A085153 -id:A085153 - cp's OEIS frontend

A143702 a(n) is the minimal values of A007947((2^n)m(2^n-m)).

2, 6, 14, 30, 30, 42, 30, 78, 182, 1110, 570, 1830, 6666, 2310, 2534, 5538, 9870, 20010, 141270, 14070, 480090, 155490, 334110, 1794858, 2463270, 2132130, 2349390

Offset: 1

Artur Jasinski, Nov 10 2008

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

For numbers m see A143700.

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

aa = {1}; bb = {1}; rr = {2}; 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, 2*r]; AppendTo[a, {x, logmax}], {x, 2, 15}]; rr (* Artur Jasinski with assistance of M. F. Hasler *)

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

A147641 Numbers B in the triples (A,B,C) that set a record in the L-function of the ABC conjecture if the search for C admits only the restricted integer subset of A009967 as described in A147642.

Original entry on oeis.org

16, 512, 12005, 6436341

Offset: 1

Artur Jasinski, Nov 09 2008

If the ABC conjecture is true this sequence is finite.

For numbers A for this case see A147643.

OEIS Index entries for sequences related to the abc conjecture.

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

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.

A086247 Differences of successive 7-smooth numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 3, 1, 1, 4, 2, 4, 3, 1, 6, 2, 3, 5, 1, 3, 6, 6, 2, 2, 5, 3, 4, 8, 5, 1, 2, 7, 5, 4, 3, 3, 10, 2, 6, 7, 5, 9, 3, 4, 4, 10, 6, 8, 1, 15, 3, 2, 5, 2, 4, 14, 10, 8, 6, 6, 15, 5, 4, 12, 7, 7, 10, 15, 3, 6, 8, 8, 5, 15, 12

Offset: 1

Reinhard Zumkeller, Jul 13 2003

A002473(n) is a term of A085153 iff a(n)=1.

			a(23) = 3 as A002473(23 + 1) - A002473(23) = 35 - 32 = 3. - _David A. Corneth_, Mar 31 2021

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A002473, A061987, A085153.

smooth7Q[n_] := n == Times@@({2, 3, 5, 7}^IntegerExponent[n, {2, 3, 5, 7}]);
Select[Range[1000], smooth7Q] // Differences (* Jean-François Alcover, Oct 17 2021 *)

a(n) = A002473(n+1) - A002473(n).

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

Original entry on oeis.org

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

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

A275156 The 108 numbers n such that n(n+1) is 17-smooth.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 24, 25, 26, 27, 32, 33, 34, 35, 39, 44, 48, 49, 50, 51, 54, 55, 63, 64, 65, 77, 80, 84, 90, 98, 99, 104, 119, 120, 125, 135, 143, 153, 168, 169, 175, 195, 220, 224, 242, 255, 272, 288, 324, 350, 351, 363, 374, 384, 440, 441, 539, 560, 594, 624, 675, 714, 728, 832, 935, 1000, 1088, 1155, 1224, 1274, 1700, 1715, 2057, 2079, 2400, 2430, 2499, 2600, 3024, 4095, 4224, 4374, 4913, 5831, 6655, 9800, 10647, 12375, 14399, 28560, 31212, 37179, 123200, 194480, 336140

Offset: 1

Jean-François Alcover, Nov 13 2016

This is the 7th row of the table A138180.

See A002071.

Cf. A002071, A145604, A138180, A145605, A002072, A085152, A085153, A252493, A252492.

pMax = 17; smoothMax = 10^12; smoothNumbers[p_?PrimeQ, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand@Log[pp[[j]], max/Times @@ (Take[pp, j - 1]^Take[aa, j - 1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; Select[(Sqrt[1 + 4*smoothNumbers[pMax, smoothMax]] - 1)/2, IntegerQ]

is(n)=my(t=510510); n*=n+1; while((t=gcd(n,t))>1, n/=t); n==1 \\ Charles R Greathouse IV, Nov 13 2016

A275164 The 167 numbers n such that n(n+1) is 19-smooth.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 32, 33, 34, 35, 38, 39, 44, 48, 49, 50, 51, 54, 55, 56, 63, 64, 65, 75, 76, 77, 80, 84, 90, 95, 98, 99, 104, 119, 120, 125, 132, 135, 143, 152, 153, 168, 169, 170, 175, 189, 195, 208, 209, 220, 224, 242, 255, 272, 285, 288, 323, 324, 342, 350, 351, 360, 363, 374, 384, 399, 440

Offset: 1

Jean-François Alcover, Nov 14 2016

See A002071.

The full list of 167 terms is given in the b-file (this is the 8th row of the table A138180).

Jean-François Alcover, Table of n, a(n) for n = 1..167

Cf. A002071, A145604, A138180, A145605, A002072, A085152, A085153, A252493, A252492,A275156.

pMax = 19; smoothMax = 10^15; smoothNumbers[p_?PrimeQ, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand@Log[pp[[j]], max/Times @@ (Take[pp, j - 1]^Take[aa, j - 1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; Select[(Sqrt[1 + 4*smoothNumbers[pMax, smoothMax]] - 1)/2, IntegerQ]

A303332 7-smooth numbers representable as the sum of two relatively prime 7-smooth numbers.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 21, 25, 27, 28, 32, 35, 36, 49, 50, 54, 64, 81, 125, 126, 128, 135, 189, 225, 245, 250, 256, 343, 375, 625, 1029, 2401, 4375

Offset: 1

Tomohiro Yamada, May 29 2018

It follows from Theorem 6.3 of de Weger's tract that there are exactly 40 terms, the largest of which is 4375 = 1 + 4374 = 5^4 * 7 = 1 + 2 * 3^7.
Indeed, de Weger determined all solutions of the equation x + y = z with x, y, z 13-smooth, x, y relatively prime and x <= y; there exist exactly 545 solutions.
Among them, exactly 63 solutions consist of 7-smooth numbers, which yields exactly 40 terms of this sequence.

			a(13) = 16 = 1 + 15 = 7 + 9 = 2^4 = 1 + 3 * 5 = 7 + 3^2.
a(25) = 81 = 1 + 80 = 25 + 56 = 32 + 49 = 3^4 = 1 + 2^4 * 5 = 5^2 + 2^3 * 7 = 2^5 + 7^2.

T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

B. M. M. de Weger, Algorithms for Diophantine Equations, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.

Cf. A085153 (subsequence)

s7 = Select[Range[10000], FactorInteger[#][[-1, 1]] <= 7 &]; Select[s7, AnyTrue[ IntegerPartitions[#, {2}, s7], GCD @@ # == 1 &] &] (* Giovanni Resta, May 30 2018 *)

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

A143702 a(n) is the minimal values of A007947((2^n)*m*(2^n-m)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A147641 Numbers B in the triples (A,B,C) that set a record in the L-function of the ABC conjecture if the search for C admits only the restricted integer subset of A009967 as described in A147642.

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A086247 Differences of successive 7-smooth numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A275156 The 108 numbers n such that n(n+1) is 17-smooth.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

PARI

A275164 The 167 numbers n such that n(n+1) is 19-smooth.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A303332 7-smooth numbers representable as the sum of two relatively prime 7-smooth numbers.

Views

Author

Keywords

Comments

Examples

References

A143702 a(n) is the minimal values of A007947((2^n)m(2^n-m)).