A139325 -id:A139325 - cp's OEIS frontend

A139328 Sums of rows of the triangle in A139325.

0, 3, 6, 10, 14, 19, 24, 30, 36, 45, 52, 60, 67, 76, 86, 96, 105, 117, 127, 138, 151, 162, 176, 189, 203, 216, 230, 246, 262, 277, 292, 308, 325, 343, 362, 376, 398, 417, 435, 451, 473, 491, 515, 535, 557, 579, 599, 622, 646, 668, 691, 712, 737, 764, 788, 815

Offset: 1

Reinhard Zumkeller, Apr 14 2008

nonn

a(n) = Sum_{k=1..n} A139325(n,k).

			a(4) = #{3,5,7}+#{11,13}+#{17,19,23}+#{29,31} = 3+2+3+2 = 10:
..1 ...3 ...5 ...7 ... primes in first row = {3,5,7},
..9 ..11 ..13 ..15 ... primes in 2nd row = {11,13},
.17 ..19 ..21 ..23 ... primes in 3rd row = {17,19,23},
.25 ..27 ..29 ..31 ... primes in 4th row = {29,31}.

Cf. A000720, A038107, A139325.

a(n) = A000720(2*n^2 - 1) - 1.

A099802 Bisection of A000720.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10, 11, 11, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 30, 30, 30, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 34

Offset: 1

N. J. A. Sloane, Nov 19 2004

Maximal number of primes possible in a string of 2n consecutive numbers. - Lekraj Beedassy, Dec 04 2004
a(n) = A139325(n,1) + 1. - Reinhard Zumkeller, Apr 14 2008
Or the number of primes <= 2n. - Juri-Stepan Gerasimov, Oct 29 2009

Cf. A099081.

with(numtheory):seq(pi(2*n),n=1..86); # Emeric Deutsch, Apr 12 2005

a(n)=primepi(2*n) \\ Charles R Greathouse IV, Sep 02 2015

[prime_pi(2*n) for n in range(1, 75)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000720(n) + A035250(n) - A010051(n). - Reinhard Zumkeller, Jul 05 2010

a(n) = A000720(2*n). - Wesley Ivan Hurt, Jun 16 2013

More terms from Emeric Deutsch, Apr 12 2005

A083415 Triangle read by rows: T(n,k) is defined as follows. Write the numbers from 1 to n^2 consecutively in n rows of length n; T(n,k) = number of primes in k-th row.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 2, 2, 1, 1, 4, 2, 2, 1, 2, 2, 2, 4, 2, 3, 2, 1, 3, 1, 2, 4, 3, 2, 2, 3, 2, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 3, 2, 1, 5, 3, 3, 3, 2, 2, 3, 2, 2, 4, 1, 5, 4, 2, 4, 2, 3, 3, 1, 4, 2, 2, 2, 6, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 2, 3, 6, 3, 4, 3, 3, 4, 2, 4

Offset: 1

N. J. A. Sloane, following a suggestion of Wouter Meeussen, Jun 10 2003

Sum(T(n,k): 1<=k<=n) = A038107(n); T(n,1)=A000720(n); T(n,2)=A060715(n) for n>1. - Reinhard Zumkeller, Jan 07 2004

			{0}
{1, 1}
{2, 1, 1} from / 1 2 3 / 4 5 6 / 7 8 9 /
{2, 2, 1, 1}
{3, 1, 2, 2, 1}
{3, 2, 2, 2, 1, 1}

Paulo Ribenboim, "The Little Book Of Big Primes," Springer-Verlag, NY 1991, page 185.

T. D. Noe, Rows n=1..100 of triangle, flattened

Cf. A083382, A083414, A092556, A092557.
Cf. A139325.
Cf. A010051.

a083415 n k = a083415_row n !! (k-1)
a083415_row n = f n a010051_list where
   f 0 _     = []
   f k chips = (sum chin) : f (k - 1) chips' where
     (chin,chips') = splitAt n chips
a083415_tabl = map a083415_row [1..]
-- Reinhard Zumkeller, Jun 10 2012

Table[PrimePi[m n]-PrimePi[(m-1) n], {n, 17}, {m, n}]

A139326 Write the first n^2 odd numbers consecutively in n rows of length n: a(n) = minimal number of primes in a row.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 3, 3, 4, 5, 4, 6, 5, 5, 5, 5, 5, 5, 4, 5, 5, 5, 6, 6, 5, 5, 5, 8, 6, 7, 7, 8, 7, 7, 8, 8, 8, 9, 9, 8, 9, 9, 10, 10, 9, 9, 8, 9, 9, 9, 10, 10, 11, 9, 11, 10, 11, 10, 11, 11, 11, 12, 12, 12, 13, 12, 13, 13, 13, 12, 13, 14, 15, 14, 12, 13, 13, 15, 14

Offset: 1

Reinhard Zumkeller, Apr 14 2008

nonn

a(n) = Min{A139325(n,k): 1<=k<=n}.

			a(4)=Min{#{3,5,7},#{11,13},#{17,19,23},#{29,31}}=Min{3,2,3,2}=2:
..1 ...3 ...5 ...7 ... primes in first row = {3,5,7},
..9 ..11 ..13 ..15 ... primes in 2nd row = {11,13},
.17 ..19 ..21 ..23 ... primes in 3rd row = {17,19},
.25 ..27 ..29 ..31 ... primes in 4th row = {29,31}.

Cf. A139327, A083382.

A139327 Write the first n^2 odd numbers consecutively in n rows of length n: a(n) = maximal number of primes in a row.

Original entry on oeis.org

0, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 8, 9, 10, 10, 10, 11, 11, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 17, 17, 17, 18, 18, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 29, 29, 29, 29, 29, 30, 30, 31, 31, 31, 32, 33, 33, 33, 33, 33

Offset: 1

Reinhard Zumkeller, Apr 14 2008

nonn

a(n) = Max{A139325(n,k): 1<=k<=n}.

			a(4)=Max{#{3,5,7},#{11,13},#{17,19,23},#{29,31}}=Max{3,2,3,2}=3:
..1 ...3 ...5 ...7 ... primes in first row = {3,5,7},
..9 ..11 ..13 ..15 ... primes in 2nd row = {11,13},
.17 ..19 ..21 ..23 ... primes in 3rd row = {17,19},
.25 ..27 ..29 ..31 ... primes in 4th row = {29,31}.

Cf. A139326.

cp's OEIS Frontend

A139328 Sums of rows of the triangle in A139325.

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A099802 Bisection of A000720.

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A083415 Triangle read by rows: T(n,k) is defined as follows. Write the numbers from 1 to n^2 consecutively in n rows of length n; T(n,k) = number of primes in k-th row.

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A139326 Write the first n^2 odd numbers consecutively in n rows of length n: a(n) = minimal number of primes in a row.

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A139327 Write the first n^2 odd numbers consecutively in n rows of length n: a(n) = maximal number of primes in a row.

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