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 1-9 of 9 results.

A109198 Minimal value of k > 0 such that n^3 + k^2 is semiprime.

Original entry on oeis.org

2, 3, 1, 8, 1, 2, 1, 8, 5, 4, 9, 4, 11, 2, 13, 2, 1, 12, 1, 12, 7, 5, 1, 2, 5, 3, 3, 10, 1, 18, 7, 4, 3, 2, 5, 2, 7, 2, 7, 2, 7, 6, 1, 2, 5, 7, 3, 4, 11, 2, 3, 16, 5, 2, 11, 2, 9, 2, 1, 40, 1, 4, 5, 8, 5, 9, 17, 2, 7, 5, 1, 6, 1, 4, 3, 14, 3, 4, 1, 18, 9, 10, 3, 6, 5, 3, 15, 4, 9, 3, 7, 10, 11, 2, 3, 12
Offset: 0

Views

Author

Jonathan Vos Post, Jun 22 2005

Keywords

Comments

When n+1 and n^2+1 are both prime, then k=1.

Examples

			a(0) = 2 because 0^3 + 1^2 = 1 is not semiprime, but 0^3 + 2^2 = 4 = 2^2 is.
a(1) = 3 because 1^3 + 1^2 and 1^3 + 2^2 are not semiprime, but 1^3 + 3^2 = 10 = 2 * 5 is semiprime.
a(59) = 40 because 59^3 + 40^2 = 206979 = 3 * 68993 and for no smaller k > 0 is 59^3 + k^2 a semiprime.
a(100) = 1 because 100^3 + 1^2 = 1000001 = 101 * 9901.

Crossrefs

Cf. A001358, A108714, A109197.

Programs

  • Mathematica
    k2sp[n_]:=Module[{n3=n^3,k=1},While[PrimeOmega[n3+k^2]!=2,k++];k]; Array[ k2sp,100,0] (* Harvey P. Dale, Oct 15 2013 *)
  • PARI
    A109198(n)={local(r);r=1;while(bigomega(n^3+r^2)<>2,r=r+1);r} \\ Michael B. Porter, May 14 2010

Formula

a(n) = minimal value of k > 0 such that n^3 + k^2 is semiprime.

A109199 Minimal value of k>0 such that n^4 + k^2 is semiprime.

Original entry on oeis.org

2, 3, 3, 1, 3, 1, 7, 1, 1, 14, 1, 1, 1, 1, 1, 2, 3, 1, 1, 5, 17, 1, 1, 1, 17, 2, 1, 10, 9, 1, 1, 4, 1, 4, 5, 1, 1, 6, 1, 1, 1, 5, 1, 4, 5, 7, 5, 6, 13, 5, 1, 14, 1, 4, 5, 2, 3, 1, 1, 14, 7, 1, 1, 4, 7, 1, 5, 4, 1, 16, 3, 1, 1, 1, 3, 4, 5, 6, 1, 10, 7, 1, 9, 4, 1, 3, 1, 16, 3, 4, 31, 15, 1, 4, 1, 3, 5, 6, 1, 4
Offset: 0

Views

Author

Jonathan Vos Post, Jun 22 2005

Keywords

Examples

			a(0) = 2 because 0^4 + 1^2 = 1 is not semiprime, but 0^4 + 2^2 = 4 = 2^2 is.
a(1) = 3 because 1^4 + 1^2 and 1^4 + 2^2 are not semiprime, but 1^4 + 3^2 = 10 = 2 * 5 is semiprime.
a(90) = 31 because 90^4 + 31^2 = 65610961 = 13 * 5046997 and for no smaller k>0 is 90^4 + k^2 a semiprime.
a(100) = 1 because 100^4 + 1^2 = 100000001 = 17 * 5882353.

Links

Crossrefs

Cf. A001358, A108714, A109197, A109198.

Programs

  • Mathematica
    n4sp[n_]:=Module[{k=1,n4=n^4},While[PrimeOmega[n4+k^2]!=2,k++];k]; Array[n4sp,100,0] (* Harvey P. Dale, Dec 03 2011 *)

Formula

a(n) = minimal value of k>0 such that n^4 + k^2 is semiprime.

A109200 Minimal value of k>0 such that n^5 + k^2 is a semiprime.

Original entry on oeis.org

2, 3, 1, 2, 7, 3, 5, 16, 3, 4, 1, 10, 1, 2, 3, 8, 1, 2, 5, 10, 3, 2, 1, 8, 5, 4, 9, 2, 9, 3, 13, 8, 15, 8, 7, 2, 5, 2, 3, 16, 3, 9, 31, 14, 3, 4, 3, 10, 11, 2, 3, 2, 9, 12, 5, 4, 3, 10, 5, 6, 11, 6, 9, 16, 5, 28, 19, 4, 3, 16, 3, 6, 7, 4, 9, 28, 9, 6, 11, 12, 7, 10, 7, 14, 29, 3, 11, 8, 3, 18, 7, 8, 3, 4
Offset: 0

Views

Author

Jonathan Vos Post, Jun 26 2005

Keywords

Examples

			a(0) = 2 because 0^5 + 1^2 = 1 is not semiprime, but 0^5 + 2^2 = 4 = 2^2 is.
a(1) = 3 because 1^5 + 1^2 and 1^5 + 2^2 are not semiprime, but 1^5 + 3^2 = 10 = 2 * 5 is semiprime.
a(2) = 1 because 2^5 + 1^2 = 33 = 3 * 11 is semiprime.
a(42) = 31 because 42^5 + 31^2 = 130692193 = 571 * 228883 and for no smaller k>0 is 42^4 + k^2 a semiprime.

Crossrefs

Cf. A001358, A108714, A109197, A109198, A109199.

Programs

  • Mathematica
    a[n_] := (For[k = 1, PrimeOmega[n^5 + k^2] != 2, k++]; k); a /@ Range[0, 93] (* Giovanni Resta, Jun 16 2016 *)

Formula

a(n) = minimal value of k>0 such that n^5 + k^2 is a semiprime.

Extensions

a(46) corrected by Giovanni Resta, Jun 16 2016

A109201 Minimal value of k>0 such that n^6 + k^2 is a semiprime.

Original entry on oeis.org

2, 3, 1, 4, 1, 3, 7, 2, 5, 10, 1, 2, 5, 6, 5, 2, 7, 6, 11, 6, 3, 5, 3, 7, 11, 2, 3, 2, 9, 10, 7, 5, 5, 5, 5, 2, 1, 2, 5, 2, 3, 2, 5, 4, 9, 4, 3, 2, 5, 11, 3, 11, 3, 3, 5, 7, 1, 4, 3, 4, 11, 4, 5, 16, 7, 2, 7, 2, 3, 25, 9, 6, 5, 2, 5, 2, 5, 2, 5, 4, 17, 20, 7, 4, 5, 4, 15, 2, 5, 6, 7, 6, 3, 5, 1, 2, 5, 8, 3
Offset: 0

Views

Author

Jonathan Vos Post, Jun 29 2005

Keywords

Comments

It seems that one or more primes nearly always occur before finding the first such semiprime for a given n. There seems to be a high correlation with the n^5 + k^2 sequence (A109200) [such as n=63] and it with the n^2 + k^2 sequence (A109197).

Examples

			a(0) = 2 because 0^6 + 1^2 = 1 is not semiprime, but 0^6 + 2^2 = 4 = 2^2 is.
a(1) = 3 because 1^6 + 1^2 and 1^6 + 2^2 are not semiprime, but 1^6 + 3^2 = 10 = 2 * 5 is semiprime.
a(2) = 1 because 2^6 + 1^2 = 65 = 5 * 13 is semiprime.
a(69) = 25 because 69^6 + 25^2 = 107918163706 = 2 * 53959081853 and for no smaller k>0 is 69^6 + k^2 a semiprime.
a(100) = 7 because 100^6 + 7^2 = 1000000000049 = 6337 * 157803377 and for no smaller k>0 is 100^6 + k^2 a semiprime.

Crossrefs

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

Formula

a(n) = minimal value of k>0 such that n^6 + k^2 is semiprime.

A109202 Minimal value of k>0 such that n^7 + k^2 is a semiprime.

Original entry on oeis.org

2, 3, 1, 2, 5, 6, 7, 4, 5, 8, 1, 6, 7, 5, 27, 16, 1, 12, 1, 2, 3, 8, 3, 6, 7, 2, 5, 2, 3, 12, 7, 4, 9, 2, 5, 6, 7, 4, 21, 2, 9, 4, 11, 6, 3, 4, 1, 2, 7, 25, 21, 14, 1, 4, 5, 4, 15, 8, 3, 22, 17, 8, 21, 10, 5, 2, 1, 14, 9, 32, 11, 6, 1, 13, 3, 2, 3, 3, 1, 2, 63, 4, 5, 10, 11, 9, 9, 4, 5, 33, 19, 6, 3
Offset: 0

Views

Author

Jonathan Vos Post, Jul 02 2005

Keywords

Comments

It seems that one or more primes nearly always occur before finding the first such semiprime for a given n. There seems to be a high correlation with the n^6 + k^2 sequence (A109201) with 24 times less than 100 the same values A109201(n) = A109202(n) for [n = 0,1,2,6,8,10,20,22,25,27,30,34,39,45,47,54,58,65,71,75,88,91,92,96].

Examples

			a(0) = 2 because 0^7 + 1^2 = 1 is not semiprime, but 0^7 + 2^2 = 4 = 2^2 is.
a(1) = 3 because 1^7 + 1^2 and 1^7 + 2^2 are not semiprime, but 1^7 + 3^2 = 10 = 2 * 5 is semiprime.
a(2) = 1 because 2^7 + 1^2 = 129 = 3 * 43 is semiprime.
a(80) = 63 because 80^7 + 63^2 = 20971520003969 = 47363 * 442782763 and for no smaller k>0 is 80^7 + k^2 a semiprime.
a(100) = 9 because 100^7 + 9^2 = 100000000000081 = 47309 * 2113762709 and for no smaller k>0 is 100^7 + k^2 a semiprime.

Crossrefs

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

Programs

  • Mathematica
    svk[n_]:=Module[{k=1,n7=n^7},While[PrimeOmega[n7+k^2]!=2,k++];k]; Array[ svk,100,0] (* Harvey P. Dale, Mar 01 2017 *)

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 *)
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