A147300 -id:A147300 - 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)

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 *)

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 *)

A147802 Least m coprime to 3 minimizing A007947(m*(3^n-m)).

Original entry on oeis.org

1, 1, 2, 1, 1, 25, 139, 289, 31, 1, 16096, 49, 7424, 588665, 619115, 83521, 8000000, 1515625, 505620842, 17643776, 244140625, 5443635008

Offset: 1

M. F. Hasler, Nov 13 2008

Related to the abc conjecture: Since m is coprime to 3, it is also coprime to 3^n and thus to 3^n-m. Thus A007947(m*(3^n-m)*3^n) = 3*A007947(m(3^n-m)).

Cf. A007947, A143700 (analog for 2^n), A147300 (general case).

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := MinimalBy[Select[Range[3^n - 2], CoprimeQ[#, 3] &], rad[# (3^n - #)] &][[1]];
Reap[Do[Print[n, " ", an = a[n]]; Sow[an], {n, 1, 16}]][[2, 1]] (* Jean-François Alcover, Mar 27 2020 *)

A147802(n, p=3) = {local(b, m=n=p^n); for(a=1, (n-1)\2, a%p || next; A007947(n-a)*A007947(a)A007947((n-a)*b=a)); b; }

a(17) from Jean-François Alcover, Mar 28 2020

a(18)-a(22) from Giovanni Resta, Mar 29 2020

A147803 Least m coprime to 5 minimizing A007947(m*(5^n-m)).

Original entry on oeis.org

1, 1, 4, 49, 128, 9, 36864, 19332, 4508, 121, 2

Offset: 1

M. F. Hasler, Nov 13 2008

The minima are given in A147800.

This is related to the abc conjecture: Since m is coprime to 5, it is also coprime to 5^n and thus to 5^n-m. Thus the squarefree kernel A007947(m*(5^n-m)*5^n) = 5*A007947(m*(5^n-m)).

Cf. A007947, A147298 (general case), A147800 (value of minima), A143700 (analog for 2^n), A147802 (analog for 3^n), A147300 (analog for any number).

A147803(n,p=5) = {my(b, m=n=p^n); for(a=1, n\2, a%p || next; A007947(n-a)*A007947(a)A007947((n-a)*b=a)); b; }

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

A147804 Least m coprime to 7 minimizing A007947(m*(7^n-m)).

Original entry on oeis.org

1, 1, 100, 1, 423, 28561, 3072, 124609, 119232

Offset: 1

M. F. Hasler, Nov 13 2008

The minima are given in A147799.

This is related to the abc conjecture: Since m is coprime to 7, it is also coprime to 7^n and thus to 7^n-m. Thus the squarefree kernel A007947(m*(7^n-m)*7^n) = 7*A007947(m(7^n-m)).

Cf. A007947, A147799 (value of minima), A143700, A147802, A147803 (analog for 2^n, 3^n, 5^n), A147300 (analog for any number).

A147804(n,p=7)={my(b, m=3*n=p^n, t); for(a=1, n\2, a%p || next; m>2*(t=A007947(a)) || next; m>(t*=A007947(n-a)) || next; m=t; b=a); b; }

Typo in title corrected by M. F. Hasler, Nov 17 2008

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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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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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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A147802 Least m coprime to 3 minimizing A007947(m*(3^n-m)).

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A147803 Least m coprime to 5 minimizing A007947(m*(5^n-m)).

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

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A147804 Least m coprime to 7 minimizing A007947(m*(7^n-m)).

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

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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.