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 A210452 Number of integers k
 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 3, 1, 3, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 4, 2, 5, 5, 4, 5, 5, 2, 4, 5, 5, 1, 5, 2, 6, 6, 5, 2, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 5, 2, 6, 3, 5, 7, 7, 7, 7, 3, 7, 7, 7, 3, 7, 4, 6, 8, 8, 8, 8, 3, 8, 8, 6, 3, 8, 8, 6, 8, 8, 3, 8, 8, 8, 7, 6, 8, 8, 3, 8, 8, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Conjecture: a(n)>0 for all n>4. This implies the twin prime conjecture since k*p is not practical for any prime p>sigma(k)+1. Zhi-Wei Sun also made the following conjectures: (1) For each integer n>197, there is a practical number k26863, the interval [1,n] contains five consecutive integers m-2, m-1, m, m+1, m+2 with m-1 and m+1 both prime, and m-2, m, m+2, m*n all 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]. Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017. Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013. EXAMPLE a(11)=1 since 5 and 7 are twin primes, and 6 and 6*11 are 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) a[n_]:=a[n]=Sum[If[PrimeQ[k-1]==True&&PrimeQ[k+1]==True&&pr[k]==True&&pr[k*n]==True, 1, 0], {k, 1, n-1}] Do[Print[n, " ", a[n]], {n, 1, 100}] CROSSREFS Cf. A005153, A210444, A210445, A071558, A208243, A208244, A208246, A208249, A219185, A209253, A209254, A219312, A219315, A219320. Sequence in context: A055020 A052435 A094701 * A240301 A289641 A209312 Adjacent sequences:  A210449 A210450 A210451 * A210453 A210454 A210455 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 20 2013 STATUS approved

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Last modified August 23 22:45 EDT 2019. Contains 326254 sequences. (Running on oeis4.)