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A258836 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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Checking the conjecture for r = n/m with 1 <= n <= m <= 1000

Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

EXAMPLE

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.

MATHEMATICA

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}]

CROSSREFS

Cf. A000040, A001359, A006512, A005153, A210479, A258803, A258811, A258838.

Sequence in context: A019191 A321355 A185145 * A292387 A208540 A019195

Adjacent sequences:  A258833 A258834 A258835 * A258837 A258838 A258839

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jun 11 2015

STATUS

approved

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Last modified July 23 03:00 EDT 2019. Contains 325230 sequences. (Running on oeis4.)