cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Artur Jasinski, Nov 10 2008

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

A143700 a(n) is the least odd number m minimizing A007947(m*(2^n-m)).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 3, 13, 169, 25, 243, 375, 11, 49, 7, 3, 18225, 71875, 4913, 1701, 144027, 1825, 3483, 2197, 9156027, 131989, 1103, 5103, 38525, 458703, 1523, 3483891, 19283525
Offset: 1

Views

Author

Artur Jasinski, Nov 10 2008

Keywords

Comments

Smallest odd number a(n) such that product of distinct prime divisors of (2^n)*a(n)*(2^n - a(n)) is the smallest available for a(n) <= 2^x - a(n) < 2^x.
Product of distinct prime divisors of (2^n)*a(n)*(2^n - a(n)) is also called radical: rad((2^n)*a(n)*(2^n - a(n))).
For numbers 2^n - a(n) see A143701.
For minimal values of rad((2^n)*a(n)*(2^n - a(n))) see A143702.
Related to the abc conjecture. - M. F. Hasler, Nov 13 2008

Crossrefs

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

Programs

  • Mathematica
    a = {{1, 1}}; aa = {1}; bb = {}; 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}]; aa (* Artur Jasinski with assistance of M. F. Hasler *)
  • PARI
    A143700(n) = {my(b=1, m=2^n-b); forstep(a=3, 2^(n-1), 2, A007947(a)*A007947(2^n-a)A007947((2^n-a)*b=a)); b; } \\ M. F. Hasler, Nov 13 2008

Extensions

a(28)-a(33) from M. F. Hasler, Nov 13 2008

A147800 Minimal value of A007947(m*(5^n-m)) with m coprime to 5.

Original entry on oeis.org

2, 6, 22, 42, 222, 366, 2046, 13962, 10626, 79926, 293262
Offset: 1

Views

Author

M. F. Hasler, Nov 13 2008

Keywords

Comments

The minima are reached for m values given in A147803.
This is related to the abc conjecture.
All terms of this sequence are even, so one could also consider A147800/2 = 1, 3, 11, 21, 111, 183, 1023, 6981, 5313, 39963, 146631, ...

Crossrefs

Cf. A007947, A147803 (m values), A143702 (analog for 2^n), A147801 (analog for 3^n), A147298 (general case).

Programs

  • PARI
    A147800(n, p=5) = {my(m=n=p^n); for(a=1, (n-1)\2, a%p || next; A007947(n-a)*A007947(a)A007947((n-a)*a)); m; }

A147801 Minimal value of A007947(m*(3^n-m)) with m coprime to 3.

Original entry on oeis.org

2, 2, 10, 10, 22, 110, 278, 238, 1054, 1342, 11066, 6118, 18734, 107030, 557270, 163030, 1440430, 2195110, 11016290, 3641210, 23250370, 38188766
Offset: 1

Views

Author

M. F. Hasler, Nov 13 2008

Keywords

Comments

Related to the abc conjecture. The minima are reached for m values given in A147802.
All terms of this sequence are even, so one could also consider A147801/2 = 1, 1, 5, 5, 11, 55, 139, 119, 527, 671, 5533, 3059, 9367, 53515, 278635, 81515, ...

Crossrefs

Cf. A007947, A147298 (general case), A143702 (analog for 2^n), A147800 (analog for 5^n), A147802.

Programs

  • PARI
    A147801(n, p=3) = {my(m=n=p^n); for(x=1, (n-1)\2, x%p || next; A007947(n-x)*A007947(x)A007947((n-x)*x)); m; }

Extensions

a(17)-a(22) from Jinyuan Wang, Aug 11 2020

A147799 Minimal value of A007947(m*(7^n-m)) with m coprime to 7.

Original entry on oeis.org

6, 6, 30, 30, 282, 2262, 17034, 36006, 71070
Offset: 1

Views

Author

M. F. Hasler, Nov 13 2008

Keywords

Comments

The minima are reached for m values given in A147804.
This is related to the abc conjecture.
All terms of this sequence are even, so one could also consider A147799/2 = 3, 3, 15, 15, 141, 1131, 8517, 18003, 35535, ... So far these terms are also multiples of 3, but this might be a coincidence.

Crossrefs

Cf. A007947, A147804 (m values); A143702, A147801, A147800 (analog for 2^n, 3^n, 5^n), A147298 (general case).

Programs

Showing 1-5 of 5 results.

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