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 1-4 of 4 results.

A109203 Minimal value of k>0 such that n^8 + k^2 is a semiprime.

Original entry on oeis.org

2, 3, 3, 14, 3, 2, 1, 5, 7, 1, 1, 4, 5, 1, 3, 7, 1, 10, 1, 11, 1, 4, 1, 6, 13, 3, 1, 20, 1, 4, 11, 4, 1, 1, 1, 16, 5, 5, 1, 4, 3, 6, 1, 1, 15, 4, 5, 1, 17, 4, 1, 1, 1, 1, 11, 4, 1, 14, 1, 10, 1, 14, 7, 4, 15, 4, 1, 4, 1, 1, 1, 9, 1, 15, 9, 8, 9, 10, 5, 14, 3, 1, 5, 6, 1, 3, 19, 14, 5, 6, 41, 4, 1, 14, 1
Offset: 0

Views

Author

Jonathan Vos Post, Jul 03 2005

Keywords

Comments

There seems to be a modest correlation with the n^7 sequence (A109202) with often the same values [n = 10,16,18,23,31,45,52,55,66,72,82,88,100]. Sometimes the same value of k occurs for the n^6 sequence (A109201), the n^7 sequence (A109202) and this n^8 sequence, for instance n=88, k=5. The statistics of these sequences is unclear, as are the asymptotics.

Examples

			a(0) = 2 because 0^8 + 1^2 = 1 is not semiprime, but 0^8 + 2^2 = 4 = 2^2 is.
a(1) = 3 because 1^8 + 1^2 and 1^8 + 2^2 are not semiprime, but 1^8 + 3^2 = 10 = 2 * 5 is semiprime.
a(2) = 3 because 2^8 + 3^2 = 265 = 5 * 53 is semiprime, but 2^8 + 1^2 and 2^8 + 2^2 are not semiprimes.
a(90) = 41 because 90^8 + 41^2 = 4304672100001681 = 6317 * 681442472693 and for no smaller k>0 is 90^8 + k^2 a semiprime.
a(100) = 9 because 100^8 + 9^2 = 10000000000000081 = 34361 * 291027618521 and for no smaller k>0 is 100^8 + k^2 a semiprime.

Crossrefs

Cf. A001358, A108714, A109197, A109198, A109199, A109200, A109201, A109202.

A109204 Minimal value of k>0 such that n^9 + k^2 is a semiprime.

Original entry on oeis.org

2, 3, 5, 10, 5, 2, 11, 4, 7, 2, 9, 4, 7, 5, 3, 2, 7, 16, 7, 2, 39, 2, 25, 12, 5, 7, 21, 2, 5, 3, 7, 16, 9, 17, 5, 24, 19, 4, 3, 20, 7, 6, 11, 4, 3, 4, 17, 12, 17, 2, 7, 70, 3, 3, 5, 2, 11, 16, 5, 42, 7, 4, 3, 26, 3, 9, 25, 26, 9, 5, 33, 6, 23, 12, 23, 2, 9, 6, 7, 2, 23, 4, 3, 16, 11, 16, 9, 2, 3
Offset: 0

Views

Author

Jonathan Vos Post, Jul 04 2005

Keywords

Examples

			a(0) = 2 because 0^9 + 1^2 = 1 is not semiprime, but 0^9 + 2^2 = 4 = 2^2 is.
a(1) = 3 because 1^9 + 1^2 and 1^9 + 2^2 are not semiprime, but 1^9 + 3^2 = 10 = 2 * 5 is semiprime.
a(2) = 5 because 2^9 + 5^2 = 537 = 3 * 179 is semiprime, but 2^9 plus no smaller square is.
a(51) = 70 because 51^9 + 70^2 = 2334165173095351 = 43063 * 54203496577 and for no smaller k>0 is 51^9 + k^2 a semiprime.
a(100) = 7 because 100^9 + 7^2 = 1000000000000000049 = 157 * 6369426751592357 and for no smaller k>0 is 100^9 + k^2 a semiprime.

Links

Crossrefs

Cf. A001358, A108714, A109197, A109198, A109199, A109200, A109201, A109202, A109203.

Programs

  • Mathematica
    a[n_] := (For[k = 1, PrimeOmega[n^9 + k^2] != 2, k++]; k); a /@ Range[0, 88] (* Giovanni Resta, Jun 17 2016 *)
  • PARI
    a(n) = my(k=1); while(bigomega(n^9+k^2)!=2, k++); k \\ Felix Fröhlich, Jun 17 2016

Extensions

a(15) corrected by Giovanni Resta, Jun 17 2016

A109205 Minimal value of k>0 such that n^10 + k^2 is a semiprime.

Original entry on oeis.org

2, 3, 7, 4, 1, 4, 5, 2, 5, 10, 3, 2, 11, 7, 9, 8, 1, 10, 7, 4, 7, 4, 5, 2, 5, 3, 1, 20, 3, 9, 7, 2, 7, 5, 21, 4, 5, 2, 3, 4, 3, 4, 25, 3, 3, 13, 31, 2, 7, 24, 7, 2, 5, 2, 1, 4, 9, 7, 5, 4, 23, 9, 17, 8, 29, 8, 17, 2, 3, 10, 13, 2, 13, 7, 5, 4, 11, 8, 5, 10, 17, 4, 21, 5, 31, 4, 5, 4, 13, 2, 7, 4, 25
Offset: 0

Views

Author

Jonathan Vos Post, Jul 05 2005

Keywords

Examples

			a(0) = 2 because 0^10 + 1^2 = 1 is not semiprime, but 0^10 + 2^2 = 4 = 2^2 is.
a(1) = 3 because 1^10 + 1^2 and 1^10 + 2^2 are not semiprime, but 1^10 + 3^2 = 10 = 2 * 5 is semiprime.
a(2) = 7 because 2^10 + 7^2 = 1073 = 29 * 37 is semiprime, but 2^10 plus no smaller square is.
a(99) = 56 because 99^10 + 56^2 = 90438207500880452137 = 3733 * 24226682963000389 and for no smaller k>0 is 99^10 + k^2 a semiprime.
a(100) = 17 because 100^10 + 17^2 = 100000000000000000289 = 181 *
552486187845303869 and for no smaller k>0 is 100^10 + k^2 a semiprime.

Links

Crossrefs

Cf. A001358, A108714, A109197, A109198, A109199, A109200, A109201, A109202, A109203, A109204.

Programs

  • Mathematica
    mvk[n_]:=Module[{c=n^10,k=1},While[PrimeOmega[c+k^2]!=2,k++];k]; Array[ mvk,100,0] (* Harvey P. Dale, Aug 01 2021 *)

A109206 Minimal value of k>0 such that n^11 + k^2 is a semiprime.

Original entry on oeis.org

2, 3, 1, 2, 3, 6, 1, 4, 9, 8, 13, 4, 1, 2, 3, 8, 7, 6, 5, 28, 3, 4, 5, 6, 5, 2, 9, 4, 9, 6, 29, 2, 15, 7, 5, 48, 5, 5, 33, 8, 7, 24, 17, 4, 15, 14, 11, 4, 5, 8, 9, 10, 7, 6, 31, 8, 3, 4, 5, 18, 13, 34, 5, 2, 5, 18, 35, 12, 15, 2, 27, 6, 31, 5, 3, 34, 5, 9, 7, 2, 3, 4, 13, 14, 23, 2, 15, 22, 21, 48
Offset: 0

Views

Author

Jonathan Vos Post, Jul 06 2005

Keywords

Comments

It seems that one or more primes almost always occur before finding the first such semiprime for a given n. There seems to be a modest correlation with the n^10 sequence (A109205) with often the same values [n = 0,1,15,21,22,24,31,36,58,81,94]. Or differs by 10 [n = 10,12,60,65,67, 86, 92,100]. Or 20 [n = 41, 46] or 30 [n = 38, 54,75]. Sometimes A109206(n) = A109205(n) = A109204(n) [n = 58,81]. Is it obvious that there must be a k for each n and not an infinite sequence of nonsemiprimes of the form n^11 + k^2?

Examples

			a(0) = 2 because 0^11 + 1^2 = 1 is not semiprime, but 0^11 + 2^2 = 4 = 2^2 is.
a(1) = 3 because 1^11 + 1^2 and 1^11 + 2^2 are not semiprime, but 1^11 + 3^2 = 10 = 2 * 5 is semiprime.
a(2) = 1 because 2^11 + 1^2 = 2049 = 3 * 683 is semiprime.
a(35) = 48 because 35^11 + 48^2 = 96549157373049179 = 401 * 240770966017579 and for no smaller k>0 is 35^11 + k^2 a semiprime.
a(100) = 37 because 100^11 + 37^2 = 10000000000000000001369 = 60089 *
166419810614255521 and for no smaller k>0 is 100^11 + k^2 a semiprime.

Crossrefs

Cf. A001358, A108714, A109197, A109198, A109199, A109200, A109201, A109202, A109203, A109204, A109205.

Programs

  • Mathematica
    mk[n_]:=Module[{n11=n^11,k=1},While[PrimeOmega[n11+k^2]!=2,k++];k]; Array[ mk,100,0] (* Harvey P. Dale, Aug 06 2012 *)
Showing 1-4 of 4 results.

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