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%I #18 Jan 21 2019 20:02:20
%S 2,6,13,26,48,75,103,135,199,270,338,443,508,581,706,878,1001,1124,
%T 1305,1413,1565,1764,1978,2299,2571,2724,2886,3052,3213,3710,4259,
%U 4581,4859,5259,5668,5954,6409,6797,7184,7696,8029,8515,9062,9325,9608,10246,11444
%N Index k of the semiprime A001358(k) = prime(n) * prime(n+1).
%C The positions of products of 2 successive primes in A001358. - _Juri-Stepan Gerasimov_, Apr 14 2010
%D Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909.
%D Derrick H. Lehmer, Guide to Tables in the Theory of Numbers Washington, D.C. 1941.
%H Donovan Johnson, <a href="/A172348/b172348.txt">Table of n, a(n) for n = 1..1000</a>
%H E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, <a href="http://name.umdl.umich.edu/ABV2766.0001.001">vol. 1</a> and <a href="http://name.umdl.umich.edu/ABV2766.0002.001">vol. 2</a>, Leipzig, Berlin, B. G. Teubner, 1909.
%F a(n) = {k: A001358(k) = A006094(n)}.
%e n=1: 6 = 2 * 3 = prime(1) * prime(2) = semiprime(2). Therefore a(1) = 2.
%e n=2: 15 = 3 * 5 = prime(2) * prime(3) = semiprime(6). Therefore a(2) = 6.
%e n=3: 35 = 5 * 7 = prime(3) * prime(4) = semiprime(13). Therefore a(3) = 13.
%p A001358 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a)= 2 then return a; end if; end do ; end if; end proc:
%p A006094 := proc(n) ithprime(n)*ithprime(n+1) ; end proc:
%p A172348 := proc(n) pp := A006094(n) ; for k from 1 do if A001358(k) = pp then return k; end if; end do ; end proc:
%p seq(A172348(n),n=1..70) ; # _R. J. Mathar_, Feb 09 2010
%t semiPrimePi[n_] := Sum[ PrimePi[n/Prime@ i] - i + 1, {i, PrimePi@ Sqrt@ n}]; semiPrimePi@# & /@ Table[ Prime[n] Prime[n + 1], {n, 47}] (* _Robert G. Wilson v_, Feb 02 2013 *)
%t nn=50000;Flatten[Module[{sp=Select[Range[nn+PrimePi[nn]],PrimeOmega[#] == 2&]},Table[ Position[sp,Prime[n]Prime[n+1]],{n,PrimePi[nn]}]]] (* _Harvey P. Dale_, Sep 07 2013 *)
%Y Cf. A001358, A006094, A006881.
%K nonn
%O 1,1
%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Feb 01 2010
%E Entries checked by _R. J. Mathar_, Feb 09 2010
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