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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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

A108780 Numbers n such that (n / sum of digits of n) is a golden semiprime.

Original entry on oeis.org

54, 135, 1122, 1386, 2244, 2805, 3366, 3927, 4488, 5301, 12369, 15318, 20445, 22977, 24534, 28623, 44979, 58941, 74169, 78588, 98892, 131454, 153363, 175272, 197181, 210693, 213206, 222132, 240792, 240999, 270891, 274122, 304580, 335038, 350267
Offset: 1

Views

Author

Jason Earls, Jun 26 2005

Keywords

Examples

			1122 is a term because 1122/6=187 and 187=11*17 and 11*phi-17 = 0.798373... < 1.

Links

Crossrefs

Cf. A001101, A108540.

Programs

  • Mathematica
    goldQ[n_] := Module[{f = FactorInteger[n]}, If[Length[f] != 2, False, If[Max[f[[;;,2]]] != 1, False, Abs[f[[2,1]] - f[[1,1]] * GoldenRatio] < 1]]]; sumDigits[n_] := Plus @@ IntegerDigits[n]; seqQ[n_] := Divisible[n, (sd = sumDigits[n])] && goldQ[n/sd]; Select[Range[360000], seqQ] (* Amiram Eldar, Nov 29 2019 *)

A107787 Golden semiprimes that are not brilliant numbers.

Original entry on oeis.org

77, 7303, 10033, 15229, 644251, 706609, 836197, 870071, 936791, 1027333, 1177993, 1261807, 1366553, 1433143, 1525441, 1608161, 62259511, 62782091, 63462979, 64441621, 64973261, 65427751, 65599201, 65837659, 66043093, 66834373, 67966247, 70635437, 70882871
Offset: 1

Views

Author

Jason Earls, Jun 14 2005

Keywords

Examples

			7303=67*109 is in the sequence because abs(67*phi-109) = 0.59172... < 1 and its prime factors have an unequal number of decimal digits.

Links

Crossrefs

Cf. A108540, A078972.

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 && IntegerLength[p] != IntegerLength[q], AppendTo[seq, p*q]], {1000}]; seq (* Amiram Eldar, Nov 29 2019 *)

Extensions

More terms from Donovan Johnson, Mar 06 2008

A108219 Numbers n such that A001414(n) is a golden semiprime, where A001414 is the sum of primes dividing n (with repetition).

Original entry on oeis.org

8, 9, 26, 44, 105, 112, 125, 126, 150, 160, 180, 192, 216, 243, 292, 568, 639, 1174, 1407, 1448, 1629, 1675, 2010, 2144, 2379, 2412, 2685, 2722, 2864, 3222, 3355, 3835, 3999, 4026, 4107, 4543, 4602, 5035, 5709, 5978, 6042, 6235, 6307, 6355, 6490, 7482
Offset: 1

Views

Author

Jason Earls, Jun 16 2005

Keywords

Comments

Numbers n such that A001414(n) and A001414(n+1) are both golden semiprimes: 8, 125, 153759, 247455, 678807, 1243499, 1243500, ... Notice that the last two terms indicate a triple. Conjecture: this subsequence is infinite.

Examples

			5709 = 3*11*173 is in the sequence because 3+11+173 = 187 = 11*17 and 11*phi-17 = 0.79837... < 1.

Links

Crossrefs

Cf. A001414, A108540.

Programs

  • Mathematica
    goldQ[n_] := Module[{f = FactorInteger[n]}, If[Length[f] != 2, False, If[Max[f[[;;,2]]] != 1, False, Abs[f[[2,1]] - f[[1,1]] * GoldenRatio] < 1]]]; sumPrimes[n_] := Plus @@ Times @@@ FactorInteger[n]; Select[Range[7500], goldQ[sumPrimes[#]] &] (* Amiram Eldar, Nov 29 2019 *)

A108706 Reverse these primes to get golden semiprimes.

Original entry on oeis.org

3037, 3631, 10271, 92251, 334651, 3302191, 3349403, 3494923, 3500897, 3574297, 3971207, 9067837, 9150139, 11914451, 16077421, 16651111, 30842221, 32333971, 34747217, 71704051, 71900987, 76642031, 78818581, 92032757, 104062963
Offset: 1

Views

Author

Jason Earls, Jun 20 2005

Keywords

Examples

			The prime 3631 is in the sequence because reversed it is 1363 = 29*47 and abs(29*phi-47) = 0.077014... < 1.

Links

Crossrefs

Cf. A108540.

Programs

  • Mathematica
    revDigits[n_] := FromDigits @ Reverse @ IntegerDigits[n]; goldQ[n_] := Module[{f = FactorInteger[n]}, If[Length[f] != 2, False, If[Max[f[[;;,2]]] != 1, False, Abs[f[[2,1]] - f[[1,1]] * GoldenRatio] < 1]]]; seqQ[n_] := PrimeQ[n] && goldQ @ revDigits[n]; Select[Range[4*10^6], seqQ] (* Amiram Eldar, Nov 29 2019 *)

Extensions

a(19)-a(25) from Donovan Johnson, Nov 11 2008
Offset corrected by Amiram Eldar, Nov 29 2019

A109875 Chen primes p such that their p + 2 counterpart is a golden semiprime.

Original entry on oeis.org

13, 587, 1361, 15227, 118967, 337721, 383267, 512891, 1027331, 1780151, 2303681, 8200391, 9310517, 14666579, 25005089, 29105981, 34824971, 38895497, 40436909, 51819461, 63462977, 65427749, 65599199, 66043091, 75552479, 94671671
Offset: 1

Views

Author

Jason Earls, Aug 31 2005

Keywords

Comments

Conjecture: sequence is infinite.

Examples

			1361 is a term because it is prime and 1363 = 29*47 and abs(29*phi - 47) = 0.07701... < 1.

Links

Crossrefs

Cf. A109611, A108540.

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 && PrimeQ[p*q - 2], AppendTo[seq, p*q - 2]], {1000}]; seq (* Amiram Eldar, Nov 29 2019 *)

Extensions

a(15)-a(26) from Donovan Johnson, Nov 17 2008
Previous Showing 11-16 of 16 results.

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