A147301 -id:A147301 - 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 *)

A147302 Numbers k where records occur in the expression log(k) / log(A147298(k)).

Original entry on oeis.org

2, 9, 81, 128, 2401, 4375, 6436343

Offset: 1

Artur Jasinski, Nov 05 2008

Numbers a(n) such that a(n)/R(m a(n)(a(n)-m)) > a(n-1)/R(g a(n-1)(a(n-1)-g)) 0 < m < a(n) and 0 < g < a(n-1).
This sequence is list of successive records in the abc conjecture.
No more terms up to 10^20.
For smallest values of function rad(m*n*(n-m)) see A147298.
For biggest values of 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.

Carlos Rivera, Puzzle 901. A question about consecutive integers, The Prime Puzzles and Problems Connection.

Cf. A085152, A085153, A147298-A147307.

logmax = 0; aa = {}; 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}]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[aa, n]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 2500}]; aa

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 *)

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

A147303 Numbers k where records occur in expression Log[A147298(k)]/Log[k] k=2,3,4,...

Original entry on oeis.org

2, 3, 6, 7, 14, 15, 22, 30, 42, 62, 66, 70, 78, 102, 114, 158, 166, 182, 186, 210, 222, 230, 255, 258, 282, 318, 330, 402, 430, 438, 462, 470, 474, 494, 498, 510, 570, 582, 598, 690, 710, 786, 798, 822, 870, 906, 930, 942, 1002, 1038, 1074, 1110, 1122, 1146, 1158

Offset: 1

Artur Jasinski, Nov 06 2008

nonn

Limit k->Infinity Log[A147298(k)]/Log[k] = 2.

Values m for which records occur, see A147301.

Cf. A147298-A147302, A147304-A147307.

logmin = 10^10; logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; hh = {}; ii = {}; jj = {}; Do[min = 10^100; max = 0; Do[If[GCD[m, n, n - m] == 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]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]]; If[(Log[n]/Log[min]) < logmin, logmin = (Log[n]/Log[min]); AppendTo[hh, {N[logmin], n, mmin, min, mmax, max}]; AppendTo[ii, n]]; AppendTo[bb, max]; AppendTo[dd, mmin], {n, 2, 1200}]; ii

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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A147302 Numbers k where records occur in the expression log(k) / log(A147298(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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A179017 Odd numbers c such that c*(c^2 - 1)/4 is squarefree.

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

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A147303 Numbers k where records occur in expression Log[A147298(k)]/Log[k] k=2,3,4,...

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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.