A108540 -id:A108540 - cp's OEIS frontend

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

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

Reinhard Zumkeller, Jun 09 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Corrected by T. D. Noe, Oct 25 2006

A108544 Primes that are factors of distinct golden semiprimes (A108540).

Original entry on oeis.org

3, 11, 37, 67, 163, 167, 241, 271, 367, 449, 457, 613, 661, 1613, 1979, 2069, 2137, 2221, 2347, 2351, 2381, 2399, 2477, 2633, 2749, 2837, 3011, 3449, 3593, 3671, 3797, 3911, 3943, 4001, 4049, 4079, 4241, 4999, 5147, 5261, 5711, 5981, 6337, 6379, 6619

Offset: 1

Reinhard Zumkeller, Jun 09 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A108541 and A108542.

Cf. A108540, A108543.

A107768 Integers pqr such that pq and qr are both golden semiprimes (A108540). Integers pqr such that p = A108541(j), q = A108542(j) = A108541(k) and r = A108542(k).

Original entry on oeis.org

30, 1309, 50209, 299423, 4329769, 4661471, 13968601, 19867823, 49402237, 90419171, 95575609, 230236057, 289003081, 4195692049, 7752275351, 8857002097, 9759031489, 10956612769, 12930672109, 12991059409, 13494943703, 13807499677, 15195694009, 18253659551, 20769940297

Offset: 1

Jonathan Vos Post, Jun 11 2005

Golden 3-almost primes.

Volumes of bricks (rectangular parallelepipeds) each of whose faces has golden semiprime area. How long a chain is possible of the form p(1) * p(2) * p(3) * ... * p(n) where each successive pair of values are factors of a golden semiprime? That is, if Zumkeller's golden semiprimes are the 2-dimensional case and the present sequence is the 3-dimensional case, is there a maximum n for an n-dimensional case?

			30 = 2 * 3 * 5, where both 2*3=6 and 3*5=15 are golden semiprimes.
1309 = 7 * 11 * 17.
50209 = 23 * 37 * 59.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A014612, A108540, A108541, A108542.

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]]; g[p_] := Module[{ p1 = f[p]}, If[p1 == 0, 0, p2 = f[p1]; If[p2 == 0, 0, p*p1*p2]]]; seq={}; p=1; Do[p = NextPrime[p]; gp = g[p]; If[gp > 0, AppendTo[seq, gp]], {300}]; seq (* Amiram Eldar, Nov 29 2019 *)

More terms from Amiram Eldar, Nov 29 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

Reinhard Zumkeller, Jun 09 2005

nonn

Complement of A108543;

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

A108200 Least positive k such that k*n + 1 is a golden semiprime (A108540).

Original entry on oeis.org

5, 7, 62, 19, 1, 31, 2, 1254, 1692, 85, 912, 49, 20796, 1, 1234, 627, 50, 846, 4, 860, 28, 456, 4076, 418, 34, 10398, 564, 21, 91250, 617, 6, 47670, 304, 25, 6218, 423, 352018, 2, 6932, 430, 9348, 14, 400, 228, 2874, 2038, 324, 209, 12, 17, 5562, 5199, 25784, 282

Offset: 1

Jason Earls, Jun 15 2005

nonn

Conjecture: for every n > 4 there exists a number k < n^[n/2] such that k*n + 1 is a golden semiprime, where [] is the floor function.

			a(3) = 62 because 62*3+1 = 187 = 11*17 and 11*phi-17 = 0.7983... < 1.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A108540.

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]]]; a[n_] := Module[{k = 1}, While[!goldQ[k * n + 1], k++]; k]; Array[a, 54] (* Amiram Eldar, Nov 29 2019 *)

A108399 Least positive k such that n^2 + k is a golden semiprime (A108540).

Original entry on oeis.org

5, 2, 6, 61, 52, 41, 28, 13, 106, 87, 66, 43, 18, 393, 364, 333, 300, 265, 228, 189, 148, 105, 60, 13, 226, 175, 122, 67, 10, 463, 402, 339, 274, 207, 138, 67, 814, 739, 662, 583, 502, 419, 334, 247, 158, 67, 538, 443, 346, 247, 146, 43, 4494, 4387, 4278, 4167, 4054

Offset: 1

Jason Earls, Jul 03 2005

Conjecture: for every n > 1 there exists a number k < n^3 such that n^2 + k is a golden semiprime.

			a(4)=61 because 4^2+61 = 77 = 7*11 and 7*phi-11 = 0.326237... < 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A108540.

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]]]; a[n_] := Module[{k = 1}, While[!goldQ[n^2 + k], k++]; k]; Array[a, 57] (* Amiram Eldar, Nov 29 2019 *)

A330092 The least prime that starts a chain of exactly n primes such that the product of each successive pair is a golden semiprime (A108540).

Original entry on oeis.org

5, 3, 2, 103, 2437, 6991, 455033, 252492571, 8276659373, 18749113741

Offset: 1

Amiram Eldar, Dec 01 2019

The question of the existence of arbitrary long chains of such primes was asked by Jonathan Vos Post in A107768.

Such chains may be called "golden chains of primes". They are analogous to Cunningham chains: this sequence is analogous to A005602, as A108541 is analogous to A005384.

			a(1) = 5 since 5 is not a lesser prime of a golden semiprime, i.e., it is not in A108541.
a(2) = 3 since 3 * 5 is a golden semiprime.
a(3) = 2 since {2, 3, 5} is a chain of 3 primes such that 2 * 3 and 3 * 5 are golden semiprimes.

Cf. A005602, A107768, A108540, A108541.

goldPrime[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]];
goldChainLength[p_] := -1 + Length @ NestWhileList[goldPrime, p, # > 0 &];
max = 7; seq = Table[0, {max}]; count = 0; p = 1; While[count < max, p = NextPrime[p]; i = goldChainLength[p]; If[i <= max && seq[[i]] < 1, count++; seq[[i]] = p]]; seq

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

Reinhard Zumkeller, Jun 09 2005

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001622, A084127, A108540, A108541, A108543, A108544.

Subsequence of A108539.

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 *)

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

A108539 a(n) = that prime p such that p/prime(n) is nearest to phi, the golden ratio: phi=(1+sqrt(5))/2.

Original entry on oeis.org

3, 5, 7, 11, 17, 23, 29, 31, 37, 47, 53, 59, 67, 71, 79, 83, 97, 97, 109, 113, 113, 127, 137, 149, 157, 163, 167, 173, 179, 181, 211, 211, 223, 223, 241, 241, 257, 263, 271, 281, 293, 293, 311, 313, 317, 317, 337, 359, 367, 373, 379, 389, 389, 409, 419, 421, 433

Offset: 1

Reinhard Zumkeller, Jun 09 2005

nonn

a(n)*prime(n) is a term of A108540 iff abs(phi-a(n)/prime(n))<1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Golden Ratio

Cf. A001622, A007067, A108540.

A108542 is a subsequence.

a[n_] := Module[{x = GoldenRatio * Prime[n]}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; If[x - p1 < p2 - x, p1, p2]]; Array[a, 57] (* Amiram Eldar, Nov 28 2019 *)

Corrected by T. D. Noe, Oct 25 2006

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

Reinhard Zumkeller, Jun 09 2005

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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 *)

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

cp's OEIS Frontend

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

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A108544 Primes that are factors of distinct golden semiprimes (A108540).

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A107768 Integers p*q*r such that p*q and q*r are both golden semiprimes (A108540). Integers p*q*r such that p = A108541(j), q = A108542(j) = A108541(k) and r = A108542(k).

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

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A108200 Least positive k such that k*n + 1 is a golden semiprime (A108540).

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A108399 Least positive k such that n^2 + k is a golden semiprime (A108540).

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A330092 The least prime that starts a chain of exactly n primes such that the product of each successive pair is a golden semiprime (A108540).

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A108542 Greater prime factor of n-th golden semiprime.

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A108539 a(n) = that prime p such that p/prime(n) is nearest to phi, the golden ratio: phi=(1+sqrt(5))/2.

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A107768 Integers pqr such that pq and qr are both golden semiprimes (A108540). Integers pqr such that p = A108541(j), q = A108542(j) = A108541(k) and r = A108542(k).