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.

A147307 Numbers A of the constrained search for ABC records described in A147306.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 5, 19, 41, 125, 23, 1, 1, 1, 95
Offset: 1

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Author

Artur Jasinski, Nov 09 2008

Keywords

Comments

The sequences A147305, a(n) and A147307 are steered by searching for records in the ABC conjecture along increasing C confined as described in A147306, the main entry for these three sequences.

Crossrefs

Cf. A085152, A085153, A147298-A147307.

Formula

a(n)+A147305(n) = A147306(n). gcd(a(n),A147305(n))=1.

Extensions

Edited by R. J. Mathar, Aug 24 2009

A147306 Numbers C in successive records of the merit function of the ABC conjecture considering only C from A033845.

Original entry on oeis.org

6, 12, 18, 24, 36, 48, 54, 144, 384, 486, 648, 2304, 3888, 5832, 279936
Offset: 1

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Author

Artur Jasinski, Nov 09 2008

Keywords

Comments

In a variant of the ABC conjecture (see A120498) we look at triples (A,B,C) restricted to A+B=C, gcd(A,B)=1, and at the merit function L(A,B,C)=log(C)/log(rad(A*B*C)), where rad() is the squarefree kernel A007947, as usual. Watching for records in L() as C runs through the integers generates A147302. In this sequence here, we admit only the C of the sequence A033845, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the L-values.
If the ABC conjecture is true this sequence is finite.
The associated numbers B for this case are A147305, the associated A are A147307.

Examples

			(A,B,C) = (1,5,6) defines the first record, L=0.5268.. (A,B,C)=(1,11,12) reaches L=0.5931..
(A,B,C) = (1,17,18) reaches L=0.6249.. The first C-number selected from A033845 that does not generate a new record is 72.

Crossrefs

Cf. A085152, A085153, A147298-A147307.

Programs

  • Maple
    Digits := 120 : A007947 := proc(n) local f,p; f := ifactors(n)[2] ; mul( op(1,p),p=f) ; end:
L := proc(A,B,C) local rad; rad := A007947(A*B*C) ; evalf(log(C)/log(rad)) ; end:
isA033845 := proc(n) local f,p; f := ifactors(n)[2] ; for p in f do if not op(1,p) in {2,3} then RETURN(false) ; fi; od: RETURN( (n mod 2 = 0 ) and (n mod 3 = 0 ) ) ; end:
crek := -1 : for C from 3 do if isA033845(C) then for A from 1 to C/2 do B := C-A ; if gcd(A,B) = 1 then l := L(A,B,C) ; if l > crek then print(C) ; crek := l ; fi; fi; od: fi; od: # R. J. Mathar, Aug 24 2009

Extensions

Edited by R. J. Mathar, Aug 24 2009

A143701 a(n) is the least odd number 2^n - m minimizing A007947(m*(2^n - m)).

Original entry on oeis.org

1, 3, 7, 15, 27, 63, 125, 243, 343, 999, 1805, 3721, 8181, 16335, 32761, 65533, 112847, 190269, 519375, 1046875, 1953125, 4192479, 8385125, 16775019, 24398405, 66976875, 134216625
Offset: 1

Views

Author

Artur Jasinski, Nov 10 2008

Keywords

Comments

a(n) is the smallest odd number such that the product of distinct prime divisors of (2^n)*a(n)*(2^n-a(n)) is the smallest for the range a(n) <= 2^x - a(n) < 2^x.
The product of distinct prime divisors of m*(2^n-m) is also called the radical of that number: rad(m*(2^n-m)).

Crossrefs

Cf. A007947, A085152, A085153.
Cf. A147298, A147299, A147300, A147302, A147303, A147305, A147306, A147307.
Cf. A147638, A147639, A147640, A147641, A147642, A147643.

Programs

  • Mathematica
    aa = {1}; bb = {1}; rr = {}; Do[logmax = 0; k = 2^x; w = Floor[(k - 1)/2]; Do[m = FactorInteger[n (k - n)]; rad = 1; Do[rad = rad m[[s]][[1]], {s, 1, Length[m]}]; log = Log[k]/Log[rad]; If[log > logmax, bmin = k - n; amax = n; logmax = log; r = rad], {n, 1, w, 2}]; Print[{x, amax}]; AppendTo[aa, amax]; AppendTo[bb, bmin]; AppendTo[rr, r]; AppendTo[a, {x, logmax}], {x, 2, 15}]; bb (* Artur Jasinski with assistance of M. F. Hasler *)

Formula

a(n) = 2^n - A143700(n).

Extensions

a(1) added by Jinyuan Wang, Aug 11 2020
Showing 1-3 of 3 results.

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