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A258836
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Least practical number q with q-1 and q+1 twin prime such that n = q'/q for some practical number q' with q'-1 and q'+1 twin prime.
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7
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4, 6, 4, 18, 6, 12, 6, 30, 12, 6, 18, 6, 150, 30, 4, 12, 60, 4, 12, 12, 42, 30, 240, 18, 6, 12, 4, 270, 12, 6, 42, 6, 6, 30, 12, 12, 180, 6, 60, 6, 30, 150, 30, 30, 4, 18, 2550, 4, 18, 12, 42, 6, 150, 30, 12, 60, 4, 6, 60, 4, 462, 180, 1230, 18, 30, 108, 60, 180, 12, 6, 30, 6, 570, 420, 462, 180, 6, 4, 198, 42, 522, 600, 1050, 42, 12, 12, 4, 60, 432, 18, 12, 60, 30, 60, 6, 12, 150, 60, 30, 6
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OFFSET
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1,1
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COMMENTS
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Conjecture: a(n) exists for any n > 0. Moreover, any positive rational number r can be written as q'/q, where q and q' are terms of A258838 (i.e., q is practical with q-1 and q+1 twin prime, and q' is practical with q'-1 and q'+1 twin prime).
This implies that there are infinitely many "sandwiches of the second kind" (i.e., triples {q-1,q,q+1} with q practical and q-1 and q+1 twin prime).
I have verified the conjecture for all those rational numbers r = n/m with m,n = 1,...,1000. -Zhi-Wei Sun, Jun 15 2015
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LINKS
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EXAMPLE
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a(1) = 4 since 1 = 4/4 with 4 practical and 4-1 and 4+1 twin prime.
a(2) = 6 since 2 = 12/6, 6 is practical with 6-1 and 6+1 twin prime, and 12 is practical with 12-1 and 12+1 twin prime.
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MATHEMATICA
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f[n_]:=FactorInteger[n]
Pow[n_, i_]:=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
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_]:=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
SW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&pr[n]
Do[k=0; Label[bb]; k=k+1; If[PrimeQ[Prime[k]+2]&&pr[Prime[k]+1]&&SW[n*(Prime[k]+1)], Goto[aa], Goto[bb]];
Label[aa]; Print[n, " ", Prime[k]+1]; Continue, {n, 1, 100}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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