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 1-3 of 3 results.

A202157 a(n) = smallest k having at least two prime divisors d such that (d + n) | ( k + n).

Original entry on oeis.org

63, 18, 45, 50, 75, 66, 63, 102, 75, 50, 165, 198, 147, 258, 165, 110, 663, 182, 399, 442, 147, 242, 705, 678, 455, 786, 483, 182, 1015, 950, 1023, 988, 363, 506, 637, 1446, 1083, 322, 885, 590, 1155, 1443, 1935, 2118, 627, 770, 3243, 2502, 1407, 2706, 845
Offset: 1

Views

Author

Michel Lagneau, Dec 13 2011

Keywords

Comments

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree: 63, 18, 45, ...

Examples

			a(8) = 102 because the prime divisors of 102 are 2, 3 and 17;
(2 + 8) | (102 + 8) = 110 = 10*11;
(3 + 8) | 110 = 11*10.

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.

Links

Crossrefs

Cf. A006972, A029591, A202158, A202159.

Programs

  • Maple
    with(numtheory):for n from 1 to 52 do:i:=0:for k from 1 to 5000 while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>=2 then i:=1:printf(`%d, `,k):else fi:od:od:
  • Mathematica
    numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 1, k++]; k]; Array[a, 50] (* Amiram Eldar, Sep 09 2019 *)

Formula

a(n) >= n^2 + 4n + 6. [Charles R Greathouse IV, Dec 13 2011]

A202159 a(n) = smallest k having at least four prime divisors d such that (d + n) | (k + n).

Original entry on oeis.org

8855, 11590, 27885, 122360, 16555, 10290, 6545, 61642, 71799, 65195, 14245, 142788, 63635, 580930, 39585, 21098, 69003, 258482, 59885, 378952, 8715, 266090, 133285, 690501, 27335, 704790, 1017423, 299222, 187891, 771650, 293405, 1638598, 282315, 553610, 227205
Offset: 1

Views

Author

Michel Lagneau, Dec 13 2011

Keywords

Comments

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree.

Examples

			a(3) = 27885 because the prime divisors of 27885 are 3, 5, 11, 13  =>
(3 + 3)| (27885 + 3) = 27888 = 6*4648;
(5 + 3) | 27888 = 8*3486;
(11 + 3) | 27888 = 14*1992;
(13 + 3) | 27888 = 16*1743.

Links

Crossrefs

Cf. A006972, A029591, A202157, A202158.

Programs

  • Maple
    with(numtheory):for n from 1 to 33 do:i:=0:for k from 1 to 5000000  while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>3 then i:=1:printf(`%d, `,k):else fi:od:od:
  • Mathematica
    numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 3, k++]; k]; Array[a, 35] (* Amiram Eldar, Sep 09 2019 *)

A202160 a(n) = smallest k having at least five prime divisors d such that (d + n) | (k + n).

Original entry on oeis.org

588455, 179998, 460317, 6265805, 1236235, 287274, 949025, 1436932, 794871, 2013650, 3797365, 1169688, 3739827, 1587586, 6872565, 7706270, 1529983, 7351242, 2528045, 5247970, 487179, 10920965, 1316497, 121894476, 1404455, 5814874, 12223653, 2260412, 8022531
Offset: 1

Views

Author

Michel Lagneau, Dec 13 2011

Keywords

Comments

The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
In this sequence, the majority of terms are not squarefree.

Examples

			a(3) = 460317 because the prime divisors of 460317 are 3, 11, 13, 29, 37  =>
(3 + 3) | (460317 + 3) = 460320 = 6*76720;
(11 + 3) | 460320 = 14*32880;
(13 + 3) | 460320 = 16*28770;
(29+3)  |  460320 = 32*14385;
(37+3) | 460320 = 40*11508.

Links

Crossrefs

Cf. A006972, A029591, A202157, A202158, A202159.

Programs

  • Maple
    with(numtheory):for n from 1 to 23 do:i:=0:for k from 1 to 10^8 while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>4 then i:=1: printf ( "%d %d \n",n,k):else fi:od:od:
  • Mathematica
    numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 4, k++]; k]; Array[a, 30] (* Amiram Eldar, Sep 09 2019 *)
Showing 1-3 of 3 results.

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