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A165570 Successively better golden semiprimes. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Amiram Eldar, Table of n, a(n) for n = 1..48

FORMULA

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

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 *)

CROSSREFS

Cf. A108539, A165569, A165571, A165572.

Sequence in context: A177122 A108540 A232170 * A260117 A220360 A096565

Adjacent sequences:  A165567 A165568 A165569 * A165571 A165572 A165573

KEYWORD

nonn

AUTHOR

Antti Karttunen, Sep 22 2009

EXTENSIONS

a(16)-a(23) from Donovan Johnson, May 13 2010

a(24)-a(26) from Amiram Eldar, Nov 28 2019

STATUS

approved

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Last modified June 16 19:49 EDT 2021. Contains 345068 sequences. (Running on oeis4.)