A222532 -id:A222532 - cp's OEIS frontend

A163846 Starting from a(1)=5, a(n+1) is the smallest prime > a(n) such that 2*a(n)-a(n+1) is also prime.

5, 7, 11, 17, 23, 29, 41, 53, 59, 71, 83, 107, 113, 137, 167, 197, 227, 257, 263, 269, 281, 293, 317, 353, 359, 401, 419, 449, 467, 491, 503, 557, 593, 599, 641, 683, 719, 761, 821, 881, 941, 953, 977, 983, 1013, 1049, 1151, 1193, 1223, 1229, 1277, 1361, 1433

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

nonn

This is: select the smallest prime a(n+1) = a(n)+d such that at a(n)-d is another prime at the same distance to but at the opposite side of a(n).
From Zhi-Wei Sun, Feb 25 2013: (Start)
By induction, a(n)==2 (mod 3) for all n>2.
For a prime p>3 define g(p) as the least prime q>p such that 2p-q is also prime. Construct a simple (undirected) graph G as follows: The vertex set is the set of all primes greater than 3, and there is an edge connecting the vertices p and q>p if and only if g(p)=q.
Conjecture: The graph G constructed above consists of exactly two trees: one containing 7 and all odd primes congruent to 2 modulo 3, and another one containing all primes congruent to 1 modulo 3 except 7. (End)

			The first candidate for a(2) is the prime 5+2=7, which is selected since 5-2=3 is also prime.
The first candidate for a(3) is the prime 7+4=11, which is selected since 7-4=3 is also prime.
The first candidate for a(4) is the prime 11+2=13, which is not selected since 11-2=9 is composite.
The second candidate for a(4) is the prime 11+4=17, which is selected since 11-4=7 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A163847, A222532.

DeltaPrimePrevNext[n_]:=Module[{d, k1, k2}, k1=n-1; k2=n+1; While[ !PrimeQ[k1] || !PrimeQ[k2], k2++; k1-- ]; d=k2-n]; lst={}; p=5; Do[If[p-DeltaPrimePrevNext[p]>1, AppendTo[lst, p]; p=p+DeltaPrimePrevNext[p]], {n,6!}]; lst
k=3
n=1
Do[If[m==3, Print[n, " ", 5]]; If[m==k, n=n+1; Do[If[PrimeQ[2Prime[m]-Prime[j]]==True, k=j; Print[n, " ", Prime[j]]; Goto[aa]], {j, m+1, PrimePi[2Prime[m]]}]]; Label[aa]; Continue, {m, 3, 1000}] (* Zhi-Wei Sun, Feb 25 2013 *)
np[n_]:=Module[{nxt=NextPrime[n]},While[!PrimeQ[2n-nxt],nxt=NextPrime[nxt]]; nxt]; NestList[np, 5, 60] (* Harvey P. Dale, Feb 28 2013 *)

Edited by R. J. Mathar, Aug 29 2009

A163847 Starting from a(1)=13, a(n+1) is the smallest prime > a(n) such that 2*a(n) - a(n+1) is also prime.

Original entry on oeis.org

13, 19, 31, 43, 67, 73, 79, 97, 127, 151, 163, 199, 241, 271, 313, 349, 367, 397, 421, 433, 457, 541, 619, 631, 643, 673, 727, 811, 853, 877, 967, 997, 1087, 1123, 1129, 1171, 1213, 1297, 1303, 1327, 1423, 1447, 1471, 1483, 1543, 1597, 1627, 1657, 1693

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

nonn

This is: select the prime a(n+1) = a(n)+d such that at a(n)-d is another prime at the same distance to but at the opposite side of a(n).

It seems all these primes are in the class 1 (mod 6), that is, in A002476 as opposed to A007528.

			For a(2), the first candidate is the prime 17=13+4, which is not selected because 13-4=9 is not prime.
The next larger candidate is the prime 13+6=19, which is selected as a(2) because 13-6=7 is also prime.
For a(3) the first candidate is the prime 19+4=23, which is not selected because 19-4=15 is not prime.
The next candidate is the prime 19+10=29, which is not selected because the 19-10=9 is not prime.
The next larger candidate, the prime 19+12=31 is selected as a(3), because 19-12=7 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A163846, A222532.

A163847 := proc(n) option remember; if n = 1 then 13; else for a from procname(n-1)+2 by 2 do if isprime(a) and isprime( 2*procname(n-1)-a) then RETURN(a) ; fi; od: fi; end:
seq(A163847(n),n=1..80) ; # R. J. Mathar, Aug 29 2009

DeltaPrimePrevNext[n_]:=Module[{d, k1, k2}, k1=n-1; k2=n+1; While[ !PrimeQ[k1] || !PrimeQ[k2], k2++; k1-- ]; d=k2-n]; lst13={}; p=13; Do[If[p-DeltaPrimePrevNext[p]>1, AppendTo[lst13, p]; p=p+DeltaPrimePrevNext[p]],{n, 7!}]; lst13
(* Second program: *)
k=6
n=1
Do[If[m==6, Print[n, " ", 13]]; If[m==k, n=n+1; Do[If[PrimeQ[2Prime[m]-Prime[j]]==True, k=j; Print[n, " ", Prime[j]]; Goto[aa]], {j, m+1, PrimePi[2Prime[m]]}]]; Label[aa]; Continue, {m, 6, 1000}] (* Zhi-Wei Sun, Feb 25 2013 *)
sp[p_]:=Module[{p1=NextPrime[p]},While[!PrimeQ[2p-p1],p1=NextPrime[p1]];p1]; NestList[ sp,13,50] (* Harvey P. Dale, Aug 09 2023 *)

first(n) = { my(res = vector(n)); res[1] = 13; for(x=2, n, forprime(p=res[x-1]+1, , if(ispseudoprime(2*res[x-1] - p), res[x]=p; break()))); res; } \\ Iain Fox, Nov 18 2017

Definition and comment rephrased by R. J. Mathar, Aug 29 2009

A222566 a(1)=2; for n>0, a(n+1) is the least prime p>a(n) such that 2*(a(n)+1)-p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 89, 97, 107, 109, 113, 127, 149, 151, 167, 173, 181, 191, 193, 197, 199, 227, 229, 233, 239, 241, 251, 263, 271, 281, 283, 311, 313, 317, 353, 359, 367, 383, 389, 397, 443, 449, 457, 467, 479, 499, 509, 521, 523, 557, 569, 571, 587, 599, 601, 617, 619, 641, 643, 647, 653, 661, 677, 683, 691, 701, 727, 773, 787, 857, 859, 863, 907, 929, 941, 947, 967, 983, 991, 1013, 1019, 1021, 1031, 1033, 1049, 1051, 1091, 1093

Offset: 1

Zhi-Wei Sun, Feb 25 2013

nonn

For a prime p define g(p) as the least prime q>p such that 2*(p+1)-q is prime. Construct a simple (undirected) graph G as follows: The vertex set of G is the set of all primes, and for the vertices p and q>p there is an edge connecting p and q if and only if g(p)=q. Clearly G contains no cycle.

Conjecture: The graph G constructed above is connected and hence it is a tree!

			a(2)=3 since 2(2+1)=3+3, and a(3)=5 since 2(3+1)=5+3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A000040, A222532, A163846, A163847.

k=1
n=1
Do[If[m==1,Print[n," ",2]];If[m==k,n=n+1;Do[If[PrimeQ[2(Prime[m]+1)-Prime[j]]==True,k=j;Print[n," ",Prime[j]];Goto[aa]],{j,m+1,PrimePi[2Prime[m]]}]];
Label[aa];Continue,{m,1,1000}]

A222603 a(1)=1; for n>0, a(n+1) is the least practical number q>a(n) such that 2(a(n)+1)-q is practical.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 20, 24, 30, 32, 36, 42, 54, 56, 60, 66, 78, 80, 84, 90, 104, 120, 162, 176, 192, 210, 224, 234, 260, 270, 272, 276, 294, 320, 330, 342, 378, 380, 384, 390, 392, 396, 414, 416, 420, 450, 462, 464, 468, 476, 486, 510, 512, 522, 546, 594, 620, 630, 702, 704, 714, 726, 728, 744, 750, 798, 800, 810, 812, 816, 920, 924, 930, 966, 968, 972, 980, 990, 992, 1014, 1040, 1050, 1088, 1122, 1232, 1242, 1254, 1280, 1290, 1302, 1316, 1332, 1350, 1352, 1380, 1386, 1458, 1518, 1520

Offset: 1

Zhi-Wei Sun, Feb 26 2013

nonn

By a result of Melfi, each positive even number can be written as the sum of two practical numbers.
For a practical number p, define h(p) as the least practical number q>p such that 2(p+1)-q is practical. Construct a simple (undirected) graph H as follows: The vertex set of H is the set of all practical numbers, and for two vertices p and q>p there is an edge connecting p and q if and only if h(p)=q. Clearly H contains no cycle.
Conjecture: The graph H constructed above is connected and hence it is a tree.

			a(4)=6 since 2(a(3)+1)=10=6+4 with 4 and 6 both practical, and 6>a(3)=4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].

Cf. A005153, A222532, A163846, A163847, A222566.

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)
k=1
n=1
Do[If[m==1,Print[n," ",1]];If[m==k,n=n+1;Do[If[pr[2j]==True&&pr[2m+2-2j]==True,k=2j;Print[n," ",2j];Goto[aa]],{j,Ceiling[(m+1)/2],m}]];
Label[aa];Continue,{m,1,1000}]

A198472 a(n)=q(n) if 4 | q(n)-2, and a(n)=q(n)/2 if 4 | q(n), where q(n) is the least practical number q>n with 2(n+1)-q practical.

Original entry on oeis.org

2, 2, 2, 6, 6, 4, 4, 6, 6, 8, 6, 18, 8, 18, 8, 18, 18, 10, 10, 12, 12, 14, 12, 30, 14, 30, 14, 30, 30, 16, 16, 18, 18, 20, 18, 42, 20, 42, 20, 42, 42, 54, 24, 24, 28, 54, 24, 28, 30, 54, 28, 32, 54, 28, 28, 30, 30, 32, 30, 66, 32, 66, 32, 66, 66, 78, 36, 36, 40, 78, 36, 40, 42, 78, 40, 44, 78, 40, 40, 42, 42, 44, 42, 90, 44, 90, 44, 90, 90, 52, 48, 48, 50, 50, 48, 52, 50, 54, 50, 56

Offset: 1

Zhi-Wei Sun, Feb 27 2013

nonn

Conjecture: If b(1)>=4 is an integer and b(k+1)=a(b(k)) for k=1,2,3,..., then b(n)=4 for some n>0.

This conjecture has the same flavor as the Collatz conjecture.

			a(20)=12 since 2(20+1)=24+18 with 24 and 18 both practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].

Cf. A005153, A163846, A163847, A222603, A222532, A222566.

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)
Do[Do[If[pr[2k]==True&&pr[2n+2-2k]==True,Print[n," ",2k/(1+Mod[k-1,2])];Goto[aa]],{k,Ceiling[(n+1)/2],n}];
Label[aa];Continue,{n,1,100}]

A198472(n) = forstep(q=n+++bittest(n,0),9e9,2, is_A005153(q) && is_A005153(2*n-q) && return(if(q%4,q,q\2))) \\ M. F. Hasler, Feb 27 2013

A213187 a(n) = (p+1)/2 if 4 | p+1, and p otherwise, where p is the least prime > n with 2(n+1)-p prime.

Original entry on oeis.org

2, 2, 5, 5, 4, 4, 6, 6, 13, 6, 13, 13, 17, 17, 10, 17, 10, 10, 12, 12, 16, 12, 29, 16, 29, 16, 37, 29, 16, 16, 41, 37, 37, 41, 41, 37, 24, 41, 22, 41, 22, 22, 24, 24, 61, 24, 53, 61, 53, 30, 61, 53, 61, 34, 30, 61, 73, 30, 61, 61, 36, 34, 34, 36, 36, 34, 42, 36, 73, 36, 73, 73, 89, 40, 40, 42, 42, 40, 89, 42, 97, 42, 89, 97, 89, 101, 97, 89, 97, 52, 101, 97, 109, 101, 52, 97, 54, 101, 52, 101

Offset: 1

Zhi-Wei Sun, Feb 28 2013

nonn

Conjecture: If b(1)>2 is an integer, and b(k+1)=a(b(k)) for k=1,2,3,..., then b(n)=4 for some n>0.

For example, if we start from b(1)=45 then we get the sequence 45, 61, 36, 37, 24, 16, 17, 10, 6, 4, 5, 4, ...

			a(8)=6 since 2(8+1)=11+5 with (11+1)/2=6;
a(9)=13 since 2(9+1)=13+7.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A002372, A222566, A222532.

Do[Do[If[PrimeQ[2n+2-Prime[k]]==True,Print[n," ",If[Mod[Prime[k],4]==3,(Prime[k]+1)/2,Prime[k]]];Goto[aa]],{k,PrimePi[n]+1,PrimePi[2n]}];
Label[aa];Continue,{n,1,100}]
nxt[{n_,a_}]:=Module[{p=NextPrime[n]},While[!PrimeQ[2(n+1)-p],p = NextPrime[ p]];{n+1,If[Divisible[p+1,4],(p+1)/2,p]}]; Rest[ Transpose[ NestList[ nxt,{1,2},110]][[2]]] (* Harvey P. Dale, May 30 2016 *)

a(n)=my(q=nextprime(n+1)); while(!isprime(2*n+2-q),q=nextprime(q+1)); if(q%4<3,q,(q+1)/2) \\ Charles R Greathouse IV, Feb 28 2013

cp's OEIS Frontend

A163846 Starting from a(1)=5, a(n+1) is the smallest prime > a(n) such that 2*a(n)-a(n+1) is also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A163847 Starting from a(1)=13, a(n+1) is the smallest prime > a(n) such that 2*a(n) - a(n+1) is also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A222566 a(1)=2; for n>0, a(n+1) is the least prime p>a(n) such that 2*(a(n)+1)-p is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A222603 a(1)=1; for n>0, a(n+1) is the least practical number q>a(n) such that 2(a(n)+1)-q is practical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A198472 a(n)=q(n) if 4 | q(n)-2, and a(n)=q(n)/2 if 4 | q(n), where q(n) is the least practical number q>n with 2(n+1)-q practical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A213187 a(n) = (p+1)/2 if 4 | p+1, and p otherwise, where p is the least prime > n with 2(n+1)-p prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI