A203069 -id:A203069 - cp's OEIS frontend

A346609 The successive odd nonprimes produced during the construction of A203069.

1, 9, 15, 9, 15, 25, 15, 9, 15, 25, 21, 9, 15, 21, 25, 33, 35, 39, 45, 49, 45, 33, 35, 45, 49, 45, 49, 57, 55, 57, 63, 65, 63, 65, 69, 75, 77, 75, 77, 75, 77, 85, 87, 85, 87, 91, 93, 95, 99, 105, 111, 115, 111, 105, 111, 115, 105, 99, 105, 115, 121, 123, 125, 129, 133, 135, 133, 129

Offset: 1

Harvey P. Dale and N. J. A. Sloane, Aug 16 2021

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A203069, A346610, A346611.

A346610 First differences of A203069.

Original entry on oeis.org

7, -1, -5, 11, -1, -9, 3, 3, 7, -11, -1, 7, -1, 5, 3, -1, 5, 1, 3, -7, -5, 7, 3, 1, -5, 9, -1, -1, 3, 3, -1, -1, 3, 1, 5, -3, 1, 1, -3, 5, 3, -1, -1, 3, 1, 1, 1, 3, 3, 3, 1, -5, -1, 7, -3, -7, 1, 5, 5, 1, 1, 1, 3, 1, 1, -3, -1, 5, 3, -1, -1, 3, 1, 5, -3, 1, 1, -3, 5, 1, 3, 1, 1

Offset: 1

Harvey P. Dale and N. J. A. Sloane, Aug 16 2021

sign

In the first 100000 terms, the only differences between terms of A203069 are -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11.

N. J. A. Sloane, Table of n, a(n) for n = 1..9999

Cf. A203069, A346609, A346611.

A346611 a(n) = (A203069(n)-n)/2.

Original entry on oeis.org

0, 3, 2, -1, 4, 3, -2, -1, 0, 3, -3, -4, -1, -2, 0, 1, 0, 2, 2, 3, -1, -4, -1, 0, 0, -3, 1, 0, -1, 0, 1, 0, -1, 0, 0, 2, 0, 0, 0, -2, 0, 1, 0, -1, 0, 0, 0, 0, 1, 2, 3, 3, 0, -1, 2, 0, -4, -4, -2, 0, 0, 0, 0, 1, 1, 1, -1, -2, 0, 1, 0, -1, 0, 0, 2, 0, 0, 0, -2, 0, 0, 1, 1, 1, -1, -2

Offset: 1

Harvey P. Dale and N. J. A. Sloane, Aug 16 2021

sign

The terms are surprisingly small.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A203069, A346609, A346610.

A249918 Inverse permutation to A203069.

Original entry on oeis.org

1, 4, 7, 12, 11, 8, 3, 2, 9, 14, 13, 6, 5, 22, 15, 10, 17, 16, 21, 26, 23, 18, 19, 24, 25, 20, 29, 28, 27, 30, 33, 32, 31, 34, 35, 40, 37, 38, 39, 36, 41, 44, 43, 42, 45, 46, 47, 48, 57, 58, 49, 54, 53, 50, 59, 56, 51, 52, 55, 60, 61, 62, 63, 68, 67, 64, 65

Offset: 1

Reinhard Zumkeller, Jan 14 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A203069, A249920.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a249918 = (+ 1) . fromJust . (`elemIndex` a203069_list)

A264030 Primes p such that A203069(p) is also prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 37, 41, 43, 47, 53, 61, 71, 73, 89, 97, 103, 107, 109, 127, 131, 137, 139, 151, 163, 167, 173, 191, 193, 197, 211, 223, 227, 233, 239, 241, 257, 263, 269, 277, 283, 293, 311, 313, 317, 331, 347, 349, 359, 367, 379, 397, 419, 433, 443, 449

Offset: 1

Bill McEachen, Nov 01 2015

nonn

This provides an ordered subset of the prime numbers.

			A203069(3)=7, both are prime numbers, so the 1st entry=3.
A203069(53)=53, both are prime numbers, so the 12th entry=53.

Cf. A203069.

More terms from Michel Marcus, Nov 02 2015

A055266 a(n) + a(n+1) is never prime; lexicographically earliest such sequence of distinct positive integers.

Original entry on oeis.org

1, 3, 5, 4, 2, 6, 8, 7, 9, 11, 10, 12, 13, 14, 16, 17, 15, 18, 20, 19, 21, 23, 22, 24, 25, 26, 28, 27, 29, 31, 32, 30, 33, 35, 34, 36, 38, 37, 39, 41, 40, 42, 43, 44, 46, 45, 47, 48, 50, 49, 51, 53, 52, 54, 56, 55, 57, 58, 59, 60, 61, 62, 63, 65, 64, 66, 67, 68, 70, 71, 69, 72

Offset: 1

Henry Bottomley, May 09 2000

See A253074 for an essentially identical sequence (with a proof that the sequence is a permutation).

Sequence A253074 is defined in the same way, but starting with 0. This happens to produce the same sequence from the next term on. This is the case (M,N) = (2,0) in the family of sequences where M consecutive terms yield N primes in their pairwise sums, see the wiki page for other examples. - M. F. Hasler, Nov 26 2019

			a(3) = 5 because 1 and 3 have already been used and both 3 + 2 = 5 and 3 + 4 = 7 are prime while 3 + 5 = 8 is not prime.

Robert Israel, Table of n, a(n) for n = 1..10000
M. F. Hasler, Prime sums from neighboring terms, OEIS wiki, Nov. 23, 2019
Index entries for sequences that are permutations of the natural numbers

Cf. A055265, A010051, A249920 (inverse), A203069, A253074.

import Data.List (delete)
a055266 n = a055266_list !! (n-1)
a055266_list = 1 : f 1 [2..] where
   f u vs = g vs where
     g (w:ws) | a010051' (u + w) == 0 = w : f w (delete w vs)
              | otherwise = g ws
-- Reinhard Zumkeller, Jan 14 2015

N:= 1000; # to get a[n] for n up to N
A:= {1};
a[1]:= 1;
for n from 2 to N do
  mA:= max(A);
  R:= {$1..mA} minus A;
  for x in R do
     if not isprime(a[n-1]+x) then
       a[n]:= x;
       break
     fi
   od:
   if not assigned(a[n]) then
     for x from mA+1 do
       if not isprime(a[n-1]+x) then
         a[n]:= x;
         break
       fi
     od
   fi;
   A:= A union {x};
od:
seq(a[n],n=1..N); # Robert Israel, Jun 03 2014

f[ s_ ]:=Block[ {k=1,a=s[ [ -1 ] ]},While[ Or[ MemberQ[ s,k ],PrimeQ[ a+k ] ],k++ ];Append[ s,k ] ];Nest[ f,{1},121 ] (* Zak Seidov, Oct 21 2009 *)
a={1};z=Range[2,2002];z=Complement[z,a];While[Length[z]>1,If[!PrimeQ[z[[1]]+Last[a]],AppendTo[a,z[[1]]],If[!PrimeQ[z[[2]]+Last[a]],AppendTo[a,z[[2]]],AppendTo[a,z[[3]]]]];z=Complement[z,a]];Print[a] (* significantly faster *) (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)

v=[1]; n=1; while(n<100, if(!isprime(n+v[#v])&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0); n++); v \\ Derek Orr, Jun 08 2015

A055266_upto(n=99, u=1, U, a)={vector(n, n, n=u; while(bittest(U, n-u)|| isprime(a+n), n++); if(n>u, U+=1<<(n-u), U>>=-u+u+=valuation(U+2, 2)); a=n) + if(default(debug), print([u]))} \\ Optional args allow to tweak computation. If debug > 0, print least unused number at the end. - M. F. Hasler, Nov 25 2019

a(n) = A253074(n+1) (as long as A253074(1) = 0). - M. F. Hasler, Nov 26 2019

Corrected by Zak Seidov, Oct 21 2009

Name edited by M. F. Hasler, Nov 26 2019

A203077 Alternating-parity rearrangement of natural numbers: a(n) is the smallest number such that a(n-1)^2 + a(n)^2 is odd and composite.

Original entry on oeis.org

1, 8, 9, 2, 11, 10, 5, 12, 3, 4, 7, 6, 13, 14, 17, 16, 15, 18, 19, 22, 21, 20, 25, 30, 27, 24, 23, 26, 29, 28, 31, 32, 35, 38, 33, 34, 37, 36, 39, 42, 41, 40, 45, 44, 43, 46, 47, 50, 49, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 63, 62, 61, 66, 67, 64, 65

Offset: 1

Zak Seidov, Dec 29 2011

nonn

The maximum between a(n) and the n-th integer appears to be +-6. In the first 10k terms, the distribution of differences, from -6 to 6 is: 27, 140, 1350, 7002, 1282, 168, 31. Therefore I conjecture that Lim_{n->infinity} a(n) = n.

			1^2 + 8^2 = 65 composite, 8^2 + 9^2 = 145 composite, 9^2 + 2^2 = 85 composite.

Cf. A203069.

f[s_List] := Block[{k = If[ OddQ[ s[[-1]]], 2, 3], m = s[[-1]]}, While[a = k^2 + m^2; MemberQ[s, k] || PrimeQ[a] || EvenQ[a], k += 2]; Append[s, k]]; Nest[f, {1}, 70] (* Robert G. Wilson v, Jan 02 2012 *)

cp's OEIS Frontend

A346609 The successive odd nonprimes produced during the construction of A203069.

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A346610 First differences of A203069.

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A346611 a(n) = (A203069(n)-n)/2.

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A249918 Inverse permutation to A203069.

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Haskell

A264030 Primes p such that A203069(p) is also prime.

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Extensions

A055266 a(n) + a(n+1) is never prime; lexicographically earliest such sequence of distinct positive integers.

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Haskell

Maple

Mathematica

PARI

PARI

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Extensions

A203077 Alternating-parity rearrangement of natural numbers: a(n) is the smallest number such that a(n-1)^2 + a(n)^2 is odd and composite.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica