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 1-9 of 9 results.

A119754 Prime numbers in their order of occurrence and generated by A119752, the sequence of even numbers defined recursively by a(1)=2 and a(i) + a(j) + 1 is prime for all i,j.

Original entry on oeis.org

5, 11, 17, 17, 23, 29, 47, 53, 59, 89, 227, 233, 239, 269, 449, 641, 647, 653, 683, 863, 1277, 1277, 1283, 1289, 1319, 1499, 1913, 2549, 4007, 4013, 4019, 4049, 4229, 4643, 5279, 8009, 675407, 675413, 675419, 675449, 675629, 676043, 676679, 679409
Offset: 1

Views

Author

Walter Kehowski, Jun 17 2006, Jun 19 2006, Jun 25 2006

Keywords

Comments

The primes are p(i) + p(j) + 1, j<=i, are not distinct for the (i,j) pairs (2,2),(3,1), with prime=17 and (6,6),(7,1), with prime=1277.

Examples

			a(1)=2, a(2)=8 so 2+2+1=5, 8+2+1=11, 8+8+1=17 so the first three elements are 5, 11, 17.

Programs

  • Maple
    EP:=[]: P:=[]: for w to 1 do for n from 0 to 12^6 do s:=6*n+2; Q:=map(z->s+z+1,[op(EP),s]); if andmap(isprime,Q) then EP:=[op(EP),s]; P:=[op(P),op(Q)]; print(s); print(Q); fi; od od; EP; P;

Formula

Let a(n) be the sequence defined recursively by a(1)=2 and a(n) is the first even number greater than a(n-1) such that 2*a(n)+1 is prime and a(i) + a(n) + 1 is prime for all i<=n-1. Then p(n) is the n-th prime in the lexicographic order a(i) + a(j) + 1, i>=j.

A127257 a(n) is the initial element of the sequence A(n) defined exactly like A119752 but with the additional condition that each of its elements must not be contained in any of the sequences A(k) for k < n.

Original entry on oeis.org

2, 6, 18, 20, 50, 68, 74, 78, 90, 96, 134, 138, 156, 200, 204, 228, 278, 288, 296, 306, 326, 338, 378, 384, 398, 404, 440, 464, 468, 504, 510, 524, 530, 546, 600, 608, 644, 660, 704, 726, 740, 774, 828, 854, 870, 930, 938, 944, 966, 986, 1008, 1034, 1068
Offset: 1

Views

Author

Walter Kehowski, Jan 10 2007

Keywords

Comments

Recall that A119752 is the sequence defined recursively by a(1)=2 and a(k) is the first even number greater than a(k-1) such that 2a(k)+1 is prime and a(k)+a(j)+1 is prime for all 1<=j
a(n)=A(n,1), the first element of each sequence A(n) defined recursively as follows. Let A(1)=A119752, that is, A(1,k)=A119752(k). Then A(n) is the sequence defined recursively as follows: (1) A(n,1) is the first even number not in any A(m), 1<=m
Let us say that a positive integer t is eventually in A(n) if 2^p*t, p>=1, is in A(n). For example, if t=19, then A(36,1)=2^5*19. The only numbers not eventually in some A(n) are all powers 2^p such that 2^(p+1)+1 is not prime, (so p is not 0, 1, 3, 7, 15) and Sierpinski numbers of the second kind, namely odd integers t such that 2^p*t+1 is composite for all p>=1. The smallest known example is t=78,557.

Examples

			a(1)=2 is the first element of A119752=2, 8, 14, 44, 224, 638, 1274, 4004, 675404, ... so a(2)=6 since 6 is the first even number not in A119752 such that 2*6+1 is prime. Furthermore, A(2)=6, 30, 36, 120, 300, 426, 8700, 71910,...

Links

Crossrefs

Cf. A119751, A119752, A127256.

Programs

  • PARI
    isok(va, k, n) = if (isprime(2*k+1), for (i=1, n-1, if (! isprime(va[i]+k+1), return(0))); return(1));
avec(start, lim) = my(list=List(), n=2, ok=1); listput(list, start); while(ok, my(k=list[n-1]+2); while (!isok(list, k, n), k+=2; if (k>lim, ok=0; break)); if (ok, listput(list, k)); n++;); Vec(list);
findfirst(v, lim) = forstep (n=2, lim, 2, if (isprime(2*n+1) && !vecsearch(v, n), return(n)));
lista(lim) = my(list = List()); listput(list, 2); my(v=avec(2, lim)); while (1, my(new = findfirst(v, lim)); if (! new, break); my(w = avec(new, lim)); v = vecsort(concat(v, w)); listput(list, new);); Vec(list); \\ Michel Marcus, Mar 06 2023

A120342 Sequence of pairs numerator(s(n)), denominator(s(n)) where s(n) is the n-th partial sum of 1/A119752(n).

Original entry on oeis.org

1, 2, 5, 8, 39, 56, 443, 616, 1783, 2464, 51819, 71456, 4720633, 6502496, 4722257, 6502496, 797359442331, 1097952952096, 878673909876949097, 1209921096197797984, 1351031156635237614515155
Offset: 1

Views

Author

Walter Kehowski, Jun 23 2006

Keywords

Comments

There are only 12 terms of A119752 known and s(12) is 0.72622446726779027806723655668104871423264641644675 to 50 decimal places. What is sum(1/A119752(k),k=1..infinity)?

Examples

			a(5)=39, a(6)=56 since s(3)=1/2+1/8+1/14=39/56.

Crossrefs

Cf. A119752, A119754.

Formula

a(2n-1) = numerator(s(n)), a(2n)=denominator(s(n)), where s(n)=sum(1/A119752(k),k=1..n).

Extensions

This is not really a sequence. The standard OEIS convention would be to split this into two cross-referenced sequences with keyword "frac". - N. J. A. Sloane, Jul 22 2006

A119751 a(1) = 1; a(n) = first odd number greater than a(n-1) such that a(n) + a(i) + 1 is prime for all i=1,2,...,n.

Original entry on oeis.org

1, 3, 9, 69, 429, 4089, 86529, 513099, 913569, 7914339, 6593621379, 9366241599, 456246278469, 4565283812559
Offset: 1

Views

Author

Walter Kehowski, Jun 17 2006

Keywords

Crossrefs

Cf. A113875, A119752.

Programs

  • Mathematica
    Table[If[n == 1, a[1] = 1, j = a[n - 1] + 2; While[a[n] = j; !
    AllTrue[Table[a[i] + a[n] + 1, {i, 1, n}], PrimeQ], j += 2]; j]
, {n, 1, 7}] (* Robert Price, Apr 03 2019 *)
  • PARI
    isok(va, k, n) = if (isprime(2*k+1), for (i=1, n-1, if (! isprime(va[i]+k+1), return(0))); return(1));
lista(nn) = my(va=vector(nn)); va[1]=1; for (n=2, nn, my(k=va[n-1]+2); while (!isok(va, k, n), k+=2); va[n] = k); va; \\ Michel Marcus, Mar 07 2023

Formula

2*a(n)+1 = A113875(n). - Don Reble, Aug 17 2021

Extensions

a(13)-a(14) from Donovan Johnson, Mar 23 2008
Corrected and edited by Walter Kehowski, Oct 18 2008

A037100 Lexically first set of (even) numbers, beginning with 4, such that for any two different terms, a(i) + a(j) + 1 is prime.

Original entry on oeis.org

4, 6, 12, 24, 54, 186, 3246, 25926, 169314, 412026, 541524, 37949286, 124716066, 324532464, 26678398374, 3559613215806, 30751771983294, 20116294396883346
Offset: 1

Views

Author

Carlos Rivera

Keywords

Examples

			4+54+1 is prime.

Links

Crossrefs

Cf. A093483, A103828, A119752, A133761.

Extensions

Edited and extended with a(15)-a(18) by Don Reble, Feb 13 2019

A127256 a(n) is the initial element of the sequence A(n) defined exactly like A119751 but with the additional condition that each of its elements must not be contained in any of the sequences A(k) for k < n.

Original entry on oeis.org

1, 5, 15, 23, 33, 41, 53, 75, 89, 99, 105, 113, 153, 155, 165, 189, 215, 239, 249, 261, 281, 293, 323, 341, 363, 371, 375, 405, 411, 419, 431, 473, 519, 543, 545, 561, 575, 629, 659, 699, 725, 741, 743, 765, 785, 803, 831, 849, 893, 905, 915, 923, 933, 935
Offset: 1

Views

Author

Walter Kehowski, Jan 10 2007

Keywords

Comments

a(n)=A(n,1), the first element of each sequence A(n) defined recursively as follows. Recall that A119751 is the sequence defined recursively by a(1)=1 and a(k) is the first odd number greater than a(k-1) such that 2a(k)+1 is prime and a(k)+a(j)+1 is prime for all 1<=jA119751, that is, A(1,k)=A119751(k). Then A(n) is the sequence defined recursively as follows: (1) A(n,1) is the first odd number not in any A(m), 1<=m

Examples

			a(1)=1 is the first element of A119751=1, 3, 9, 69, 429, 4089, 86529, 513099, ... so a(2)=5 since 5 is the first odd number not in A119751 such that 2*5+1 is prime. Furthermore, A(2)=5, 11, 35, 95, 221, 551, 1271, 5705,...

Crossrefs

Cf. A119751, A119752, A127257.

A120343 Sequence of pairs numerator(s(n)), denominator(s(n)) where s(n) is the n-th partial sum of 1/A119754(n).

Original entry on oeis.org

1, 5, 16, 55, 327, 935, 382, 935, 9721, 21505, 303414, 623645, 14884103, 29311315, 818168774, 1553499695, 49825457361, 91656482005, 4526122187134, 8157426898445, 1035587163377863, 1851735905947015, 243143544972989094
Offset: 1

Views

Author

Walter Kehowski, Jun 23 2006

Keywords

Comments

There are only 78 terms of A119754 known and s(78) is 0.58817497337288808165551740612048298329310794973566 to 50 decimal places. What is sum(1/A119754(k),k=1..infinity)?

Examples

			a(5)=327, a(6)=935 since s(3)=1/5+1/11+1/17=327/935.

Crossrefs

Cf. A119752, A119754.

Formula

a(2n-1) = numerator(s(n)), a(2n)=denominator(s(n)), where s(n)=sum(1/A119754(k),k=1..n).

Extensions

This is not really a sequence. The standard OEIS convention would be to split this into two cross-referenced sequences with keyword "frac". - N. J. A. Sloane, Jul 22 2006

A120402 a(1)=2; a(n)=first even number greater than a(n-1) such that 2*a(n)-1 is prime and a(i)+a(n)-1 is prime for all 1<=i<=n-1.

Original entry on oeis.org

2, 4, 10, 70, 430, 4090, 86530, 513100, 913570, 7914340, 6593621380, 9366241600
Offset: 1

Views

Author

Walter Kehowski, Jul 02 2006

Keywords

Comments

All elements after the first are 4 mod 6. In base 12 the sequence is 2, 4, X, 5X, 2EX, 244X, 420XX, 208E24, 38082X, 2798084, where X is 10 and E is eleven.

Examples

			a(2)=4 since 4 is the first even number > a(1)=2 such that 2*4-1=7 is prime and 4+2-1=5 is prime.

Crossrefs

Cf. A119752, A119754, A119751, A119753, A103828.

Programs

  • Maple
    EP:=[2]: for w to 1 do for k from 0 to 12^8 do n:=6*k+4; p:=2*n-1; Q:=map(z-> z+n-1, EP); if isprime(p) and andmap(isprime,Q) then EP:=[op(EP),n]; print(n); fi od od;

Formula

a(1)=2; a(n) = s where s is the first even number s>a(n-1) such that 2*s-1 is prime and s+a(i)-1 is prime, 1<=i<=n-1.

Extensions

a(11)-a(12) from Bert Dobbelaere, Apr 17 2019

A120403 a(1)=3; a(n)=first odd number greater than a(n-1) such that 2*a(n)-1 is prime and a(i)+a(n)-1 is prime for all 1<=i<=n-1.

Original entry on oeis.org

3, 9, 15, 45, 225, 639, 1275, 4005, 675405, 2203959, 3075159, 6195234165, 77989711185, 4566262987329
Offset: 1

Views

Author

Walter Kehowski, Jul 02 2006

Keywords

Comments

All elements are 3 mod 6. In base 12 the sequence is 3, 9, 13, 39, 169, 453, 8X3, 2399, 286X39, 8X3533, 1043733, where X is 10 and E is eleven.

Examples

			a(2)=9 since 9 is the first odd number > a(1)=3 such that 2*9-1=17 is prime and 9+3-1=13 is prime.

Crossrefs

Cf. A119752, A119754 (resulting primes), A119751, A119753, A103828.

Programs

  • Maple
    OP:=[3]: for w to 1 do for k from 0 to 12^8 do n:=6*k+3; p:=2*n-1; Q:=map(z-> z+n-1, OP); if isprime(p) and andmap(isprime,Q) then OP:=[op(OP), n]; print(n); fi od od;

Formula

a(1)=3; a(n) = s where s is the first odd number s>a(n-1) such that 2*s-1 is prime and s+a(i)-1 is prime, 1<=i<=n-1.
a(n) = A119752(n) + 1. - Chandler

Extensions

a(12)-a(14) from Ray Chandler, Apr 04 2010
Showing 1-9 of 9 results.

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

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