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%I #49 May 23 2021 02:38:46
%S 0,2,1,1,2,2,1,2,3,2,2,3,3,3,3,2,4,3,3,4,4,4,4,4,4,4,4,5,5,6,4,5,3,6,
%T 6,7,5,5,6,4,8,5,6,6,8,6,8,5,7,5,11,4,6,9,7,8,9,8,7,7,9,7,8,7,12,5,9,
%U 9,11,9,7,7,12,10,10,9,9,9,6,11,10,11,9,12,11,12,9,10,11,12,10,13,9,11,10,12
%N Number of primes p between triangular numbers T(n) < p <= T(n+1).
%C It is conjectured that for n > 0, a(n) > 0. See also A190661. - _John W. Nicholson_, May 18 2011
%C If the above conjecture is true, then for any k > 1 there is a prime p > k such that p <= (n+1)(n+2)/2, where n = floor(sqrt(2k)+1/2). Ignoring the floor function we can obtain a looser (but nicer) lower bound of p <= 1 + k + 2*sqrt(2k). - _Dmitry Kamenetsky_, Nov 26 2016
%H T. D. Noe, <a href="/A066888/b066888.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = pi(n*(n+1)/2) - pi(n*(n-1)/2).
%F a(n) equals the number of occurrences of n+1 in A057062. - _Esko Ranta_, Jul 29 2011
%e Write the numbers 1, 2, ... in a triangle with n terms in the n-th row; a(n) = number of primes in n-th row.
%e Triangle begins
%e 1 (0 primes)
%e 2 3 (2 primes)
%e 4 5 6 (1 prime)
%e 7 8 9 10 (1 prime)
%e 11 12 13 14 15 (2 primes)
%t Table[PrimePi[(n^2 + n)/2] - PrimePi[(n^2 - n)/2], {n, 96}] (* _Alonso del Arte_, Sep 03 2011 *)
%t PrimePi[#[[2]]]-PrimePi[#[[1]]]&/@Partition[Accumulate[Range[0,100]],2,1] (* _Harvey P. Dale_, Jun 04 2019 *)
%o (PARI) { tp(m)=local(r, t); r=1; for(n=1,m,t=0; for(k=r,n+r-1,if(isprime(k),t++)); print1(t","); r=n+r; ) }
%o (PARI) {tpf(m)=local(r, t); r=1; for(n=1,m,t=0; for(k=r,n+r-1,if(isprime(k),t++); print1(k" ")); print1(" ("t" prime)"); print(); r=n+r;) }
%Y Cf. A083382.
%Y Essentially the same as A065382 and A090970.
%Y Cf. A000217, A000040, A014085, A190661.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jun 06 2003
%E More terms from _Vladeta Jovovic_ and _Jason Earls_, Jun 06 2003
%E Offset corrected by _Daniel Forgues_, Sep 05 2012
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