A119751 -id:A119751 - cp's OEIS frontend

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.

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

Walter Kehowski, Jan 10 2007

nonn

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

			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,...

Cf. A119751, A119752, A127257.

A120340 Sequence of pairs numerator(s(n)), denominator(s(n)) where s(n) is the n-th partial sum of 1/A119751(n).

Original entry on oeis.org

1, 1, 4, 3, 13, 9, 302, 207, 43255, 29601, 58966432, 40346163, 1700782246897, 1163704379409, 687683872186748, 470524470741039, 209415712651415308217, 143285523403473419397, 552462361711804327497414320

Offset: 1

Walter Kehowski, Jun 23 2006

nonn

There are only 11 terms of A119751 known and s(11) is 1.4615274858438132136894180577690622581842692760227 to 50 decimal places. What is sum(1/A119751(k),k=1..infinity)?

			a(5)=13, a(6)=9 since s(3)=1+1/3+1/9=13/9.

Cf. A119751, A119753.

a(2n-1) = numerator(s(n)), a(2n)=denominator(s(n)), where s(n)=sum(1/A119751(k),k=1..n).

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

A119752 a(1)=2; a(n)=first even number greater than a(n-1) such that a(i)+a(n)+1 is prime for all i=1,2,...,n.

Original entry on oeis.org

2, 8, 14, 44, 224, 638, 1274, 4004, 675404, 2203958, 3075158, 6195234164, 77989711184, 4566262987328

Offset: 1

Walter Kehowski, Jun 17 2006

Cf. A119751.

R:= [2]: count:= 1:
for k from 4 by 2 while count < 11 do
  if isprime(2*k+1) and andmap(isprime, [seq(R[i]+k+1,i=1..count)]) then
     R:= [op(R),k]; count:= count+1
  fi
od:
R; # Robert Israel, Mar 06 2023

Table[If[n == 1, a[1] = 2, 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, 9}] (* Robert Price, Apr 03 2019 *)

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]=2; for (n=2, nn, my(k=va[n-1]+2); while (!isok(va, k, n), k+=2); va[n] = k); va; \\ Michel Marcus, Mar 06 2023

a(13)-a(14) from Donovan Johnson, Mar 23 2008

A113875 Slowest growing sequence of primes having the prime-pairwise-average property: if i

Original entry on oeis.org

3, 7, 19, 139, 859, 8179, 173059, 1026199, 1827139, 15828679, 13187242759, 18732483199, 912492556939, 9130567625119

Offset: 1

T. D. Noe, Jan 26 2006

Assuming the prime k-tuples conjecture, Granville shows (in section 2.4) that this sequence is infinite.

			The pairwise averages of {3,7,19} are the primes {5,11,13}.

Andrew Granville, Prime number patterns, The American Mathematical Monthly, Vol. 115, No. 4 (2008), pp. 279-296; alternative link.

Cf. A113832, A115760, A119751.

s={3, 7}; i=5; Do[While[ !And@@PrimeQ[(s+Prime[i])/2], i++ ]; AppendTo[s, Prime[i]]; i++, {n, 3, 10}]; s

a(n) = 2*A119751(n)+1. - Don Reble, Aug 17 2021

More terms from Don Reble and Giovanni Resta, Feb 15 2006

a(14) from Amiram Eldar, Jun 27 2024

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

Walter Kehowski, Jan 10 2007

nonn

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.

			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,...

Eric Weisstein's World of Mathematics, Sierpinski Number of the Second Kind.

Cf. A119751, A119752, A127256.

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

A120341 Sequence of pairs numerator(s(n)), denominator(s(n)) where s(n) is the n-th partial sum of 1/A119753(n).

Original entry on oeis.org

1, 3, 8, 15, 71, 105, 886, 1155, 12673, 15015, 255802, 285285, 18447227, 20255235, 1366902806, 1478632155, 109463953829, 116811940245, 15332301522476, 16236859694055, 6624458815881211, 6998086528137705, 2875388753804702068

Offset: 1

Walter Kehowski, Jun 23 2006

nonn

There are only 66 terms of A119753 known and s(66) is 0.95578109643864566820083634846514203365246166270989 to 50 decimal places. What is sum(1/A119753(k),k=1..infinity)?

			a(5)=71, a(6)=105 since s(3)=1/3+1/5+1/7=71/105.

Cf. A119751, A119753.

a(2n-1) = numerator(s(n)), a(2n)=denominator(s(n)), where s(n)=sum(1/A119753(k),k=1..n).

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

Walter Kehowski, Jul 02 2006

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.

			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.

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

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;

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.

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

Walter Kehowski, Jul 02 2006

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.

			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.

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

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;

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

a(12)-a(14) from Ray Chandler, Apr 04 2010

Showing 1-9 of 9 results.

cp's OEIS Frontend

A119753 Prime numbers in their order of occurrence and generated by A119751, the sequence of odd numbers defined recursively by a(1)=1 and a(i) + a(j) + 1 is prime for all i,j.

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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.

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A120340 Sequence of pairs numerator(s(n)), denominator(s(n)) where s(n) is the n-th partial sum of 1/A119751(n).

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A119752 a(1)=2; a(n)=first even number greater than a(n-1) such that a(i)+a(n)+1 is prime for all i=1,2,...,n.

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A113875 Slowest growing sequence of primes having the prime-pairwise-average property: if i

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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.

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PARI

A120341 Sequence of pairs numerator(s(n)), denominator(s(n)) where s(n) is the n-th partial sum of 1/A119753(n).

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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.

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Maple

Formula

Extensions

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.

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