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 21-26 of 26 results.

A213677 Numbers n such that n and n + 12 are prime and there is a power of two in the interval (n, n+12).

Original entry on oeis.org

5, 7, 11, 29, 31, 59, 61, 127, 251, 509, 1019, 1021, 262139, 1048571, 2147483647
Offset: 1

Views

Author

Brad Clardy, Mar 04 2013

Keywords

Comments

It is a conjecture that this is a finite sequence. A search was conducted out to 2^1500.

Crossrefs

A046133, A221211.

Programs

  • Magma
    //Program finds primes separated by an even number (called gap) which
//have a power of two between them. Program starts with the smallest
//power of two above gap. Primes less than this starting point can be
//checked by inspection.
gap:=12;
start:=Ilog2(gap)+1;
for i:= start to 1000 do
    powerof2:=2^i;
    for k:=powerof2-gap+1 to powerof2-1 by 2 do
        if (IsPrime(k) and IsPrime(k+gap)) then k;
        end if;
    end for;
end for;

A056777 Composite numbers k such that both phi(k+12) = phi(k) + 12 and sigma(k+12) = sigma(k) + 12.

Original entry on oeis.org

65, 209, 11009, 38009, 680609, 2205209, 3515609, 4347209, 10595009, 12006209, 31979009, 89019209, 169130009, 244766009, 247590209, 258084209, 325622009, 357777209, 377330609, 441630209, 496175609, 640343009, 1006475609
Offset: 1

Views

Author

Labos Elemer, Aug 17 2000

Keywords

Comments

It is easy to show that if p, p+2, p+6 and p+8 are all prime (a prime quadruple as defined in A007530, which lists the values of p) with x=p(p+8), x+12=(p+2)(p+6), then x is in the sequence. I conjecture that all members of the sequence are of this form. - Jud McCranie, Oct 11 2000
Numbers so far are all congruent to 65 (mod 72). - Ralf Stephan, Jul 07 2003

Examples

			k = 209 = 11*19, k + 12 = 221 = 13*17, phi(k + 12) = 192 = 180 + 12 = phi(k) + 12, also sigma(221) = 252 = sigma(209) + 12 = 240 + 12.
phi(65) + 12 = 60 = phi(65 + 12), sigma(65) + 12 = 96 = sigma(65 + 12), 65 is composite.

Crossrefs

Cf. A000010, A001838, A015917, A054902, A046133.

Programs

  • PARI
    isok(n) = !isprime(n) && (sigma(n+12) == sigma(n)+12) && (eulerphi(n+12)==eulerphi(n)+12); \\ Michel Marcus, Jul 14 2017

Extensions

More terms from Jud McCranie, Oct 11 2000

A086137 Number of primes between p and p+8 if p is prime, i.e., number of primes between 8+A023202(n) and A023202(n).

Original entry on oeis.org

2, 2, 2, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 0, 2, 0, 1, 1, 0, 0, 0, 1, 0, 1
Offset: 1

Views

Author

Labos Elemer, Jul 29 2003

Keywords

Examples

			a(n)=0,1,2 correspond to {p,p+8} prime-pairs either consecutive or pairs with various d-patterns as follows:
a(n)=0 to 89[8]97; a(n)=1 for 29[2,6]37, 53[6,2];
a(n)=2 for 101[2,4,2]109 and once to 3[2,2,4]11.

Crossrefs

Cf. A023302, A046133, A048136, A052165.

Programs

  • Mathematica
    cp[x_,y_] := Count[Table[PrimeQ[i],{i,x,y}],True] Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d],k=k+1; Print[cp[s+1,s+d-1]]],{n,1,1000}]; k

A106062 Primes p such that 1*p + 12 and 12*p + 1 are primes.

Original entry on oeis.org

5, 19, 29, 31, 59, 61, 71, 89, 101, 139, 199, 229, 269, 271, 281, 409, 479, 601, 631, 661, 761, 811, 929, 1009, 1021, 1049, 1051, 1091, 1181, 1279, 1291, 1361, 1459, 1499, 1511, 1601, 1609, 1709, 1889, 2099, 2141, 2339, 2381, 2399, 2411, 2539, 2579, 2729
Offset: 1

Views

Author

Zak Seidov, May 07 2005

Keywords

Crossrefs

Intersection of A046133 and A075704. - Michel Marcus, Jan 20 2018

Programs

  • Magma
    [p: p in PrimesUpTo(10000)| IsPrime(p+12) and IsPrime(12*p+1)]; // Vincenzo Librandi, Nov 13 2010
  • Mathematica
    Select[Prime[Range[220]], PrimeQ[12#+1]&&PrimeQ[1#+12]&]

A257107 Composite numbers n such that n'=(n+12)', where n' is the arithmetic derivative of n.

Original entry on oeis.org

16, 65, 88, 209, 11009, 38009, 680609, 2205209, 2860198, 3515609, 4347209, 5365387, 5809361, 10595009, 12006209, 31979009, 83255059, 89019209, 152915402, 169130009, 172147423, 225869899, 244766009, 247590209, 258084209, 325622009, 357777209, 377330609
Offset: 1

Views

Author

Paolo P. Lava, Apr 17 2015

Keywords

Comments

If the limitation of being composite is removed we also have the numbers p such that if p is prime then p + 12 is prime too (A046133).

Examples

			16' = (16 + 12)' = 28' = 32;
65' = (65 + 12)' = 77' = 18.

Crossrefs

Cf. A003415, A046133, A226779.

Programs

  • Maple
    with(numtheory); P:= proc(q,h) local a,b,n,p;
for n from 1 to q do if not isprime(n) then a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=(n+h)*add(op(2,p)/op(1,p),p=ifactors(n+h)[2]);
if a=b then print(n); fi; fi; od; end: P(10^9,12);
  • Mathematica
    a[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]];
Select[Range@ 100000, And[CompositeQ@ #, a@# == a[# + 12]] &] (* Michael De Vlieger, Apr 22 2015, after Michael Somos at A003415 *)

Extensions

a(16)-a(28) from Lars Blomberg, May 06 2015

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

Original entry on oeis.org

3, 5, 3, 11, 7, 5, 17, 13, 7, 3, 29, 19, 11, 5, 3, 41, 37, 13, 11, 7, 5, 59, 43, 17, 23, 13, 7, 3, 71, 67, 23, 29, 19, 11, 5, 3, 101, 79, 31, 53, 31, 17, 17, 7, 5, 107, 97, 37, 59, 37, 19, 23, 13, 11, 3, 137, 103, 41, 71, 43, 29, 29, 31, 13, 11, 7, 149, 109
Offset: 1

Views

Author

Felix Fröhlich, Jul 28 2019

Keywords

Comments

The same as A231608 except that A231608 gives the upward antidiagonals of the array, while this sequence gives the downward antidiagonals.
Conjecture: All values are nonzero, i.e., for any even integer e there are infinitely many primes p such that p + e is also prime.
The conjecture is true if Polignac's conjecture is true.

Examples

			The array starts as follows:
3,  5, 11, 17, 29, 41, 59,  71, 101, 107, 137, 149, 179, 191
3,  7, 13, 19, 37, 43, 67,  79,  97, 103, 109, 127, 163, 193
5,  7, 11, 13, 17, 23, 31,  37,  41,  47,  53,  61,  67,  73
3,  5, 11, 23, 29, 53, 59,  71,  89, 101, 131, 149, 173, 191
3,  7, 13, 19, 31, 37, 43,  61,  73,  79,  97, 103, 127, 139
5,  7, 11, 17, 19, 29, 31,  41,  47,  59,  61,  67,  71,  89
3,  5, 17, 23, 29, 47, 53,  59,  83,  89, 113, 137, 149, 167
3,  7, 13, 31, 37, 43, 67,  73,  97, 151, 157, 163, 181, 211
5, 11, 13, 19, 23, 29, 41,  43,  53,  61,  71,  79,  83,  89
3, 11, 17, 23, 41, 47, 53,  59,  83,  89, 107, 131, 137, 173
7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 229, 241
5,  7, 13, 17, 19, 23, 29,  37,  43,  47,  59,  73,  79,  83

Links

Crossrefs

Cf. A231608.
Cf. A001359 (row 1), A023200 (row 2), A023201 (row 3), A023202 (row 4), A023203 (row 5), A046133 (row 6), A153417 (row 7), A049488 (row 8), A153418 (row 9), A153419 (row 10), A242476 (row 11), A033560 (row 12), A252089 (row 13), A252090 (row 14), A049481 (row 15), A049489 (row 16), A252091 (row 17), A156104 (row 18), A271347 (row 19), A271981 (row 20), A271982 (row 21), A272176 (row 22), A062284 (row 25), A049490 (row 32), A020483 (column 1).

Programs

  • PARI
    row(n, terms) = my(i=0); forprime(p=1, , if(i>=terms, break); if(ispseudoprime(p+2*n), print1(p, ", "); i++))
array(rows, cols) = for(x=1, rows, row(x, cols); print(""))
array(12, 14) \\ Print initial 12 rows and 14 columns of the array
Previous Showing 21-26 of 26 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.