A222579 -id:A222579 - cp's OEIS frontend

A225889 Least prime p_m such that n = p_m-p_{m-1}+...+(-1)^(m-k)*p_k for some 0

3, 5, 7, 5, 7, 11, 13, 11, 11, 17, 19, 17, 17, 23, 17, 23, 23, 31, 23, 41, 23, 41, 31, 47, 29, 47, 37, 59, 41, 59, 37, 59, 43, 67, 37, 67, 43, 67, 43, 73, 61, 83, 53, 83, 47, 101, 61, 97, 53, 97, 59, 97, 59, 103, 61, 109, 67, 127, 67, 131

Offset: 1

Zhi-Wei Sun, May 19 2013

nonn

By a conjecture of the author, a(n) <= 2*n+2.2*sqrt(n), and moreover a(n) <= n+4.6*sqrt(n) if n is odd. Clearly a(n)>n. We guess that a(2n)/(2n) --> 2 as n tends to the infinity.
Note that this sequence is different from A222579 which involves a stronger conjecture of the author.
Zhi-Wei Sun also conjectured that any positive even integer m can be written in the form p_n-p_{n-1}+...+(-1)^{n-k}*p_k with k < n and 2m-3.6*sqrt(m+1) < p_n < 2m+2.2*sqrt(m).

			a(7) = 13 since 7 = 13-11+7-5+3.
a(20) = 41 since 20 = 41-37+31-29+23-19+17-13+11-7+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), no.8, 2794-2812.

Cf. A000040, A222579.

s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
Do[Do[If[s[j]-(-1)^(j-i)*s[i]==m,Print[m," ",Prime[j]];Goto[aa]],{j,PrimePi[m]+1,PrimePi[2m+2.2Sqrt[m]]},{i,0,j-2}];
Print[m," ",counterexample];Label[aa];Continue,{m,1,100}]

A222580 Number of ways to write n=p_m-p_{m-1}+...+(-1)^{m-k}p_k with k

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 2, 3, 2, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 3, 3, 1, 1, 2, 4, 2, 1, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 6, 1, 2, 3, 4, 2, 3, 3, 2, 3, 4, 2, 4, 2, 1, 1, 4, 3, 4, 2, 4, 1, 3, 3, 2, 4, 4, 2, 3, 2, 3, 3, 3, 3, 2, 5, 1, 3, 4, 7, 4, 2, 3, 2, 1, 5, 2, 4, 2, 7, 3, 3, 3, 4, 5, 6

Offset: 1

Zhi-Wei Sun, Feb 25 2013

nonn

Conjecture: All the terms are positive.

See also the comments related to A222579.

			a(9)=2 since 9=11-7+5=19-17+13-11+7-5+3 with 12, 4, 20, 2 all practical.
a(806)=1 since 806=p_{358}-p_{357}+...+p_{150}-p_{149} with p_{358}=2411<=3*806=2418, and 2412 and p_{149}-1=858 are both practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On functions taking only prime values, arXiv:1202.6589.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A000040, A005153, A210479, A222579.

f[n_]:=f[n]=FactorInteger[n]
Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n],s+1],1]<=DivisorSigma[1,Product[Pow[n,i],{i,1,s}]]+1,0,1],{s,1,Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
pp[k_]:=pp[k]=pr[Prime[k]+1]==True
pq[k_]:=pq[k]=pr[Prime[k]-1]==True
s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
a[n_]:=a[n]=Sum[If[pp[j]==True&&pq[i+1]==True&&s[j]-(-1)^(j-i)*s[i]==n,1,0],{j,PrimePi[n]+1,PrimePi[3n]},{i,0,j-2}]
Table[a[n],{n,1,100}]

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.

Original entry on oeis.org

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}]

cp's OEIS Frontend

A225889 Least prime p_m such that n = p_m-p_{m-1}+...+(-1)^(m-k)*p_k for some 0

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A222580 Number of ways to write n=p_m-p_{m-1}+...+(-1)^{m-k}p_k with k

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica