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%I #100 May 27 2024 16:12:16
%S 7,13,19,23,25,31,33,37,43,47,49,53,55,61,63,67,73,75,79,83,85,89,91,
%T 93,97,103,109,113,115,117,119,121,123,127,131,133,139,141,143,145,
%U 151,153,157,159,163,167,169,173,175,181,183,185,187,193
%N Numbers that are not the difference of two primes.
%C Conjecturally, odd numbers k such that k+2 is composite.
%C Is this the same as A068780(2n-1) - 1? - _J. Stauduhar_, Aug 23 2012
%C A092953(a(n)) = 0. - _Reinhard Zumkeller_, Nov 10 2012
%C It seems that the sequence contains the squares of all primes except for 2 and 3. - _Ivan N. Ianakiev_, Aug 29 2013 [It does: For every prime p > 3, note that p^2 == 1 (mod 3), so p^2 cannot be q - r where q and r are primes. (If it were, then since p^2 is odd, q and r could not both be odd primes; r would have to be the even prime, 2, which would mean that p^2 = q - 2, so q = p^2 + 2 == 0 (mod 3), i.e., 3 would divide q, so q would not be prime -- a contradiction.) - _Jon E. Schoenfield_, May 03 2024]
%C Integers d such that A123556(d) = 1, that is, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has only one element. For each such d, the unique element of all the first largest APs with 1 element is A342309(d) = 2. - _Bernard Schott_, Jan 08 2023
%C If it exists, the least even term is > 10^12 (see 1st comment in A020483). - _Bernard Schott_, Jan 09 2023
%D F. Smarandache, Properties of Numbers, 1972. (See Smarandache odd sieve.)
%H T. D. Noe, <a href="/A007921/b007921.txt">Table of n, a(n) for n = 1..10000</a>
%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/3940-a1880-np-en-pa">A1880. NP en PA</a> (prime numbers in arithmetic progression) (in French).
%H C. Dumitrescu and V. Seleacu, editors, <a href="https://web.archive.org/web/20060220001800/http://www.gallup.unm.edu/~smarandache/SNAQINT.txt">Some Notions and Questions in Number Theory, Vol. I</a>.
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>, 4th ed., 1993, Problem 94.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression">Primes in arithmetic progression</a>.
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>.
%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>.
%p filter := d -> irem(d, 2) <> 0 and not isprime(2+d) : select(filter, [`$`(1 .. 200)]); # _Bernard Schott_, Jan 08 2023
%t Lim=200;nn=10;seq:=Complement[Range[Lim],Union[Flatten[Differences/@Subsets[Prime[Range[nn]],{2}]]]];Until[AllTrue[seq,OddQ],nn++];seq (* _James C. McMahon_, May 04 2024 *)
%o (PARI) is(n)=n%2 && !isprime(n+2) \\ On Polignac's conjecture; _Charles R Greathouse IV_, Jun 28 2013
%o (Haskell)
%o a007921 n = a007921_list !! (n-1)
%o a007921_list = filter ((== 0) . a010051' . (+ 2)) [1, 3 ..]
%o -- _Reinhard Zumkeller_, Jul 03 2015
%o (Python)
%o from sympy import isprime
%o print([n for n in range(1, 200) if n%2 and not isprime(n + 2)]) # _Indranil Ghosh_, Jun 15 2017, after _Charles R Greathouse IV_
%Y Cf. A048859.
%Y Complement of A030173. Cf. A001223.
%Y Cf. also A005408, A010051.
%Y Cf. A123556, A342309.
%Y Largest AP of prime numbers with k elements: this sequence (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), A206042 (k=8), A206043 (k=9), A206044 (k=10), A206045 (k=11).
%K nonn,easy,nice
%O 1,1
%A R. Muller
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