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 A182138 Irregular triangle T, read by rows, in which row n lists the distances between n and the two primes whose sum makes 2n in decreasing order (Goldbach conjecture). 6
 0, 0, 1, 2, 0, 1, 4, 0, 5, 3, 4, 2, 7, 3, 8, 6, 0, 7, 5, 1, 10, 6, 0, 9, 3, 8, 4, 2, 13, 3, 14, 12, 6, 0, 13, 11, 5, 1, 12, 0, 17, 9, 3, 16, 10, 8, 2, 19, 15, 9, 20, 18, 6, 0, 19, 17, 13, 7, 5, 22, 18, 12, 6, 21, 15, 3, 20, 16, 14, 10, 4, 25, 15, 9, 24, 18, 12, 0, 23, 17, 13, 11, 7, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,4 COMMENTS The Goldbach conjecture is that for any even integer 2n>=4, at least one pair of primes p and q exist such that p+q=2n. The present numbers listed here are the distances d between each prime and n, the half of the even integer 2n: d=n-p=q-n with p <= q. See the link section for plots I added. - Jason Kimberley, Oct 04 2012 Each nonzero entry d of row n is coprime to n. For otherwise n+d would be composite. - Jason Kimberley, Oct 10 2012 LINKS Alois P. Heinz, Rows n = 2..600, flattened OEIS (Plot 2), Plot of (n, d) Subplots for fixed p: OEIS (Plot 2), A067076 vs A098090 (p=3). OEIS (Plot 2), A089038 vs A089253 (p=5). OEIS (Plot 2), A105760 vs A089192 (p=7). ... OEIS (Plot 2), A153143 vs A097932 (p=19). Wikipedia, Goldbach's conjecture FORMULA T(n,i) = n - A184995(n,i). - Jason Kimberley, Sep 25 2012 EXAMPLE n=2, 2n=4, 4=2+2, p=q=2 -> d=0. n=18, 2n=36, four prime pairs have a sum of 36: 5+31, 7+29, 13+23, 17+19, with the four distances d being 13=18-5=31-18, 11=18-7=29-18, 5=18-13=23-18, 1=18-17=19-18. Triangle begins: 0; 0; 1; 2, 0; 1; 4, 0; 5, 3; 4, 2; 7, 3; 8, 6, 0; MAPLE T:= n-> seq(`if`(isprime(p) and isprime(2*n-p), n-p, NULL), p=2..n): seq(T(n), n=2..40); # Alois P. Heinz, Apr 16 2012 PROG (PARI) for(n=2, 18, forprime(p=2, n, if(isprime(2*n-p), print1(n-p", ")))) \\ Charles R Greathouse IV, Apr 16 2012 (MAGMA) A182138:= func; &cat[A182138(n):n in [2..30]]; // Jason Kimberley, Oct 01 2012 CROSSREFS Cf. A045917 (row lengths), A047949 (first column), A047160 (last elements of rows). Cf. A184995. Sequence in context: A273821 A108643 A133838 * A258123 A121583 A228924 Adjacent sequences:  A182135 A182136 A182137 * A182139 A182140 A182141 KEYWORD easy,nonn,tabf AUTHOR Jean COHEN, Apr 16 2012 STATUS approved

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Last modified May 17 06:30 EDT 2021. Contains 343965 sequences. (Running on oeis4.)