. D. Noe, Oct 25 2006

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

A165569 The indexing sequence for successively better golden semiprimes.

Original entry on oeis.org

1, 2, 4, 8, 9, 10, 25, 71, 103, 115, 157, 231, 329, 1783, 1835, 4476, 5128, 12462, 16274, 25035, 42174, 72589, 85968, 147666, 613726, 1088825, 1112415, 3125316, 3929736, 5742036, 7639447, 25716100, 32780150, 48132247, 76049401, 100464259, 108803364, 186018939
Offset: 1

Views

Author

Antti Karttunen, Sep 22 2009

Keywords

Links

Crossrefs

The corresponding semiprimes are given by A165570(n) = A165571(n)*A165572(n).
Cf. A108539.

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={}; k=0; dm = 1; p1 = 1; Do[p1 = NextPrime[p1]; k++; p2 = f[p1]; d = Abs[p2/p1 - GoldenRatio]; If[d < dm, dm = d; AppendTo[seq, k]], {10^4}]; seq (* Amiram Eldar, Nov 28 2019 *)

Formula

a(1)=1, and for n>1, a(n) = first such i>a(n-1) that abs(phi - A108539(i)/A000040(i)) < abs(phi - A108539(a(n-1))/A000040(a(n-1))), where phi = (1+sqrt(5))/2 (Golden ratio).

Extensions

a(16)-a(38) 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

A108545 Primes that are not factors of golden semiprimes (A108540).

Original entry on oeis.org

13, 43, 53, 61, 71, 73, 83, 89, 113, 137, 139, 151, 179, 191, 197, 199, 223, 229, 233, 239, 251, 257, 269, 281, 307, 317, 331, 337, 347, 349, 359, 373, 401, 421, 431, 443, 461, 463, 467, 479, 491, 499, 521, 523, 541, 547, 557, 569, 577, 587, 599, 601, 617
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 09 2005

Keywords

Comments

Complement of A108543;
abs(a(i) - a(j) * phi) > 1 for all i and j, where phi = golden ratio = (1+sqrt(5))/2.

Links

Crossrefs

Cf. A001622, A108539, A108540, A108542, A108543, A108544.
Showing 1-9 of 9 results.

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