A257232 -id:A257232 - cp's OEIS frontend

A257233 Multiplicity sequence for the alternating row sums of triangle A257232.

1, 7, 6, 6, 4, 2, 6, 2, 4, 6, 4, 2, 4, 2, 6, 2, 4, 6, 2, 4, 4, 2, 4, 2, 2, 4, 6, 6, 4, 2, 2, 2, 2, 2, 4, 4, 2, 6, 2, 2, 2, 6, 2, 4, 2, 4, 4, 2, 4, 2, 6, 2, 2, 2, 6, 6, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4

Offset: 1

Wolfdieter Lang, Apr 19 2015

The sequence of alternating row sums of A257232 gives one times 1, seven times 2, 6 times 3, six times 4, four times 5, etc., in this order.

Is a(n) = A067970(n-1), n>=3 ? - R. J. Mathar, Apr 24 2015

Cf. A257232.

The number m appears a(m) times in the sequence of row sums of triangle A257232 for m = 1, 2, ..., that is in sum(((-1)^(k-1))*(n - (k-1) - [isprime(k)]), k = 1..n) with [isprime(k)] = A010051(k) for n >= 1.

A256885 a(n) = n*(n + 1)/2 - pi(n), where pi(n) = A000720(n) is the prime counting function.

Original entry on oeis.org

1, 2, 4, 8, 12, 18, 24, 32, 41, 51, 61, 73, 85, 99, 114, 130, 146, 164, 182, 202, 223, 245, 267, 291, 316, 342, 369, 397, 425, 455, 485, 517, 550, 584, 619, 655, 691, 729, 768, 808, 848, 890, 932, 976, 1021, 1067, 1113, 1161, 1210, 1260, 1311, 1363, 1415, 1469, 1524

Offset: 1

Wesley Ivan Hurt, Apr 11 2015

Number of lattice points (x,y) in the region 1 <= x <= n, 1 <= y <= n - A010051(n); see example.

This sequence gives the row sums of the triangle A257232. - Wolfdieter Lang, Apr 21 2015

			10 .                             x
9  .                          x  x
8  .                       x  x  x
7  .                    .  x  x  x
6  .                 x  x  x  x  x
5  .              .  x  x  x  x  x
4  .           x  x  x  x  x  x  x
3  .        .  x  x  x  x  x  x  x
2  .     .  x  x  x  x  x  x  x  x
1  .  x  x  x  x  x  x  x  x  x  x
0  .__.__.__.__.__.__.__.__.__.__.
   0  1  2  3  4  5  6  7  8  9  10

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Counting Function
Wikipedia, Prime-counting function

Cf. A000217, A000720, A010051, A014684, A113638, A257232.

a256885 n = a000217 n - a000720 n  -- Reinhard Zumkeller, Apr 21 2015

[n*(n + 1)/2 - #PrimesUpTo(n): n in [1..60] ]; // Vincenzo Librandi, Apr 12 2015

with(numtheory)[pi]: A256885:=n->n*(n+1)/2-pi(n): seq(A256885(n), n=1..100);

Table[n (n + 1)/2 - PrimePi[n], {n, 1, 50}]

vector(80, n, n*(n+1)/2 - primepi(n)) \\ Michel Marcus, Apr 13 2015

a(n) = A000217(n) - A000720(n).
a(n) - a(n-1) = A014684(n), n >= 2.
a(n) = Sum_{i=1..n} A014684(i).
a(n) = 1 + Sum_{i=2..n}(i - A000720(i) + A000720(i-1)).

Edited, following the hint by Reinhard Zumkeller to change the offset. - Wolfdieter Lang, Apr 22 2015

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A257233 Multiplicity sequence for the alternating row sums of triangle A257232.

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A256885 a(n) = n*(n + 1)/2 - pi(n), where pi(n) = A000720(n) is the prime counting function.

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