A225889 -id:A225889 - cp's OEIS frontend

A226115 Least positive integer not of the form p_m - p_{m-1} + ... +(-1)^(m-k)*p_k with 0 < k < m <= n, where p_j denotes the j-th prime.

1, 2, 3, 6, 7, 10, 11, 14, 18, 18, 20, 20, 24, 24, 28, 28, 34, 34, 40, 40, 42, 42, 46, 46, 46, 54, 56, 56, 58, 58, 60, 64, 78, 78, 80, 80, 94, 94, 98, 98, 104, 104, 106, 106, 106, 106, 118, 118, 118, 118, 122, 122, 140, 140, 146, 146, 152, 152, 158, 158

Offset: 1

Zhi-Wei Sun, May 27 2013

nonn

Conjecture: sqrt(2*a(n)) > sqrt(p_n)-0.7 for all n > 0, and a(n) is even for any n > 7.

Note that f(n) = sqrt(2*a(n))-sqrt(p_n)+0.7 is approximately equal to 0.000864 at n = 651. It seems that f(n) > 0.1 for any other value of n.

			a(4) = 6,  since 2,3,5,7 are the initial four primes, and 1=3-2, 2=5-3, 3=7-5+3-2, 4=5-3+2, 5=7-5+3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), 2794-2812.

Cf. A000040, A225889, A222579, A222580.

s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
R[j_]:=R[j]=Union[Table[s[j]-(-1)^(j-i)*s[i],{i,0,j-2}]]
t=1
Do[Do[Do[If[MemberQ[R[j],m]==True,Goto[aa]],{j,PrimePi[m]+1,n}];Print[n," ",m];t=m;Goto[bb];
Label[aa];Continue,{m,t,Prime[n]-1}];Print[n," ",counterexample];Label[bb],{n,1,100}]

A226318 Positive integers n with p_{n+1}-p_n = 2 and p_{n+3}-p_{n+2} = 2, where p_k denotes the k-th prime.

Original entry on oeis.org

3, 5, 26, 33, 41, 43, 81, 140, 142, 171, 176, 234, 286, 294, 313, 318, 428, 458, 473, 475, 484, 577, 579, 584, 671, 743, 772, 862, 870, 872, 891, 934, 957, 1030, 1115, 1165, 1167, 1169, 1230, 1339, 1351, 1404, 1462, 1548, 1621, 1651, 1707, 1823, 1833, 1867, 1923, 2021, 2052, 2066, 2068, 2121, 2151, 2199, 2309, 2362

Offset: 1

Zhi-Wei Sun, Jun 03 2013

nonn

An old conjecture of de Polignac asserts that for any positive even integer d there are infinitely many n>0 with p_{n+1}-p_n = d.
The author has formulated the following further extension.
Conjecture: For any positive even integers d_1,...,d_k, there are infinitely many positive integers n such that p_{n+2j-1}-p_{n+2j-2} = d_j for all j=1,...,k.
For example,
p_{35209566+2j-1}-p_{35209566+2j-2} = 2 for all j = 1,...,7,
p_{19726689+2j-1}-p_{19726689+2j-2} = 6 for all j = 1,...,8,
and p_{297746+2j-1}-p_{297746+2j-2} = 2j for j = 1,2,3,4,5.

			a(1) = 3 and a(2) = 5 since {p_3,p_4}={5,7}, {p_5,p_6}={11,13} and {p_7,p_8}={17,19} are twin prime pairs.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), 2794-2812.
Zhi-Wei Sun, On primes in arithmetic progressions, preprint, arXiv:1304.5988.
Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, Pages 1121-1174 from Volume 179 (2014), Issue 3.

Cf. A000040, A001359, A006512, A225889, A226115.

[n: n in [1..2500] | (NthPrime(n+1)-NthPrime(n)) eq 2 and (NthPrime(n+3)-NthPrime(n+2)) eq 2]; // Vincenzo Librandi, Jun 28 2015

n=0
Do[If[Prime[k+1]-Prime[k]==2&&Prime[k+3]-Prime[k+2]==2,n=n+1;
Print[n," ",k]],{k,1,100}]
PrimePi[#]&/@Transpose[Select[Partition[Prime[Range[2500]],4,1],#[[4]]- #[[3]] == #[[2]]-#[[1]]==2&]][[1]] (* Harvey P. Dale, Nov 20 2013 *)

cp's OEIS Frontend

A226115 Least positive integer not of the form p_m - p_{m-1} + ... +(-1)^(m-k)*p_k with 0 < k < m <= n, where p_j denotes the j-th prime.

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