A247665 -id:A247665 - cp's OEIS frontend

A249058 a(n) = number of primes less than the square root of the (2^n)-th prime.

0, 0, 1, 2, 4, 5, 7, 9, 12, 17, 24, 32, 45, 61, 82, 114, 154, 215, 293, 404, 557, 766, 1057, 1459, 2025, 2800, 3880, 5379, 7470, 10368, 14414, 20030, 27864, 38745, 53982, 75206, 104799, 146151, 203821, 284381, 396976, 554303, 774256, 1081749, 1511871, 2113506

Offset: 0

Russ Cox, Oct 19 2014

nonn

Hugo Pfoertner, Table of n, a(n) for n = 0..78 (terms 0..56 from Jens Kruse Andersen)

Cf. A033844. Related to analysis of A247665.

Mathematica
```
PrimePi[Sqrt[Prime[2^n]]]
```

a(n) = primepi(sqrt(A033844(n))). - Jens Kruse Andersen, Oct 20 2014

More terms from Jens Kruse Andersen, Oct 20 2014

A249692 a(1)=2; thereafter, a(n) is the smallest number not occurring earlier such that Kronecker(a(k), a(n)) = -1 for the next n indices k = n+1, n+2, ..., 2n.

Original entry on oeis.org

2, 3, 5, 7, 13, 33, 20, 73, 47, 193, 113, 683, 103, 433, 45, 562, 1313, 10307, 4013, 12613, 9133, 10643, 5537, 31307, 16727, 50923, 66463, 195227, 92237, 229913, 125, 342763, 2248

Offset: 1

Michel Lagneau, Nov 04 2014

Kronecker(i,j) is an extension of the Jacobi symbol to all integers. The sequence with the condition Kronecker(a(k), a(n)) = -1/+1 is given by A247665.

			a(1) = 2 because the next term is 3 and k(2,3) = -1;
a(2) = 3 because the next two terms are (5,7) => k(3,5) = -1 and k(3,7) = -1;
a(3) = 5 because the next three terms are (7,13,33) => k(5,7) = -1, k(5,13) = -1 and k(5,33) = -1.

Cf. A247665.

m=33; v=vector(m); u=vectorsmall(25000*m); for(n=1, m, for(i=2, 10^9, if(!u[i], for(j=(n+1)\2, n-1, if(kronecker(v[j], i)==1 || kronecker(v[j], i)==0, next(2))); v[n]= i; u[i]=1; break))); v

A254003 For n>=1, let B_n(k) be the sequence defined by b(1)=1, b(2)=p_1, ..., b(n+2)=p_(n+1), thereafter (for k>=n+3) the smallest number not occurring earlier having at least one common factor with b(k-(n+1)), but none with b(k-1)b(k-2)...*b(k-n); a(n) is m such that B_n(m)=6.

Original entry on oeis.org

10, 14, 83, 157, 1190, 206

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Jan 22 2015

In particular, A098550(10)=6, A247225(14)=6.

On the other hand, it seems likely that 6 never appears in A247665 (which has a different although related definition). - N. J. A. Sloane, Apr 26 2015

Cf. A098550, A247225, A247665.

A249778 a(1)=2; thereafter, a(n) is the smallest number not occurring earlier such that Kronecker(a(k), a(n)) = 1 for the next n indices k = n+1, n+2, ..., 2n.

Original entry on oeis.org

2, 7, 3, 4, 11, 25, 9, 19, 16, 49, 17, 59, 67, 121, 81, 169, 43, 115, 64, 191, 293, 361, 289, 587, 269, 841, 863, 961, 1031, 1369, 441, 1681, 1867, 2209, 1849, 2809, 65, 529, 256, 643, 3481, 3517, 1639, 1877, 3721, 4489, 5041, 5329, 5591, 6241, 3557, 6889, 7921

Offset: 1

Michel Lagneau, Nov 05 2014

nonn

Kronecker(i,j) is an extension of the Jacobi symbol to all integers. The sequence with the condition Kronecker(a(i), a(n)) = -1/+1 is given by A247665.

			a(1) = 2 because the next term is 7 and k(2,7) = 1;
a(2) = 7 because the next two terms are (3,4) => k(7,3) = 1 and k(7,4) = 1;
a(3) = 3 because the next three terms are (4,11,25) => k(3,4)= 1, k(3,11) = 1 and k(3,25) = 1.

Cf. A247665, A249692.

m=55; v=vector(m); u=vectorsmall(1000*m); for(n=1, m, for(i=2, 10^9, if(!u[i], for(j=(n+1)\2, n-1, if(kronecker(v[j], i)==-1 || kronecker(v[j], i)==0, next(2))); v[n]= i; u[i]=1; break))); v

A259018 Lexicographically first permutation of the nonnegative integers such that Sum_{k=n..2n} a(k) is a square, starting with a(1)=0.

Original entry on oeis.org

0, 1, 2, 6, 3, 5, 4, 7, 8, 9, 10, 21, 11, 30, 12, 13, 14, 16, 15, 18, 17, 19, 20, 50, 22, 32, 23, 60, 24, 45, 25, 28, 26, 75, 27, 34, 29, 36, 31, 35, 33, 38, 37, 92, 39, 100, 40, 43, 41, 74, 42, 47, 44, 57, 46, 48, 49, 84, 51, 52, 53, 90, 54, 55, 56, 58, 93, 59

Offset: 1

Michel Lagneau, Jun 16 2015

nonn

The corresponding squares are 1, 9, 16, 25, 36, 64, 100, 121, 144, 169, 196, 256, 289, 361, 400, 441, 529, 576, 625, 676, 729, 841, 961, 1024, 1089, 1156, 1225, 1296, 1369, ...

This is a permutation of the integers.

			a(1) = 0 plus the next single term 1 is 1 = 1^2;
a(2) = 1 plus the next two terms (2,6) is 9 = 3^2;
a(3) = 2 plus the next three terms (6,3,5) is 16 = 4^2;
a(4) = 6 plus the next four terms (3,5,4,7) is 25 = 5^2.

Index entries for sequences that are permutations of the natural numbers

Cf. A000290, A247665.

nn:=100:T:=array(1..nn):T[1]:=0:T[2]:=1:kk:=2:lst:={0,1}:
for n from 2 to nn do:
  ii:=0:
    for k from 2 to 1000 while(ii=0)do:
     if {k} intersect lst = {}
     then
     ii:=1:lst:=lst union {k}:kk:=kk+1:T[kk]:=k:
     else
     fi:
    od:
     jj:=0:n0:=nops(lst):s:=sum('T[i]', 'i'=n..n0):
      for p from 1 to 100 while(jj=0) do:
        z:=sqrt(s+p):
         if z = floor(z) and {p} intersect lst={}
         then
         jj:=1:lst:=lst union {p}:kk:=kk+1:T[kk]:=p:
         else
         fi:
       od:
od:
print(T):

A259019 Lexicographically first permutation of the nonnegative integers such that Sum_{k=n..2n} a(k) is a prime number, with a(1)=0.

Original entry on oeis.org

0, 2, 1, 4, 3, 5, 6, 11, 7, 9, 8, 13, 10, 15, 12, 16, 14, 23, 17, 20, 18, 25, 19, 21, 22, 31, 24, 30, 26, 29, 27, 35, 28, 34, 32, 38, 33, 48, 36, 37, 39, 41, 40, 44, 42, 53, 43, 50, 45, 46, 47, 55, 49, 52, 51, 57, 54, 66, 56, 60, 58, 63, 59, 62, 61, 78, 64, 84

Offset: 1

Michel Lagneau, Jun 16 2015

nonn

Previous name: a(1)=0; for n>1, a(n) is the least number not yet used having the property that a(n) added with the next n terms is a prime number.
The corresponding primes are 2, 7, 13, 29, 41, 59, 79, 101, 127, 157, 191, 223, 263, 307, 347, 397, 443, 499, 557, 613, 673, 739, 809, 883, 953, 1033, 1103, 1187, 1277, 1367, 1459, 1553, 1657, 1777, ...
This is a permutation of the integers. - Michel Marcus, Jun 21 2015

			a(1)= 0 plus the next single term 2 is 2 (a prime);
a(2)= 2 plus the next two terms (1,4) is 7 (a prime);
a(3)= 1 plus the next three terms (4,3,5) is 13 (a prime);
a(4)= 4 plus the next four terms (3,5,6,11) is 29 (a prime);
a(5)= 3 plus the next five terms (5,6,11,7,9) is 41 (a prime).

Index entries for sequences that are permutations of the natural numbers

Cf. A000040, A247665, A259018.

nn:=100:T:=array(1..nn):T[1]:=0:T[2]:=2:kk:=2:lst:={0,2}:
for n from 2 to nn do:
  ii:=0:
    for k from 1 to 1000 while(ii=0)do:
     if {k} intersect lst = {}
     then
     ii:=1:lst:=lst union {k}:kk:=kk+1:T[kk]:=k:
     else
     fi:
    od:
     jj:=0:n0:=nops(lst):s:=sum('T[i]', 'i'=n..n0):
      for p from 1 to 100 while(jj=0) do:
        z:=s+p:
         if type(z,prime)=true and {p} intersect lst={}
         then
         jj:=1:lst:=lst union {p}:kk:=kk+1:T[kk]:=p:
         else
         fi:
       od:
od:
print(T):

Name edited by Jon E. Schoenfield, Sep 12 2017

cp's OEIS Frontend

A249058 a(n) = number of primes less than the square root of the (2^n)-th prime.

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A249692 a(1)=2; thereafter, a(n) is the smallest number not occurring earlier such that Kronecker(a(k), a(n)) = -1 for the next n indices k = n+1, n+2, ..., 2n.

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PARI

A254003 For n>=1, let B_n(k) be the sequence defined by b(1)=1, b(2)=p_1, ..., b(n+2)=p_(n+1), thereafter (for k>=n+3) the smallest number not occurring earlier having at least one common factor with b(k-(n+1)), but none with b(k-1)*b(k-2)*...*b(k-n); a(n) is m such that B_n(m)=6.

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A249778 a(1)=2; thereafter, a(n) is the smallest number not occurring earlier such that Kronecker(a(k), a(n)) = 1 for the next n indices k = n+1, n+2, ..., 2n.

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Programs

PARI

A259018 Lexicographically first permutation of the nonnegative integers such that Sum_{k=n..2n} a(k) is a square, starting with a(1)=0.

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Maple

A259019 Lexicographically first permutation of the nonnegative integers such that Sum_{k=n..2n} a(k) is a prime number, with a(1)=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A254003 For n>=1, let B_n(k) be the sequence defined by b(1)=1, b(2)=p_1, ..., b(n+2)=p_(n+1), thereafter (for k>=n+3) the smallest number not occurring earlier having at least one common factor with b(k-(n+1)), but none with b(k-1)b(k-2)...*b(k-n); a(n) is m such that B_n(m)=6.