A198472 - OEIS

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

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OFFSET

1,1

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.

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].

EXAMPLE

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

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}]

PROG

(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

CROSSREFS

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

Sequence in context: A078584 A242010 A283673 * A171698 A339385 A080969

Adjacent sequences: A198469 A198470 A198471 * A198473 A198474 A198475

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 27 2013

STATUS

approved