A147298 -id:A147298 - cp's OEIS frontend

A147305 Numbers B of the constrained search for ABC records described in A147306.

5, 11, 17, 23, 35, 47, 49, 125, 343, 361, 625, 2303, 3887, 5831, 279841

Offset: 1

Artur Jasinski, Nov 09 2008

The sequences a(n), A147306 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.

Cf. A085152, A085153, A147298-A147307.

A147307(n)+a(n) = A147306(n). gcd(A147307(n),a(n))=1.

Edited and 25 replaced by 35 - R. J. Mathar, Aug 24 2009

A172121 Complement to A172120. Related to the ABC conjecture.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 17, 18, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Artur Jasinski, Jan 26 2010

nonn

Numbers k for which the maximum of the function log(k)/log(N(x,k-x,k)) occurs only for a single value of x (x < k-x, function N(x,k-x,k) is the squarefree kernel of x*(k-x)*k and gcd(x,k-x,k)=1).

Equivalent description without the use of the logarithmic function: Let R(n,k) = rad(n*k*(n-k)) if n is prime to k and otherwise +oo. Also let L(n) = [R(n,k) for k = 1..n]. Then m is in this list <=> min(L(m)) occurs exactly once in L(m). (All minima are listed in A147298.) - Peter Luschny, Aug 05 2019

			Using the equivalent description the rows for prime numbers begin:
[2]  [2]
[3]  [6]
[5]  [10, 30]
[7]  [42, 70, 42]
[11] [110, 66, 66, 154, 330]
[13] [78, 286, 390, 78, 130, 546]
[17] [34, 510, 714, 442, 510, 1122, 1190, 102]
[19] [114, 646, 114, 570, 1330, 1482, 798, 418, 570]
2, 3, 5 and 17 are on the list because the minimum in their row is unique, 7, 11, 19 do not occur because the minimum is more than once in the row.

Cf. A007947, A147638, A147306, A147639, A147640, A172120, A147298.

rad := n -> mul(k, k in numtheory:-factorset(n)):
g := (n, k) -> `if`(igcd(n, k) = 1, 1, infinity):
L := n -> [seq(g(n, k)*rad(n*k*(n-k)), k=1..n/2)]:
isA172121 := n -> nops([ListTools:-SearchAll(min(L(n)), L(n))]) = 1:
select(isA172121, [$1..87]); # Peter Luschny, Aug 05 2019

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
isok(n) = {my(v = vector(n\2, k, if (gcd([k, n, n-k]) == 1, rad(k*(n-k)*n), oo))); if (#v, #select(x->(x==vecmin(v)), v) == 1);} \\ Michel Marcus, Aug 06 2019

Offset 1 and name corrected by Michel Marcus, Aug 04 2019

Prepended 2 to the list by Peter Luschny, Aug 06 2019

A147798 Minimal value of A007947(m*(11^n-m)) with m coprime to 11.

Original entry on oeis.org

6, 30, 30, 390, 3162, 2730, 17706

Offset: 1

M. F. Hasler, Dec 06 2008

The minima are reached for m values given in A147805.

Cf. A007947, A147298, A147799, A147800, A147801, A147802, A147803, A147804, A147805.

A147798(n, p=11) = {my(m=n=p^n); for(a=1, n\2, a%p || next; m > 2*A007947(a) || next; m > A007947(n-a)*A007947(a) || next; m = A007947(n-a)*A007947(a) ); m; }

A147805 Least m coprime to 11 minimizing A007947(m*(11^n-m)).

Original entry on oeis.org

2, 1, 81, 16, 8748, 91, 1489347

Offset: 1

M. F. Hasler, Dec 06 2008

The values of the minima are given in A147798.

Cf. A007947, A147298, A147798, A147799, A147800, A147801, A147802, A147803, A147804.

/* If A147798(n) has already been computed and is stored in a147798[n]: */
A147805(n)={my(m=a147798[n]); for(a=1, n=11^n, a%11 || next; m>2*A007947(a) || next; m==A007947(a)*A007947(n-a) && return(a)) }
/* else use code from A147804 : A147805(n)=A147804(n,11) */

A179019 a(n) = (A179017(n)+1)/2.

Original entry on oeis.org

2, 3, 6, 7, 11, 15, 22, 30, 31, 34, 35, 39, 42, 43, 47, 58, 62, 66, 67, 70, 71, 78, 79, 83, 87, 94, 102, 103, 106, 107, 110, 111, 114, 115, 119, 130, 134, 139, 142, 143, 146, 155, 159, 166, 174, 178, 179, 183, 186, 187, 191, 195, 202, 206, 210, 211, 214, 215, 218, 219

Offset: 1

Artur Jasinski, Jun 24 2010

nonn

For numbers a and c, see A172186 and A179017. Numbers b are this sequence.
These numbers c, with distribution a+b=c such that a=(c-1)/2 and b=(c+1)/2, have minimal possible values with function L(a,b,c) = log(c)/log(N[a,b,c]) = log(c)/log((c^2-1)c/4).
There exist no numbers or distributions for which L < log(c)/log((c^2-1)c/4). - Artur Jasinski

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

Cf. A172120, A172121, A147298, A147299, A147300, A147301, A147302, A147306, A147307, A147638, A147639, A147640, A147641, A147642, A147643, A143700, A179017, A172186.

aa = {}; Do[If[(GCD[x, (x - 1)/2] == 1) && (GCD[x, (x + 1)/2] == 1) && (GCD[(x - 1)/2, (x + 1)/2] == 1), If[SquareFreeQ[(x^2 - 1) x/4], AppendTo[aa, (x + 1)/2]]], {x, 2, 1000}]; aa

a(n) = A179017(n) - A172186(n). - Hugo Pfoertner, Mar 22 2020

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

Artur Jasinski, Nov 10 2008

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)).

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

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 *)

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

a(1) added by Jinyuan Wang, Aug 11 2020

A143703 a(n) = A143702(n)/2.

Original entry on oeis.org

1, 3, 7, 15, 15, 21, 15, 39, 91, 555, 285, 915, 3333, 1155, 1267, 2769, 4935, 10005, 70635, 7035, 240045, 77745, 167055, 897429, 1231635, 1066065, 1174695

Offset: 1

Artur Jasinski, Nov 10 2008

The product of distinct prime divisors of m*(2^n-m) is also called the radical of that number: rad(m*(2^n-m)).

Cf. A007947, A085152, A085153, A147298-A147307, A147638-A147643.

aa = {1}; bb = {1}; rr = {1}; 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}]; rr (* Artur Jasinski with assistance of M. F. Hasler *)

a(1) added by Jinyuan Wang, Aug 11 2020

cp's OEIS Frontend

A147305 Numbers B of the constrained search for ABC records described in A147306.

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A172121 Complement to A172120. Related to the ABC conjecture.

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A147798 Minimal value of A007947(m*(11^n-m)) with m coprime to 11.

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A147805 Least m coprime to 11 minimizing A007947(m*(11^n-m)).

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A179019 a(n) = (A179017(n)+1)/2.

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

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A143703 a(n) = A143702(n)/2.

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