%I A006093 M1006
%S A006093 1,2,4,6,10,12,16,18,22,28,30,36,40,42,46,52,58,60,66,70,72,78,82,88,
%T A006093 96,100,102,106,108,112,126,130,136,138,148,150,156,162,166,172,178,
%U A006093 180,190,192,196,198,210,222,226,228,232,238,240,250,256,262,268,270
%N A006093 Primes minus 1.
%C A006093 These are also the numbers that cannot be written as i*j + i + j (i,j 
               >= 1) - Rainer Rosenthal (r.rosenthal(AT)web.de), Jun 24 2001; Henry 
               Bottomley, Jul 06 2002
%C A006093 The values of k for which sum((-1)^j*binomial(k, j)*binomial(k-1-j, n-j)/
               (j+1), j=0..n) produces an integer for all n such that n < k. Setting 
               k=10 yields [0, 1, 4, 11, 19, 23, 19, 11, 4, 1, 0] for n = [ -1, 
               0, 1, 2, 3, 4, 5, 6, 7, 8, 9], so 10 is in the sequence. Setting 
               k=3 yields [0, 1, .5, .5] for n = [ -1, 0, 1, 2], so 3 is not in 
               the sequence. - Dug Eichelberger (dug(AT)mit.edu), May 14 2001
%C A006093 n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible. - Robert 
               G. Wilson v (rgwv(AT)rgwv.com), Jun 22 2002.
%C A006093 Records for Euler totient function phi.
%C A006093 Using Wilson's theorem, also n such that (n+1) divides (n!+1) - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Aug 20 2002
%C A006093 n such that phi(n^2)==phi(n^2+n) - Jon Perry (perry(AT)globalnet.co.uk), 
               Feb 19 2004
%C A006093 Numbers having only the trivial perfect partition consisting of a(n) 
               1's. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 23 2006
%C A006093 Numbers n such that the sequence {binomial coefficient C(k,n), k >= n 
               } contains exactly one prime. - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 02 2007
%C A006093 Record values of A143201: a(n)=A143201(A001747(n+1)) for n>1. [From Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 12 2008]
%C A006093 Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jul 10 2009: (Start)
%C A006093 The first N terms can be generated by the following sieving process:
%C A006093 start with {1, 2, 3, 4, ..., N-1, N};
%C A006093 for i:=1 until SQRT(N) do
%C A006093 (if (i is not striked out) then
%C A006093 (for j:=2*i+1 step i+1 until N do
%C A006093 (strike j from the list)));
%C A006093 remaining numbers = {a(n): a(n)<=N}. (End)
%C A006093 a(n) = partial sums of A075526(n-1) = Sum_(1...n) A075526(n-1) = Sum_(1...n) 
               [A008578(n+1) - A008578(n)] = Sum_(1...n) [A158611(n+1) - A158611(n)] 
               for n >= 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Aug 04 2009]
%C A006093 Or, largest divisor of nth prime minus smallest divisor of nth prime. 
               [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Sep 07 2009].
%C A006093 Also, phi(prime(n)); nth prime minus number of perfect partitions of 
               nth prime. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 
               10 2009]
%C A006093 A006093 U A072668 = A000027. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 
               Oct 22 2009]
%D A006093 Archimedeans Problems Drive, Eureka, 40 (1979), 28.
%D A006093 M. Gardner, The Colossal Book of Mathematics, pp. 31 W.W.Norton & Co. 
               NY 2001.
%D A006093 M. Gardner, Mathematical Circus, pp. 251-2, Alfred A.Knopf NY 1979.
%D A006093 Problem E 3065, American Mathematical Monthly, 1984, p. 649.
%D A006093 Problem E 3065, American Mathematical Monthly, No. 4, 1987, pp. 378.
%D A006093 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006093 T. D. Noe, <a href="b006093.txt">Table of n, a(n) for n=1..10000</a>
%H A006093 <a href="Sindx_Si.html#sieve">Index entries for sequences generated by 
               sieves</a> [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jul 10 2009]
%F A006093 a(n)=A000040(n)-A000012(n). {From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 
               Sep 07 2009].
%F A006093 a(n)=A000010(A000040(n)) [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 
               Oct 10 2009] [Corrected by R. J. Mathar, Dec 05 2009]
%t A006093 Table[Prime[n]-1,{n,1,30}] - Vladimir Orlovsky, Apr 27 2008
%Y A006093 a(n) = K(n, 1) and A034693(K(n, 1)) = 1 for all n. The subscript n refers 
               to this sequence and K(n, 1) is the index in A034693 - Labos E.
%Y A006093 Cf. A000040, A034693, A034694. Different from A075728.
%Y A006093 Complement of A072668 (composite numbers minus 1), A072670(a(n))=0.
%Y A006093 Essentially the same as A039915.
%Y A006093 Cf. A084920, A006093, A050997, A008864, A060800, A131991, A131992, A131993.
%Y A006093 Cf. A000010, A000012, A002033. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 
               Oct 10 2009]
%Y A006093 Sequence in context: A128984 A075728 A146886 this_sequence A127965 A117891 
               A072752
%Y A006093 Adjacent sequences: A006090 A006091 A006092 this_sequence A006094 A006095 
               A006096
%K A006093 nonn,easy,nice,new
%O A006093 1,2
%A A006093 N. J. A. Sloane (njas(AT)research.att.com).