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.

Previous Showing 11-15 of 15 results.

A228027 Primes of the form 4^k + 9.

Original entry on oeis.org

13, 73, 1033, 262153, 1073741833, 73786976294838206473, 4835703278458516698824713
Offset: 1

Views

Author

Vincenzo Librandi, Aug 11 2013

Keywords

Comments

Subsequence of A104070. - Elmo R. Oliveira, Nov 28 2023

Examples

			262153 is a term because 4^9 + 9 = 262153 is prime.

Links

Crossrefs

Cf. A000040, A217350 (corresponding k's).
Cf. Primes of the form r^k + h: A092506 (r=2, h=1), A057733 (r=2, h=3), A123250 (r=2, h=5), A104066 (r=2, h=7), A104070 (r=2, h=9), A057735 (r=3, h=2), A102903 (r=3, h=4), A102870 (r=3, h=8), A102907 (r=3, h=10), A290200 (r=4, h=1), A228026 (r=4, h=3), this sequence (r=4, h=9), A182330 (r=5, h=2), A228029 (r=5, h=6), A102910 (r=5, h=8), A182331 (r=6, h=1), A104118 (r=6, h=5), A104115 (r=6, h=7), A104065 (r=7, h=4), A228030 (r=7, h=6), A228031 (r=7, h=10), A228032 (r=8, h=3), A228033 (r=8, h=5), A144360 (r=8, h=7), A145440 (r=8, h=9), A228034 (r=9, h=2), A159352 (r=10, h=3), A159031 (r=10, h=7).

Programs

  • Magma
    [a: n in [0..200] | IsPrime(a) where a is 4^n+9];
  • Mathematica
    Select[Table[4^n + 9, {n, 0, 200}],PrimeQ]

Formula

a(n) = 4^A217350(n) + 9. - Elmo R. Oliveira, Nov 28 2023

Extensions

Corrected cross-references - Robert Price, Aug 01 2017

A228033 Primes of the form 8^k + 5.

Original entry on oeis.org

13, 2787593149816327892691964784081045188247557, 15177100720513508366558296147058741458143803430094840009779784451085189728165691397
Offset: 1

Views

Author

Vincenzo Librandi, Aug 11 2013

Keywords

Comments

a(4) = 8^64655 + 5 = 1.919...*10^58389 is too large to include. - Amiram Eldar, Jul 23 2025

Crossrefs

Cf. A217355 (associated n).
Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=8, h=5), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

Programs

  • Magma
    [a: n in [1..300] | IsPrime(a) where a is 8^n+5];
  • Mathematica
    Select[Table[8^n + 5, {n, 4000}], PrimeQ]

A228028 Primes of the form 5^n + 4.

Original entry on oeis.org

5, 29, 15629, 9765629
Offset: 1

Views

Author

Vincenzo Librandi, Aug 11 2013

Keywords

Links

Crossrefs

Cf. A124621 (associated n).
Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A228027 (k=4, h=9), A182330 (k=5, h=2), this sequence (k=5, h=4), A228029 (k=5, h=6), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), A228030 (k=7, h=6), A228031 (k=7, h=10), A228032 (k=8, h=3), A228033 (k=8, h=5), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

Programs

  • Magma
    [a: n in [0..200] | IsPrime(a) where a is  5^n+4];
  • Mathematica
    Select[Table[5^n + 4, {n, 0, 200}], PrimeQ]

Extensions

Corrected cross-references - Robert Price, Aug 01 2017

A193109 Least k such that 2^x + k produces primes for x=1..n and composite for x=n+1.

Original entry on oeis.org

0, 1, 9, 3, 225, 15, 65835, 1605, 19425, 2397347205, 153535525935
Offset: 1

Views

Author

Arkadiusz Wesolowski, Jul 21 2011

Keywords

Comments

All terms except the first four are congruent to 15 mod 30.
a(10) was found in 2005 by T. D. Noe and a(11) was found in the same year by Don Reble.
Other known values: a(13) = 29503289812425.
a(12) > 10^13. - Tyler Busby, Feb 19 2023

Crossrefs

Another version of A110096.
Cf. A008597, A057733, A092506, A104070, A144487.

Programs

  • Mathematica
    Table[k = 0; While[i = 1; While[i <= n && PrimeQ[2^i + k], i++]; i <= n || PrimeQ[2^i + k], k++]; k, {n, 9}] (* T. D. Noe, Jul 21 2011 *)
  • PARI
    is(k, n) = for(x=1, n, if(!isprime(k+2^x), return(0))); 1;
a(n) = {my(s=2); forprime(p=3, n, if(znorder(Mod(2, p))==(p-1), s*=p)); forstep(k=s*(n>1)/2, oo, s, if(is(k, n) && !isprime(k+2^(n+1)), return(k))); } \\ Jinyuan Wang, Jul 30 2020

A363464 Numbers k in A052294 with arithmetic derivative k' (A003415) in A052294.

Original entry on oeis.org

6, 9, 10, 14, 18, 20, 21, 22, 24, 25, 33, 34, 35, 38, 40, 42, 44, 48, 49, 52, 62, 65, 66, 68, 69, 70, 76, 80, 84, 88, 91, 93, 94, 96, 100, 104, 110, 115, 117, 118, 121, 132, 133, 134, 138, 140, 143, 144, 145, 148, 152, 155, 158, 164, 174, 182, 185, 186, 188, 192
Offset: 1

Views

Author

Marius A. Burtea, Jul 08 2023

Keywords

Comments

If p > 2 is in A092506 then m = 2*p and u = 4*p are terms. Indeed, if p = 2^k + 1, k >= 1, m = 2*(2^k + 1) = 2^(k+1) + 2^1 has two 1's in its binary expansion, and m' = p+2 = 2^k + 3 = 2^k + 2^1 + 1 has three 1's in its binary expansion. Similarly u = 4*(2^k + 1) = 2^(k+2) + 2^2 and u' = 4*p + 4 = 2^(k+2) + 2^3.
If p is in A057733 then the number m = 2*p is a term. Indeed, if p = 2^k + 3, k >= 1, m = 2*(2^k + 3) = 2^(k+1) + 2^2 + 2 has three 1's in its binary expansion, and m' = p+2 = 2^k + 5 = 2^k + 2^2 + 1 has three 1's in its binary expansion.
If p > 7 is in A057733 then the number m = 4*p is a term. Indeed, if p = 2^k + 3, k >= 3, m = 4*(2^k + 3) = 2^(k+2) + 2^3 + 2 has three 1's in its binary expansion, and m' = 4*(p + 1) = 4*(2^k + 4) = 2^(k+2) + 2^4 has two 1's in its binary expansion.
If p is in A123250 then the number m = 4*p is a term. Indeed, if p = 2^k + 5, k >= 1, m = 4*(2^k + 5) = 2^(k+2) + 2^4 + 2^2 has three 1's its binary expansion, and m' = 4*(p+1) = 4*(2^k + 6) = 2^(k+2) + 2^4 + 2^2 has three 1's in its binary expansion.
If p is in A104070 then the number m = 4*p is a term. Indeed, if p = 2^k + 9, k >= 1, m = 4*(2^k + 9) = 2^(k+2) + 2^5 + 2^2 has three 1's its binary expansion, and m' = 4*(p+1) = 4*(2^k + 10) = 2^(k+2) + 2^5 + 2^3 has three 1's in its binary expansion.

Examples

			6 = 110_2 has two 1's, 6' = 5 = 101_2 has two 1's, so 6 is a term.
9 = 101_2 has two 1's, 9' = 6 = 110_2 has two 1's, so 9 is a term.
10 = 1010_2 has two 1's, 10' = 7 = 111_2 has three 1's, so 10 is a term.
18 = 10010_2 has two 1's, 18' = 21 = 10101_2 has three 1's, so 18 is a term.

Crossrefs

Cf. A000120, A003415, A052294, A057733, A081092, A092506, A104070, A123250.

Programs

  • Magma
    fp:=func; f:=func; [n:n in [1..200]| fp(n) and fp(Floor(f(n)))];
  • Mathematica
    pernQ[n_] := PrimeQ[DigitCount[n, 2, 1]]; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[200], And @@ pernQ[{#, d[#]}] &] (* Amiram Eldar, Jul 10 2023 *)
Previous Showing 11-15 of 15 results.

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