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 1-10 of 12 results. Next

A230224 Number of ways to write 2*n = p + q + r + s with p <= q <= r <= s such that p, q, r, s are primes in A230223.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Oct 12 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 17.

Examples

			a(21) = 1 since 2*21 = 7 + 7 + 11 + 17, and 7, 11, 17 are primes in A230223.
a(27) = 1 since 2*27 = 7 + 11 + 17 + 19, and 7, 11, 17, 19 are primes in A230223.

Links

Crossrefs

Cf. A002375, A068307, A230223, A230140, A230141, A230219.

Programs

  • Mathematica
    RQ[n_]:=n>5&&PrimeQ[3n-4]&&PrimeQ[3n-10]&&PrimeQ[3n-14]
SQ[n_]:=PrimeQ[n]&&RQ[n]
a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&RQ[Prime[k]]&&SQ[2n-Prime[i]-Prime[j]-Prime[k]],1,0],
{i,1,PrimePi[n/2]},{j,i,PrimePi[(2n-Prime[i])/3]},{k,j,PrimePi[(2n-Prime[i]-Prime[j])/2]}]
Table[a[n],{n,1,100}]

A237890 Primes p such that p^2 + 4 and p^2 + 10 are also primes.

Original entry on oeis.org

3, 7, 13, 97, 487, 613, 743, 827, 883, 1117, 1987, 2477, 2887, 3863, 4483, 5153, 5557, 5683, 5923, 5953, 6287, 7643, 7937, 8093, 9323, 10343, 12377, 13033, 13063, 14087, 14767, 15373, 16937, 17713, 17987, 18257, 19013, 19333, 19753, 19853, 20287, 20873, 21673
Offset: 1

Views

Author

K. D. Bajpai, Feb 15 2014

Keywords

Examples

			7 is prime and appears in the sequence because 7^2+4 = 53 and 7^2+10 = 59 are also primes.
97 is prime and appears in the sequence because 97^2+4 = 9413 and 97^2+10 = 9419 are also primes.

Links

Crossrefs

Cf. A000040, A023200, A046136, A230223.

Programs

  • Maple
    KD := proc() local a,b,d;  a:=ithprime(n);  b:=a^2+4; d:=a^2+10;  if isprime (b) and isprime(d) then RETURN (a); fi;  end: seq(KD(), n=1..5000);
  • Mathematica
    Select[Prime[Range[5000]], PrimeQ[#^2 + 4] && PrimeQ[#^2 + 10] &]
  • PARI
    s=[]; forprime(p=2, 25000, if(isprime(p^2+4) && isprime(p^2+10), s=concat(s, p))); s \\ Colin Barker, Feb 15 2014

A236302 Primes p such that p+8, p+86, p+864 are prime.

Original entry on oeis.org

23, 743, 983, 1163, 1373, 1613, 2663, 4013, 4643, 6113, 6863, 7583, 7673, 8513, 10313, 10853, 11243, 12503, 12713, 15233, 15263, 25733, 25763, 28703, 39623, 40763, 42743, 46133, 54623, 56093, 61643, 63353, 65003, 67733, 68813, 70373, 70913, 71933, 78893, 86453
Offset: 1

Views

Author

K. D. Bajpai, Apr 21 2014

Keywords

Comments

All the terms in the sequence are congruent to 2 mod 3.
The constants in the definition (8, 86 and 864) are the concatenation of successive even digits 8,6 and 4.

Examples

			a(1) = 23 is a prime: 23+8 = 31, 23+86 = 109 and 23+864 = 887 are also prime.
a(2) = 743 is a prime: 743+8 = 751, 743+86 = 829 and 743+864 = 1607 are also prime.

Links

Crossrefs

Cf. A000040, A023200, A046136, A230223, A237890.

Programs

  • Maple
    KD:= proc() local a,b,d,e,f; a:= ithprime(n); b:=a+8;d:=a+86;e:=a+864; if isprime(b)and isprime(d)and isprime(e) then return (a) :fi; end: seq(KD(), n=1..15000);
  • Mathematica
    KD = {}; Do[p = Prime[n];If[PrimeQ[p + 8] && PrimeQ[p + 86] && PrimeQ[p + 864],AppendTo[KD, p]], {n, 15000}]; KD
c=0; p=Prime[n]; Do[If[PrimeQ[p+8]&&PrimeQ[p+86]&&PrimeQ[p+864],c=c+1;Print[c,"  ",p]], {n,1,5*10^6}]; (*b-file*)
  • PARI
    s=[]; forprime(p=2, 90000, if(isprime(p+8) && isprime(p+86) && isprime(p+864), s=concat(s, p))); s \\ Colin Barker, Apr 21 2014

A230230 Number of ways to write 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 5, 1, 5, 5, 3, 4, 5, 3, 2, 6, 4, 3, 6, 3, 3, 6, 3, 5, 6, 3, 6, 5, 4, 4, 9, 5, 4, 9, 5, 3, 9, 4, 4, 6, 4, 5, 6, 5, 5, 10, 4, 8, 10, 3, 7, 12, 3, 6, 11, 5, 7, 8, 3, 4, 6, 6, 4, 7, 2, 7, 9, 2, 10, 9, 3, 9, 8, 3, 5, 14, 8, 4, 12, 5, 5, 11, 5, 6, 8, 3, 8, 7, 4, 9, 11, 3
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 12 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 3.
This is stronger than Goldbach's conjecture for even numbers. If 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime, then 6*n is the sum of the two primes 3*p - 10 and 3*q + 10.
Conjecture verified for 2*n up to 10^9. - Mauro Fiorentini, Jul 08 2023

Examples

			a(5) = 1 since 2*5 = 7 + 3 with 3*7 - 10 = 11 and 3*3 + 10 = 19 both prime.
a(16) = 1 since 2*16 = 13 + 19 with 3*13 - 10 = 29 and 3*19 + 10 = 67 both prime.

Links

Crossrefs

Cf. A002375, A187757, A199920, A227923, A230227, A230140, A230141, A230217, A230219, A230223, A230224.

Programs

  • Mathematica
    PQ[n_]:=n>3&&PrimeQ[3n-10]
SQ[n_]:=PrimeQ[n]&&PrimeQ[3n+10]
a[n_]:=Sum[If[PQ[Prime[i]]&&SQ[2n-Prime[i]],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

A230227 Primes p with 3*p - 10 also prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 59, 61, 67, 79, 83, 89, 97, 101, 107, 109, 131, 137, 151, 157, 163, 167, 173, 191, 193, 199, 223, 229, 251, 257, 269, 277, 283, 307, 313, 317, 331, 347, 353, 367, 373, 397, 401, 409
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 12 2013

Keywords

Comments

Conjecture: For any integer n > 4 not equal to 76, we have 2*n = p + q for some terms p and q from the sequence.
This is stronger than Goldbach's conjecture for even numbers.

Examples

			a(1) = 5 since 3*5 - 10 = 5 is prime.

Links

Crossrefs

Cf. A000040, A002375, A230138, A230140, A230141, A230217, A230219, A230223, A230224, A230230.

Programs

  • Mathematica
    PQ[p_]:=PQ[p]=p>3&&PrimeQ[3p-10]
m=0
Do[If[PQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,80}]
Select[Prime[Range[100]],PrimeQ[3#-10]&] (* Harvey P. Dale, Jun 28 2015 *)

A236304 Primes p such that p+12, p+1234 and p+123456 are also prime.

Original entry on oeis.org

127, 907, 3037, 3457, 5737, 7057, 11047, 15427, 15667, 21517, 24697, 30307, 38287, 38317, 39607, 40177, 46477, 47797, 48787, 51157, 52177, 57667, 65587, 70627, 70867, 71887, 72997, 74857, 75277, 80317, 99817, 100447, 103657, 106747, 128437, 130087, 132157
Offset: 1

Views

Author

K. D. Bajpai, Apr 21 2014

Keywords

Comments

All the terms in the sequence are congruent to 1 mod 3.
The constants in the definition (12, 1234 and 123456) are the concatenation of digits 1,2,3,4,5 and 6.

Examples

			a(1) = 127 is a prime: 127+12 = 139, 127+1234 = 1361 and 127+123456 = 123583 are also prime.
a(2) = 907 is a prime: 907+12 = 919, 907+1234 = 2141 and 907+123456 = 124363 are also prime.

Links

Crossrefs

Cf. A000040, A023200, A046136, A230223, A236302, A237890.

Programs

  • Maple
    KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+12;d:=a+1234;e:=a+123456; if isprime(b)and isprime(d)and isprime(e)  then return (a) :fi; end: seq(KD(), n=1..15000);
  • Mathematica
    KD={}; Do[p=Prime[n]; If[PrimeQ[p+12]&&PrimeQ[p+1234]&&PrimeQ[p+123456], AppendTo[KD,p]], {n,15000}];KD
c=0; p=Prime[n]; Do[If[PrimeQ[p+12]&&PrimeQ[p+1234]&&PrimeQ[p+123456], c=c+1; Print[c,"  ",p]],{n,1,5*10^6}];(*b-file*)
  • PARI
    s=[]; forprime(p=2, 140000, if(isprime(p+12) && isprime(p+1234) && isprime(p+123456), s=concat(s, p))); s \\ Colin Barker, Apr 22 2014

A241485 Primes p such that p+2, p+222 and p+2222 are also prime.

Original entry on oeis.org

17, 29, 59, 71, 281, 461, 827, 1151, 1277, 1289, 1487, 1667, 1877, 1931, 2687, 2789, 2801, 3359, 3557, 3851, 4049, 4229, 4259, 4481, 4649, 5417, 5519, 5657, 5867, 5879, 6089, 6131, 6299, 6359, 6779, 6791, 7127, 7211, 8291, 8837, 9719, 10067, 10937, 13397, 13679
Offset: 1

Views

Author

K. D. Bajpai, Apr 23 2014

Keywords

Comments

All the terms in the sequence are congruent to 2 mod 3.
The constants in the definition (2, 222 and 2222) are the concatenation of digit 2.

Examples

			a(1) = 17 is a prime: 17+2 = 19, 17+222 = 239 and 17+2222 = 2239 are also prime.
a(2) = 29 is a prime: 29+2 = 31, 29+222 = 251 and 29+2222 = 2251 are also prime.

Links

Crossrefs

Cf. A000040, A023200, A046136, A230223, A236302, A237890.

Programs

  • Maple
    KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+2;d:=a+222;e:=a+2222; if isprime(b)and isprime(d)and isprime(e)  then return (a) :fi; end: seq(KD(), n=1..5000);
  • Mathematica
    KD={}; Do[p=Prime[n];If[PrimeQ[p+2]&&PrimeQ[p+222]&&PrimeQ[p+2222], AppendTo[KD,p]], {n,5000}]; KD
(*For b-file*) c=0;p=Prime[n];Do[If[PrimeQ[p+2]&&PrimeQ[p+222]&&PrimeQ[p+2222],c=c+1; Print[c,"  ",p]],{n,1,3*10^6}];

A227899 Number of primes p < n with 3*p - 4 and n^2 + (n - p)^2 both prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 2, 1, 2, 3, 2, 3, 3, 2, 3, 2, 3, 1, 1, 3, 2, 4, 2, 3, 3, 3, 4, 1, 2, 6, 2, 4, 2, 3, 5, 4, 2, 3, 4, 4, 4, 4, 2, 1, 2, 4, 2, 4, 2, 6, 7, 5, 3, 3, 9, 2, 3, 3, 2, 4, 4, 3, 1, 2, 8, 3, 6, 2, 2, 8, 4, 7, 2, 2, 5, 2, 3, 3, 2, 8, 3, 3, 1, 4, 7, 5, 9, 2, 2, 5
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 14 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 3.

Examples

			a(5) = 1 since 5 = 3 + 2, and the three numbers 3, 3*3 - 4 = 5 and 5^2 + (5-3)^2 = 29 are all prime.

Links

Crossrefs

Cf. A069003, A185636, A204065, A227898, A230223.

Programs

  • Mathematica
    a[n_]:=Sum[If[PrimeQ[3Prime[i]-4]&&PrimeQ[n^2+(n-Prime[i])^2],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A235871 Primes p such that p+2, p+24 and p+246 are also primes.

Original entry on oeis.org

5, 17, 107, 617, 857, 1277, 1487, 2087, 3167, 3557, 4217, 6947, 7457, 7877, 10067, 12917, 13217, 14387, 15137, 17657, 20897, 21317, 22367, 22697, 27407, 27527, 27917, 28547, 29207, 29387, 30467, 31727, 32117, 33287, 33617, 35507, 36107, 47657, 49367, 49787
Offset: 1

Views

Author

K. D. Bajpai, Apr 21 2014

Keywords

Comments

All the terms in the sequence are congruent to 5 mod 6.
The constants in the definition (2, 24 and 246) are the concatenation of first even digits 2,4 and 6.

Examples

			a(2) = 17 is a prime: 17+2 = 19, 17+24 = 41 and 17+246 = 263 are also prime.
a(3) = 107 is a prime: 107+2 = 119, 107+24 = 131 and 107+246 = 353 are also prime.

Links

Crossrefs

Cf. A000040, A023200, A046136, A230223, A237890.

Programs

  • Maple
    KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+2;d:=a+24;e:=a+246; if isprime(b) and isprime(d) and isprime(e) then return (a) :fi; end: seq(KD(), n=1..15000);
  • Mathematica
    KD = {}; Do[p = Prime[n]; If[PrimeQ[p + 2] && PrimeQ[p + 24] && PrimeQ[p + 246], AppendTo[KD, p]], {n, 15000}]; KD
c = 0; p = Prime[n]; Do[If[PrimeQ[p + 2] && PrimeQ[p + 24] && PrimeQ[p + 246], c = c + 1; Print[c, " ", Prime[n]]], {n, 1, 5000000}];  (* b - file *)
  • PARI
    s=[]; forprime(p=2, 50000, if(isprime(p+2) && isprime(p+24) && isprime(p+246), s=concat(s, p))); s \\ Colin Barker, Apr 21 2014

A241486 Primes p such that p+4, p+444 and p+4444 are also prime.

Original entry on oeis.org

13, 19, 79, 103, 229, 307, 643, 853, 859, 937, 1087, 1213, 1297, 1423, 1567, 1609, 1867, 2347, 2389, 2473, 3163, 3463, 3919, 4003, 4153, 4783, 4969, 5077, 5347, 5413, 5479, 5647, 5689, 5857, 6733, 6907, 6967, 7933, 8269, 9277, 9337, 9463, 10687, 10729, 11083
Offset: 1

Views

Author

K. D. Bajpai, Apr 23 2014

Keywords

Comments

All the terms in the sequence are congruent to 1 mod 6.
The constants in the definition (4, 444 and 4444) are the concatenations of the digit 4.

Examples

			a(1) = 13 is a prime: 13+4 = 17, 13+444 = 457 and 13+4444 = 4457 are also prime.
a(2) = 19 is a prime: 19+4 = 23, 19+444 = 463 and 19+4444 = 4463 are also prime.

Links

Crossrefs

Cf. A000040, A023200, A046136, A230223, A236302, A236304, A237890, A241485.

Programs

  • Maple
    KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+4; d:=a+444; e:=a+4444;if isprime(b)and isprime(d)and isprime(e)then return (a): fi;  end: seq(KD(), n=1..5000);
  • Mathematica
    KD = {}; Do[p = Prime[n]; If[PrimeQ[p + 4] && PrimeQ[p + 444] && PrimeQ[p + 4444], AppendTo[KD, p]], {n, 5000}]; KD
(* For the b-file*) c = 0; p = Prime[n]; Do[If[PrimeQ[p + 4] && PrimeQ[p + 444] && PrimeQ[p + 4444], c = c + 1; Print[c, "  ", p]], {n, 1, 3*10^6}];
  • PARI
    s=[]; forprime(p=2, 12000, if(isprime(p+4) && isprime(p+444) && isprime(p+4444), s=concat(s, p))); s \\ Colin Barker, Apr 25 2014
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