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.

A089437 Numbers k such that 4^k + 3 is prime.

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 14, 15, 42, 114, 195, 392, 555, 852, 1004, 1185, 2001, 2030, 2031, 2276, 8610, 8967, 10362, 11366, 15927, 16514, 17877, 19122, 19898, 27728, 29156, 61275, 102981, 117663, 181560, 239922, 342789, 371226, 717200, 937776, 970406, 1102722
Offset: 1

Views

Author

Herman H. Rosenfeld (herm3(AT)pacbell.net), Dec 28 2003

Keywords

Comments

The halved even terms of A057732. - R. J. Mathar, Feb 26 2008

Examples

			4^15 + 3 = 1073741827 a prime number.

Crossrefs

Cf. A057732, A228026 (the actual primes).

Programs

Extensions

More terms from Ray Chandler, Dec 30 2003
More terms from R. J. Mathar, Feb 26 2008
a(33)-a(37) derived from A057732 by Robert Price, Apr 26 2014
a(38)-a(42) derived from A057732 by Amiram Eldar, Nov 16 2023

A122834 Primes in the new Mersenne conjecture; odd primes of the form 2^k+-1 or 4^k+-3.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 31, 61, 67, 127, 257, 1021, 4093, 4099, 8191, 16381, 65537, 65539, 131071, 262147, 524287, 1048573, 4194301, 16777213, 268435459, 1073741827, 2147483647, 2305843009213693951, 19342813113834066795298819
Offset: 1

Views

Author

T. D. Noe, Sep 12 2006

Keywords

Comments

Let p be a prime in this sequence. Call q=2^p-1 and r=(2^p+1)/3. The new Mersenne conjecture implies that either q and r are both prime or both composite.

References

  • R. K. Guy, Unsolved Problems in Number Theory, Section A3.

Links

Crossrefs

Superset of: A000668, A019434, A228026.
Cf. A000043 (n such that 2^n-1 is prime), A000978 (n such that (2^n+1)/3 is prime), A107360 (the intersection of these).

Programs

  • Mathematica
    nn=100; Union[Select[1+2^Range[16],PrimeQ], Select[ -1+2^Range[2nn],PrimeQ], Select[3+4^Range[nn],PrimeQ], Select[ -3+4^Range[nn],PrimeQ]]

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

A229222 Smallest prime p such that p contains a digit larger than 1 and the sum of the n-th powers of the decimal digits of p is a prime number.

Original entry on oeis.org

2, 23, 113, 23, 191, 223, 191, 41, 223, 113, 157, 191, 137, 113, 113, 43, 137, 191, 179, 337, 577, 223, 227, 113, 263, 113, 199, 229, 263, 199, 467, 89, 223, 179, 223, 113, 443, 683, 1279, 337, 661, 463, 827, 2281, 577, 223, 223, 661, 137, 229, 11399, 461, 577
Offset: 1

Views

Author

Michel Lagneau, Sep 16 2013

Keywords

Comments

We impose the condition that p is not in A020449 in order to avoid trivial sequences with infinite repetitions with the numbers 11 if p>1, or 101 if p>11, or 101111 if p > 101, ... for example if p > 1 the sequence is {2, 11, 11, 11, ...}, if p > 11 the sequence is {23, 23, 101, 23, 101, 101, 41, 101, 101, 101, 101, 101, ...}.
a(n) is an unification of a family of sequences mentioned hereafter:
A082101: primes of the form 2^n+3^n => 23 is in the sequence;
A057735: primes of the form 3^n+2 => 113 is in the sequence;
A153133: primes of the form 2^n+3^(n-1) => 223 is in the sequence;
A228034: primes of the form 9^n+2 => 191 is in the sequence;
A057733: primes of the form 2^n+3 => 2111 is in the sequence;
A228026: primes of the form 4^n+3 => 4111 is in the sequence;
A228034: primes of the form 9^n+2 => 191 is in the sequence;
A182330: primes of the form 5^n+2 => 151 is in the sequence;
A111974: primes of the form 2*3^n+1 => 313 is in the sequence;
A102903: primes of the form 3^n+4 => 11113 is in the sequence.
In this sequence, we observe repetitions of numbers such that 23, 113, 223, 191, 199, 223,... and this problem is very difficult, because it is probable that there exists both finite and infinite repetitions according to the numbers: for example, if we consider the number 23 of this sequence, it is probable that the number of element "23" is finite (see the comment in A082101 for the primes of form 2^k + 3^k). But, if we consider the number 113 of this sequence, is the number of the elements "113" infinite ? (see A057735 with the primes of the form 2+3^n). We observe that a(n) = 113 for n = 3, 14, 15, 24, 26,..., 123, 126, 139,..., 386, 391, 494, ....

Examples

			a(3) = 113 because 1^3+1^3+3^3 = 29 is prime.

Links

Crossrefs

Cf. A020449.

Programs

  • Maple
    with(numtheory) :lst:={11, 101, 101111, 10011101, 10101101, 10110011, 10111001, 11000111, 11100101, 11110111, 11111101 }:for n from 1 to 300 do :ii:=0:for k from 1 to 10^8 while(ii=0) do:x:=convert(k,base,10):n1:=nops(x):it:=0:jj:=0:s:= sum('x[i]^n', 'i'=1..n1):lst1:={k} intersect lst:if type(k,prime)=true and type(s,prime)=true and (lst1<>{k}) then ii:=1: printf(`%d, `,k):else fi:od:od:
  • Mathematica
    Table[p = 2; While[d = IntegerDigits[p]; Union[d][[-1]] < 2 || ! PrimeQ[Total[d^n]],  p = NextPrime[p]]; p, {n, 60}]
  • PARI
    a(n)=forprime(p=2,,my(d=digits(p)); if(vecmax(d)>1 && isprime(sum(i=1,#d,d[i]^n)), return(p))) \\ Charles R Greathouse IV, Sep 19 2013
Showing 1-4 of 4 results.

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