A163846 -id:A163846 - cp's OEIS frontend

A222532 a(1)=2; for n >= 1, a(n+1) is the least prime p_m such that a(n)=p_m-p_{m-1}+...+(-1)^{m-k}p_k for some 0

2, 5, 7, 13, 17, 23, 31, 37, 43, 53, 59, 67, 73, 83, 89, 101, 109, 113, 131, 149, 157, 163, 173, 179, 197, 223, 257, 263, 269, 277, 283, 311, 347, 389, 401, 421, 431, 487, 503, 523, 557, 569, 577, 601, 613, 641, 661, 709, 733, 739, 773, 823, 827, 857, 883, 929, 947, 953, 977, 983, 997, 1009, 1019, 1031, 1039, 1051, 1097, 1117, 1129, 1151, 1181, 1223, 1229, 1237, 1249, 1279, 1327, 1361, 1373, 1423, 1459, 1481, 1499, 1543, 1559, 1571, 1601, 1621, 1627, 1669, 1693, 1699, 1721, 1733, 1759, 1783, 1823, 1873, 1973, 2011

Offset: 1

Zhi-Wei Sun, Feb 24 2013

nonn

Conjecture: For any given prime p, if we define b(1)=p and let b(n+1) be the least prime p_m such that b(n)=p_m-p_{m-1}+...+(-1)^{m-k}p_k for some 0p adjacent if and only if q is the least prime p_m such that p=p_m-p_{m-1}+...+(-1)^{m-k}p_k for some 0

Clearly the graph T contains no cycle. The vertices on the unique path connecting 2 and 71 are listed (in order) below: 2, 5, 7, 13, 17, 23, 31, 37, 43, 53, 59, 67, 73, 83, 89, 101, 109, 113, 131, 149, 139, 107, 97, 79, 71.

			a(2)=5 and a(3)=7 since 2=5-3 and 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), no.8, 2794-2812.

Cf. A000040, A008347, A222566, A163846, A163847.

k=1
n=1
s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
Do[If[m==1,Print[n," ",2]];If[m==k,n=n+1;Do[If[s[j]-(-1)^(j-i)*s[i]==Prime[m],k=j;Print[n," ",Prime[j]];Goto[aa]],{j,m+1,PrimePi[3Prime[m]]},{i,0,j-2}]];
Label[aa];Continue,{m,1,1000}]

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

Showing 1-5 of 5 results.

cp's OEIS Frontend

A222532 a(1)=2; for n >= 1, a(n+1) is the least prime p_m such that a(n)=p_m-p_{m-1}+...+(-1)^{m-k}p_k for some 0

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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