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.

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A145650 Linking prime for the first and second member of maximal chains of primes that have at least three members.

Original entry on oeis.org

43, 197, 1307, 2371, 4561, 9941, 22573, 33203, 214507, 227611, 306853, 332993, 389167, 505907, 695059, 758441, 810023, 1072657, 1202987, 1404211, 1567487, 1621621, 2407309, 2773681, 2854331, 2932511, 3013601, 3206773, 3851423
Offset: 1

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Author

Enoch Haga, Oct 15 2008

Keywords

Comments

A sequence of consecutive primes prime(k), ..., prime(k+r), r >= 1, is called a chain of primes if i*prime(i) + (i+1)*prime(i+1)* is prime (the linking prime for prime(i) and prime(i+1), cf. A119487) for i from k to k+r-1. A chain of primes prime(k), ..., prime(k+r) is maximal if it is not part of a longer chain, i.e., if neither (k-1)*prime(k-1) + k*prime(k) nor (k+r)*prime(k+r) + (k+r+1)*prime(k+r+1) is prime.
A145651 gives the linking prime for the second and third member of maximal chains of primes that have at least three members.
Suggested by J. M. Bergot in Puzzle 463 of Carlos Rivera's Prime Puzzles & Problems Connection

Examples

			Primes 13, 17, 19, 23 have prime indices 6, 7, 8, 9. 6*13 + 7*17 = 197 is prime; 7*17 + 8*19 = 271 is prime; 8*19 + 9*23 = 359 is prime. Neither 5*11 + 6*13 = 133 nor 9*23 + 10*29 = 497 is prime, so 13, 17, 19, 23 is maximal. Hence 6*13 + 7*17 = 197, the linking prime for 13 and 17, is in the sequence.

Links

Crossrefs

Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A119487 (primes in A152117, linking primes), A152658 (beginnings of maximal chains of primes), A145651.

Programs

  • Magma
    [ n*p+(n+1)*q: n in [1..520] | (n eq 1 or not IsPrime((n-1)*PreviousPrime(p)+n*p) ) and IsPrime(n*p+(n+1)*q) and IsPrime((n+1)*q+(n+2)*r) where r is NextPrime(q) where q is NextPrime(p) where p is NthPrime(n) ]; // Klaus Brockhaus, Dec 11 2008
  • PARI
    {n=1; while(n<520, c=0; while(isprime(b=n*prime(n)+(n+1)*prime(n+1)), c++; n++; if(c==1, a=b)); if(c>1, print1(a, ",")); n++)}

Extensions

Edited by Klaus Brockhaus, Dec 10 2008

A145651 Linking prime for the second and third member of maximal chains of primes that have at least three members.

Original entry on oeis.org

83, 271, 1553, 2693, 5051, 10651, 23333, 34123, 219389, 230933, 312007, 338017, 395309, 512891, 699437, 763999, 815257, 1078127, 1208791, 1417019, 1577561, 1629083, 2420609, 2787947, 2868787, 2944429, 3038639, 3222101, 3868201
Offset: 1

Views

Author

Enoch Haga, Oct 15 2008

Keywords

Comments

A sequence of consecutive primes prime(k), ..., prime(k+r), r >= 1, is called a chain of primes if i*prime(i) + (i+1)*prime(i+1)* is prime (the linking prime for prime(i) and prime(i+1), cf. A119487) for i from k to k+r-1. A chain of primes prime(k), ..., prime(k+r) is maximal if it is not part of a longer chain, i.e., if neither (k-1)*prime(k-1) + k*prime(k) nor (k+r)*prime(k+r) + (k+r+1)*prime(k+r+1) is prime.
A145650 gives the linking prime for the first and second member of maximal chains of primes that have at least three members.
Suggested by J. M. Bergot in Puzzle 463 of Carlos Rivera's Prime Puzzles & Problems Connection

Examples

			Primes 13, 17, 19, 23 have prime indices 6, 7, 8, 9. 6*13 + 7*17 = 197 is prime; 7*17 + 8*19 = 271 is prime; 8*19 + 9*23 = 359 is prime. Neither 5*11 + 6*13 = 133 nor 9*23 + 10*29 = 497 is prime, so 13, 17, 19, 23 is maximal. Hence 7*17 + 8*19 = 271, the linking prime for 17 and 19, is in the sequence.

Links

Crossrefs

Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A119487 (primes in A152117, linking primes), A152658 (beginnings of maximal chains of primes), A145650.

Programs

  • Magma
    [ (n+1)*q+(n+2)*r: n in [1..520] | (n eq 1 or not IsPrime((n-1)*PreviousPrime(p)+n*p) ) and IsPrime(n*p+(n+1)*q) and IsPrime((n+1)*q+(n+2)*r) where r is NextPrime(q) where q is NextPrime(p) where p is NthPrime(n) ]; // Klaus Brockhaus, Dec 11 2008
  • PARI
    {n=1; while(n<520, c=0; while(isprime(b=n*prime(n)+(n+1)*prime(n+1)), c++; n++; if(c==2, a=b)); if(c>1, print1(a, ",")); n++)}

Extensions

Edited by Klaus Brockhaus, Dec 10 2008

A268467 Smallest prime that is the (sum, k*prime(k),k=m,..n+m-1) for some m, or 0 if no such m exists.

Original entry on oeis.org

2, 43, 23, 0, 1109, 1187, 929, 0, 4973, 1291, 11197, 0, 26099, 15583, 4423, 0, 42139, 10729, 21283, 0, 36899, 27179, 21563, 0, 24359, 33863, 27361, 0, 223423, 51239, 293467, 42043, 67699, 56503, 118361, 0, 80449, 94693, 136739, 0, 127837, 136991, 387913, 0, 304259, 192013, 321721, 0, 339517, 357683
Offset: 1

Views

Author

Zak Seidov, Feb 05 2016

Keywords

Comments

Smallest prime that is the sum of n consecutive terms of A033286.
Apparently a(n) exists for any odd n.
Values of m = {1, 3, 1, 0, 7, 6, 4, 0, 9, 2, 12, 0, 17, 11, 2, 0, 17, 4, 8, 0, 11, 7, 4, 0, 3, 5, 2, 0, 27, 5, 30, 1, 5, 2, 10, 0, 3, 4, 8, 0, 5, 5, 22, 0, 15, 6, 14, 0, 13, 13, ...}. - Michael De Vlieger, Feb 05 2016

Examples

			n=1: m=1 and 1*prime(1) = 1*2 = 2 = a(1),
n=2: m=3 and 3*prime(3)+4*prime(4) = 3*5+4*7 = 43 = a(2),
n=3: m=1 and 1*prime(1)+2*prime(2)+3*prime(3) = 1*2+2*3+3*15 = 23 = a(3),
n=4: no solution => a(4) = 0,
n=5: m=7 and 7*prime(7)+..11*prime(11) = 119+152+207+290+341 = 1109 = a(5).

Crossrefs

Cf. A000040, A033286, A105455, A268465, A105454, A152117, A119487.

Programs

  • Mathematica
    Table[If[# == 0, 0, Sum[k Prime@ k, {k, #, n + # - 1}]] &@(SelectFirst[Range[10^3], PrimeQ@ Sum[k Prime@ k, {k, #, n + # - 1}] &] /. x_ /; MissingQ@ x -> 0), {n, 50}] (* Michael De Vlieger, Feb 05 2016, Version 10.2 *)
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