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
Keywords
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.
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
Keywords
Comments
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.
Examples
a(4)=6 since 2(a(3)+1)=10=6+4 with 4 and 6 both practical, and 6>a(3)=4.
Links
- 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].
Programs
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Mathematica
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.
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
Keywords
Comments
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.
Examples
a(20)=12 since 2(20+1)=24+18 with 24 and 18 both practical.
Links
- 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].
Programs
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Mathematica
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}] -
PARI
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.
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
Keywords
Comments
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, ...
Examples
a(8)=6 since 2(8+1)=11+5 with (11+1)/2=6; a(9)=13 since 2(9+1)=13+7.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
Programs
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Mathematica
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 *) -
PARI
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
Comments
Examples
Links
Crossrefs
Programs
Mathematica
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}]