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 1-8 of 8 results.

A247273 Primes p such that p + m^2 is prime for all m in {2,4,6,8,10,12,14,16,18,20,22}.

Original entry on oeis.org

163, 409333, 9671300983, 186521536807, 376040154163, 459775038913, 485142116713, 773464440907, 916792710667, 982557050143, 1087801149583, 1213507492723, 1822896797857, 2131006835017, 3026318319523, 4617478214407, 5141744558017, 6552892412047, 6629618954863, 6787014897877, 7636453217677, 7788411508483, 8311114648153, 8547311473387, 8668135024957, 9206471763547
Offset: 1

Views

Author

Zak Seidov, Sep 11 2014

Keywords

Comments

All terms are == {7, 13} mod 30.
Subsequence of A145741.

Links

Crossrefs

Cf. A049492, A092120, A246842, A247269, A145741.

Programs

  • PARI
    forprime(p=1,10^12,c=0;for(i=1,11,if(ispseudoprime(p+(2*i)^2),c++));if(c==11,print1(p,", "))) \\ Derek Orr, Sep 11 2014

A145742 Composite numbers n such that all 10 numbers n+4*m^2 for m=1,2,...,10 are prime.

Original entry on oeis.org

143473, 2821813, 3171877, 27439003, 95771533, 107832187, 174499873, 315915517, 438561313, 761271373, 1010315923, 1083101167, 1247119687, 1272171217, 1494582637, 1540272427, 2760699643, 3205014607, 3615583147, 3636010807
Offset: 1

Views

Author

Farideh Firoozbakht, Oct 17 2008

Keywords

Comments

There is no further term up to 2*10^8.

Examples

			174499873 is composite and all 10 numbers 174499873+4m^2 for m=1,2,...,10 are prime.

Crossrefs

Cf. A145741, A145743.

Programs

  • Mathematica
    Select[Range[140001,364*10^7,2],CompositeQ[#]&&AllTrue[#+{4,16,36,64,100,144,196,256,324,400},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 12 2017 *)

Extensions

a(8)-a(20) from Donovan Johnson, Nov 17 2008

A145743 a(n) is the smallest composite number m such that all n numbers m+4*k^2 for k=1,2,...,n are prime.

Original entry on oeis.org

9, 15, 25, 25, 247, 583, 5377, 143473, 143473, 143473, 2821813, 62087033647, 91450844347
Offset: 1

Views

Author

Farideh Firoozbakht, Oct 17 2008

Keywords

Comments

There is no further term up to 5*10^8.

Crossrefs

Cf. A145741, A145742.

Extensions

a(12)-a(13) from Donovan Johnson, Nov 17 2008

A145896 Values of m: where m^2 begins a run of consecutive squares satisfying r=p+4*m^2 with a sequence of primes.

Original entry on oeis.org

3, 6, 2, 1, 8, 4, 7, 1, 2, 1, 1, 1, 19, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 7, 3, 4, 1, 1, 2, 7, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2
Offset: 1

Views

Author

Enoch Haga, Oct 25 2008

Keywords

Comments

Suggested by Farideh Firoozbakht's Puzzle 464 in Carlos Rivera's The Prime Puzzles & Problems Connection

Examples

			a(1)=3 because when m is 3 a sequence of three values of r end with prime 37; then r=1+4*1^1=5, prime; and r=1+4*2^2=17, prime; and r=1+4*3^2=37, prime (and the next value of m, 4, does not produce a prime because r=1+4*4^2=65). For this one value 1 is assumed prime.

Crossrefs

A145897 A145898 A145741

Programs

  • UBASIC
    10 'p464 20 N=1 30 A=3:S=sqrt(N) 40 B=N\A 50 if B*A=N then 100 60 A=A+2 70 if A<=S then 40 80 M=M+1:R=N+4*M^2:if R=prmdiv(R) and M<100 then print N;R;M:goto 80 90 if M>=1 then stop 100 M=0:N=N+2:goto 30

A145897 Starting prime (and 1): where number of consecutive squares m^2 satisfy r=p+4*m^2, prime.

Original entry on oeis.org

1, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423, 1429, 1447, 1483
Offset: 1

Views

Author

Enoch Haga, Oct 25 2008

Keywords

Comments

Suggested by Farideh Firoozbakht's Puzzle 464 in Carlos Rivera's The Prime Puzzles & Problems Connection. In this sequence Haga accepts 1 as a prime because then m^2 begins the first run of consecutive primes.
This looks like (apparent from the ad-hoc introduced leading 1) an erroneous version of A023200, because the definition says that it registers prime chains p+4*m^2, m=1,2,3,.. but apparently does not consider whether m is actually larger than 1. So 3 should be in the sequence because 3+4*1^2 is prime. - R. J. Mathar, Mar 25 2024

Examples

			a(1)=1 because when there are 3 consecutive m^2, first prime is 5 and ending prime is 37: r=1+4*1^1=5, prime; and r=1+4*2^2=17, prime; and r=1+4*3^2=37, prime (and the next value of r does not produce a prime).

Crossrefs

Cf. A145896, A145898, A145741, A049492.

Programs

  • UBASIC
    10 'p464
20 N=1
30 A=3:S=sqrt(N)
40 B=N\A
50 if B*A=N then 100
60 A=A+2
70 if A<=S then 40
80 M=M+1:R=N+4*M^2:if R=prmdiv(R) and M<100 then print N;R;M:goto 80
90 if M>=1 then stop
100 M=0:N=N+2:goto 30

A145898 Ending prime: where number of consecutive squares m^2 satisfy r = p + 4*m^2, prime.

Original entry on oeis.org

37, 151, 29, 23, 293, 107, 263, 83, 113, 107, 113, 131, 1607, 197, 239, 233, 313, 311, 317, 353, 383, 401, 443, 461, 499, 523, 503, 617, 659, 677, 743, 773, 773, 887, 857, 863, 881, 887, 911, 953, 983, 1013, 1283, 1129, 1277, 1283, 1301, 1319, 1619, 1433
Offset: 1

Views

Author

Enoch Haga, Oct 25 2008

Keywords

Comments

Farideh Firoozbakht noticed the unusually high number of 19 primes ending in a(13)=1607 in A145741 where she specified an m sequence of 1..10. My m sequence is open. 1,2,3,...,m+1, all squared.

Examples

			a(1)=37 because when m is 3, the first prime is 5 and the ending prime is 37: r=1+4*1^1=5, prime; and r=1+4*2^2=17, prime; and r=1+4*3^2=37, prime (and the next value of r does not produce a prime).

Crossrefs

Cf. A145896, A145897, A145741.

Programs

  • UBASIC
    10 'p464
20 N=1
30 A=3:S=sqrt(N)
40 B=N\A
50 if B*A=N then 100
60 A=A+2
70 if A<=S then 40
80 M=M+1:R=N+4*M^2:if R=prmdiv(R) and M<100 then print N;R;M:goto 80
90 if M>=1 then stop
100 M=0:N=N+2:goto 30

A247275 Primes p such that p + m^2 is prime for all m in {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}.

Original entry on oeis.org

163, 409333, 376040154163, 1822896797857, 9871431850597, 13491637509487, 19802478368863
Offset: 1

Views

Author

Zak Seidov, Sep 11 2014

Keywords

Comments

All terms are == {7, 13} mod 30.
Subsequence of A247273.

Crossrefs

Cf. A049492, A092120, A246842, A247269, A247273, A145741.

Programs

  • PARI
    forprime(p=1, oo, c=0; for(i=1, 12, if(ispseudoprime(p+(2*i)^2), c++)); if(c==12, print1(p, ", "))) \\ Derek Orr, Sep 11 2014

A247276 Primes p such that p + m^2 is prime for all m in {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26}.

Original entry on oeis.org

163, 409333, 13491637509487, 19802478368863
Offset: 1

Views

Author

Zak Seidov, Sep 11 2014

Keywords

Comments

All terms are == {7, 13} mod 30.
Subsequence of A247275.

Crossrefs

Cf. A049492, A092120, A246842, A247269, A145741, A247273, A247275.

Programs

  • Mathematica
    Select[Prime[Range[35000]],AllTrue[#+{4,16,36,64,100,144,196,256,324,400,484,576,676},PrimeQ]&] (* The program generates the first two terms of the sequence. To generate a(3) and a(4), increase the Range constant to 67*10^10 but the program will take a very long time to run. *) (* Harvey P. Dale, Mar 05 2025 *)
  • PARI
    forprime(p=1, 10^12, c=0; for(i=1, 13, if(ispseudoprime(p+(2*i)^2), c++);if(!ispseudoprime(p+(2*i)^2),break)); if(c==13, print1(p, ", "))) \\ Derek Orr, Sep 11 2014
Showing 1-8 of 8 results.

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