A147302 -id:A147302 - cp's OEIS frontend

A147298 Minimum of rad(m (n - m) n) for 0 < m < n, gcd(m,n) = 1, where rad(k) = A007947(k) = product of prime factors of k.

2, 6, 6, 10, 30, 42, 14, 6, 30, 66, 66, 78, 182, 210, 30, 34, 102, 114, 190, 210, 462, 322, 138, 30, 130, 30, 42, 174, 870, 186, 30, 66, 510, 210, 210, 222, 1254, 546, 390, 246, 1722, 258, 946, 330, 690, 1410, 282, 42, 70, 510, 390, 742, 210, 330, 770, 570, 1218

Offset: 2

Artur Jasinski, Nov 05 2008

nonn

Function rad(k) is used in ABC conjecture applications.
For biggest values of function rad(m n (n - m)) see A147299.
For numbers m for which rad(m n (n - m)) reached minimal value see A147300.
For numbers m for which rad(m n (n - m)) reached maximal value see A147301.
Sequence in each value Log[n]/Log[A147298(n)] reached records see A147302.

Ivan Neretin, Table of n, a(n) for n = 2..1000

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

A147298 := proc(n) local rad, g, L;
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)]:
min(L(n)) end: seq(A147298(n), n=2..58); # Peter Luschny, Aug 06 2019

logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; gg \ = {}; Do[min = 10^100; max = 0; ile = 0; Do[If[GCD[m, n, n - m] == 1, ile = ile + 1; s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; AppendTo[aa, min]; AppendTo[bb, max]; AppendTo[cc, mmax]; AppendTo[dd, mmin]; AppendTo[gg, ile]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 129}]; aa (*Artur Jasinski*)
Table[Min[Times @@ FactorInteger[#][[All, 1]] & /@ ((m = Select[Range[1, n - 1], GCD[n, #] == 1 &])*(n - m)*n)], {n, 2, 58}] (* Ivan Neretin, May 21 2015 *)

A147298(n)= local(m=n^2); for( a=1,n\2, gcd(a,n)>1 && next; A007947(n-a)*A007947(a)A007947(n-a)*A007947(a)); m*A007947(n)

A147300 a(n) = smallest value of parameter m such that the function rad(mn(n - m)) has minimal value n=2,3,4,..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 2, 1, 2, 1, 4, 5, 1, 9, 3, 1, 1, 11, 7, 1, 9, 1, 16, 1, 1, 1, 2, 1, 1, 1, 1, 25, 4, 5, 1, 1, 25, 9, 27, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 25, 11, 1, 13, 1, 4, 1, 1, 1, 2, 1, 4, 5, 23, 7, 8, 1, 27, 11, 1, 13, 14, 1, 1, 17, 1, 1

Offset: 2

Artur Jasinski, Nov 05 2008

nonn

The function rad(k) is used in ABC conjecture applications.
For smallest values of the function rad(m n (n - m)) see A147298.
For the largest values of the function rad(m n (n - m)) see A147299.
For numbers m at which rad(m*n*(n - m)) reaches minimal value see A147300.
For numbers m at which rad(m*n*(n - m)) reaches maximal value see A147301.
For sequence in which each value log(n)/log(A147298(n)) reaches records see A147302.

Cf. A085152, A085153, A147298-A147307.

logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; gg \ = {}; Do[min = 10^100; max = 0; ile = 0; Do[If[GCD[m, n, n - m] == 1, ile = ile + 1; s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; AppendTo[aa, min]; AppendTo[bb, max]; AppendTo[cc, mmax]; AppendTo[dd, mmin]; AppendTo[gg, ile]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 129}]; dd (* Artur Jasinski *)

A147299 a(n) = largest value of the function rad(mn(n - m)) n=2,3,4,..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.

Original entry on oeis.org

2, 6, 6, 30, 30, 70, 30, 42, 210, 330, 210, 546, 462, 390, 110, 1190, 462, 1482, 910, 2310, 2310, 2990, 858, 770, 4290, 546, 2730, 6090, 6630, 7378, 510, 8778, 9690, 10010, 1938, 12210, 13566, 14586, 3990, 17138, 18354, 19866, 10626, 7590, 22678

Offset: 2

Artur Jasinski, Nov 05 2008

nonn

The function rad(k) is used in ABC conjecture applications.
For smallest values of the function rad(m n (n - m)) see A147298.
For numbers m at which rad(m*n*(n - m)) reaches minimal value see A147300.
For numbers m at which rad(m*n*(n - m)) reaches maximal value see A147301.
Sequence in which each value log(n)/log(A147298(n)) reaches records see A147302.

Ivan Neretin, Table of n, a(n) for n = 2..1000

Cf. A085152, A085153, A147298-A147307.

logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; gg \ = {}; Do[min = 10^100; max = 0; ile = 0; Do[If[GCD[m, n, n - m] == 1, ile = ile + 1; s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; AppendTo[aa, min]; AppendTo[bb, max]; AppendTo[cc, mmax]; AppendTo[dd, mmin]; AppendTo[gg, ile]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 129}]; bb (* Artur Jasinski *)
Table[Max[Times @@ FactorInteger[#][[All, 1]] & /@ ((m = Range[1, n - 1])*(n - m)*n)], {n, 2, 46}] (* Ivan Neretin, May 21 2015 *)

A147301 a(n) = smallest value of parameter m such that the function rad(m n (n - m)) has maximal value n=2,3,4..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 3, 5, 5, 6, 3, 2, 5, 7, 7, 6, 7, 10, 7, 10, 11, 11, 11, 13, 13, 14, 13, 14, 15, 14, 15, 13, 17, 15, 17, 17, 19, 19, 19, 21, 21, 22, 17, 21, 19, 23, 21, 22, 23, 23, 23, 26, 23, 26, 23, 29, 29, 30, 29, 29, 31, 31, 31, 33, 33, 34, 33, 34, 35, 35, 35, 37, 37, 38

Offset: 2

Artur Jasinski, Nov 05 2008

nonn

Function rad(k) is used in ABC conjecture applications.
For smallest values of the function rad(m n (n - m)) see A147298.
For biggest values of the function rad(m n (n - m)) see A147299.
For numbers m for which rad(m n (n - m)) reaches a minimal value see A147300.
For numbers m for which rad(m n (n - m)) reaches a maximal value see A147301.
For the sequence in each value log(n)/log(A147298(n)) reached records see A147302.

Cf. A085152, A085153, A147298-A147307.

logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; gg \ = {}; Do[min = 10^100; max = 0; ile = 0; Do[If[GCD[m, n, n - m] == 1, ile = ile + 1; s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; AppendTo[aa, min]; AppendTo[bb, max]; AppendTo[cc, mmax]; AppendTo[dd, mmin]; AppendTo[gg, ile]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 129}]; cc (* Artur Jasinski *)

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

Artur Jasinski, Nov 09 2008

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.

			(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.

Cf. A085152, A085153, A147298-A147307.

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

Edited by R. J. Mathar, Aug 24 2009

A147639 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 2.

Original entry on oeis.org

4, 8, 16, 32, 64, 128, 1048576

Offset: 1

Artur Jasinski, Nov 09 2008

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 A000079, 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 A147638, the associated A are A147640.

			The case C=2 does not create a valid (A,B,C) triple, so C=4 is the first case, which sets a first record L=0.7737 with (A,B,C)=(1,3,4). The next admitted case, C=8, sets a new record L=0.7879 with (A,B,C)=(1,7,8), and so do (A,B,C)=(1,15,16) with L=0.8151. For C=32, we consider the largest L possible for A<B<C, which is (A,B,C)=(5,27,32) with L=1.0189. The value L=0.839 from (A,B,C)=(1,31,32) at the same C is smaller and discarded.

Abderrahmane Nitaj, The ABC conjecture homepage

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

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:
crek := -1 : for x from 2 do C := 2^x ; 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: od: # R. J. Mathar, Aug 28 2009

a(2) corrected by R. J. Mathar, Aug 28 2009

A147642 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 23.

Original entry on oeis.org

23, 529, 12167, 6436343

Offset: 1

Artur Jasinski, Nov 09 2008

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 form 23^x, see A009967, which avoids some early larger records that would be created by unrestricted C, and leads to a slower increase of the L-values.

For associated B for this case see A147641, for associated A see A147643.

			C= 23 is the first candidate (and therefore by definition a record). Scanning the pairs (A,B) for this C we have L-values of L(1,22,23) = 0.5035, L(2,21,23) = 0.456, ... L(6,17,23) = 0.404, L(7,16,23) = 0.542 ,... L(11,12,23) = 0.428. The largest L-value stems from (A=7,B=16) which means the representative triple of the first record is (A,B,C) = (7,16,23).
C= 23^2= 529 is the next candidate. Scanning again all (A,B) values subject to the constraints we achieve L(17,512,529) = 0.941... (Smaller ones like L(81,448,529) = 0.9123... are discarded). Since the L-value for C=529 is larger than the L-value for C=23, the next record is C=529 with representatives (A,B,C)= (17,512,529).
The third candidate is C= 23^3= 12167. This generates a maximum of L(162,12005,12167) = 1.1089... (smaller values like L(17,12150,12167) = 1.0039.. discarded) which is again larger than the maximum of the previous record (which was 0.941..) So the C-value of 12167 is again a record-holder.

OEIS Index entries for sequences related to the abc conjecture.

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

A179017 Odd numbers c such that c*(c^2 - 1)/4 is squarefree.

Original entry on oeis.org

3, 5, 11, 13, 21, 29, 43, 59, 61, 67, 69, 77, 83, 85, 93, 115, 123, 131, 133, 139, 141, 155, 157, 165, 173, 187, 203, 205, 211, 213, 219, 221, 227, 229, 237, 259, 267, 277, 283, 285, 291, 309, 317, 331, 347, 355, 357, 365, 371, 373, 381, 389, 403, 411, 419, 421

Offset: 1

Artur Jasinski, Jun 24 2010

nonn

Original title was: "Numbers c such that (c^2-1)c is square free and gcd(c-1,c,c+1)=1", but (c^2-1)c is never squarefree for odd c, and gcd(n,n+1) is always = 1. - M. F. Hasler, Nov 03 2013
These numbers c with distribution a+b=c such that a=(c-1)/2 (see A172186) and b=(c+1)/2 (see A179019) 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).
This function is minimal orbital in hypothesis (a,b,c).
There are no numbers or distributions which have value L less than log(c)/log((c^2-1)*c/4).
Equivalently, odd squarefree numbers c such that (c^2 - 1)/4 is also squarefree. - Jon E. Schoenfield, Feb 13 2023
The asymptotic density of this sequence is Product_{p prime} (1 - 3/p^2) = A206256 = 0.125486980905... (Tsang, 1985). - Amiram Eldar, Feb 26 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Kai-Man Tsang, The distribution of r-tuples of square-free numbers, Mathematika, Vol. 32, No. 2 (1985), pp. 265-275.

Cf. A172120, A172121, A147298, A147299, A147300, A147301, A147302, A147306, A147307, A147638, A147639, A147640, A147641, A147642, A147643, A143700, A172186, A179019, A206256.

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]]], {x, 2, 1000}]; aa

forstep(n=3,421,2,issquarefree(n*(n^2-1)/4)&&print1(n",")) \\ M. F. Hasler, Nov 03 2013

is(n)=n%2 && issquarefree(n) && issquarefree(n^2\4) \\ Charles R Greathouse IV, Mar 11 2014

a(n) = 2*A172186(n) + 1. - Bernard Schott, Mar 06 2023

Edited by M. F. Hasler, Nov 03 2013

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

cp's OEIS Frontend

A147298 Minimum of rad(m (n - m) n) for 0 < m < n, gcd(m,n) = 1, where rad(k) = A007947(k) = product of prime factors of k.

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A147300 a(n) = smallest value of parameter m such that the function rad(m*n*(n - m)) has minimal value n=2,3,4,..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.

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A147299 a(n) = largest value of the function rad(m*n*(n - m)) n=2,3,4,..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.

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A147301 a(n) = smallest value of parameter m such that the function rad(m n (n - m)) has maximal value n=2,3,4..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.

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

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A147639 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 2.

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A147642 Numbers C which generate successive records of the merit function of the ABC conjecture admitting only C which are powers of 23.

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A179017 Odd numbers c such that c*(c^2 - 1)/4 is squarefree.

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

A147300 a(n) = smallest value of parameter m such that the function rad(mn(n - m)) has minimal value n=2,3,4,..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.

A147299 a(n) = largest value of the function rad(mn(n - m)) n=2,3,4,..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.