A108309 -id:A108309 - cp's OEIS frontend

A176271 The odd numbers as a triangle read by rows.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Reinhard Zumkeller, Apr 13 2010

A108309(n) = number of primes in n-th row.

			From _Philippe Deléham_, Oct 03 2011: (Start)
Triangle begins:
   1;
   3,  5;
   7,  9, 11;
  13, 15, 17, 19;
  21, 23, 25, 27, 29;
  31, 33, 35, 37, 39, 41;
  43, 45, 47, 49, 51, 53, 55;
  57, 59, 61, 63, 65, 67, 69, 71;
  73, 75, 77, 79, 81, 83, 85, 87, 89; (End)

G. C. Greubel, Rows n = 1..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Nicomachus's Theorem
Wikipedia, Nikomachos von Gerasa

Cf. A000466, A000537, A000578, A001105, A002061, A005408, A014209.
Cf. A016754, A027688, A027690, A027692, A027694, A028387, A048058.
Cf. A053755, A065599, A082111, A108195, A108309, A214604, A214661.

a176271 n k = a176271_tabl !! (n-1) !! (k-1)
a176271_row n = a176271_tabl !! (n-1)
a176271_tabl = f 1 a005408_list where
   f x ws = us : f (x + 1) vs where (us, vs) = splitAt x ws
-- Reinhard Zumkeller, May 24 2012

[n^2-n+2*k-1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

A176271 := proc(n,k)
    n^2-n+2*k-1 ;
end proc: # R. J. Mathar, Jun 28 2013

Table[n^2-n+2*k-1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[n^2-n+2*k-1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n, k) = n^2 - n + 2*k - 1 for 1 <= k <= n.
T(n, k) = A005408(n*(n-1)/2 + k - 1).
T(2*n-1, n) = A016754(n-1) (main diagonal).
T(2*n, n) = A000466(n).
T(2*n, n+1) = A053755(n).
T(n, k) + T(n, n-k+1) = A001105(n), 1 <= k <= n.
T(n, 1)   = A002061(n), central polygonal numbers.
T(n, 2)   = A027688(n-1) for n > 1.
T(n, 3)   = A027690(n-1) for n > 2.
T(n, 4)   = A027692(n-1) for n > 3.
T(n, 5)   = A027694(n-1) for n > 4.
T(n, 6)   = A048058(n-1) for n > 5.
T(n, n-3) = A108195(n-2) for n > 3.
T(n, n-2) = A082111(n-2) for n > 2.
T(n, n-1) = A014209(n-1) for n > 1.
T(n, n) = A028387(n-1).
Sum_{k=1..n} T(n, k) = A000578(n) (Nicomachus's theorem).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A065599(n) (alternating sign row sums).
Sum_{j=1..n} (Sum_{k=1..n} T(j, k)) = A000537(n) (sum of first n rows).

A089610 Number of primes between n^2 and (n+1/2)^2.

Original entry on oeis.org

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

Offset: 1

Cino Hilliard, Dec 30 2003

For small values of n, these numbers exhibit higher and lower values as n increases. Conjectures: After n=17 a(n) > 1. There exists an n_1 such that a(n) is < a(n+1) for all n >= n_1.

Same as the number of primes between n^2 and n^2+n. Oppermann conjectured in 1882 that a(n)>0. - T. D. Noe, Sep 16 2008

Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., 1995, Springer, p. 248.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 183.

T. D. Noe, Table of n, a(n) for n=1..10000
Wikipedia, Oppermann's conjecture

Cf. A010051, A014085, A094189, A108309.

a089610 n = sum $ map a010051' [n^2 .. n*(n+1)]
-- Reinhard Zumkeller, Jun 07 2015

a[n_] := PrimePi[(n + 1/2)^2] - PrimePi[n^2]; Table[ a@n, {n, 100}] (* Robert G. Wilson v, May 04 2009 *)

a(n) = primepi(n^2+n) - primepi(n^2); \\ Michel Marcus, May 18 2020

A094189 Number of primes between n^2-n and n^2 (inclusive).

Original entry on oeis.org

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

Offset: 1

Jason Earls, May 25 2004

Conjecture: for n>11, a(n)>1.

Oppermann conjectured in 1882 that a(n)>0 for n>1. - T. D. Noe, Sep 16 2008

Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., 1995, Springer, p. 248.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 183.

T. D. Noe, Table of n, a(n) for n = 1..10000
Wikipedia, Oppermann's conjecture

Cf. A014085, A089610, A108309, A010051.

a094189 n = sum $ map a010051' [n*(n-1) .. n^2]
-- Reinhard Zumkeller, Jun 07 2015

Table[PrimePi[n^2]-PrimePi[n^2-n-1],{n,100}] (* Harvey P. Dale, Jul 24 2015 *)

PARI
```
a(n) = sum(k=n^2-n,n^2,isprime(k))
```

a(n)=my(s);forprime(p=n^2-n,n^2,s++);s \\ Charles R Greathouse IV, Jan 18 2016

A246826 Numbers n such that there is no prime of a prime twin pair between n^2 + n and n^2 + 3*n + 2.

Original entry on oeis.org

0, 10, 26, 30, 36, 136, 156, 433

Offset: 1

Pierre CAMI, Sep 04 2014

nonn

No more values for n = 434 to 45140.
Conjecture: the sequence is finite and given in full.
a(9), if it exists, is greater than 10^5. - Derek Orr, Sep 19 2014

			n = 0 only 1 between 0 and 2 so a(1) = 0.
n = 1 between 2 and 6, 3 is the first of twin pair 3, 5.
For n = 2 to 9 always at least one prime of a twin pair between n^2 + n and n^2 + 3*n + 2.
n = 10 no prime of a twin pair between 110 and 132 so a(2) = 10.

Cf. A091592, A108309.

a(n)=forprime(p=n^2+n,n^2+3*n+2,if(precprime(p-1)==p-2||nextprime(p+1)==p+2,return(0)));return(1)
n=0;while(n<10^5,if(a(n),print1(n,", "));n++) \\ Derek Orr, Sep 19 2014

cp's OEIS Frontend

A176271 The odd numbers as a triangle read by rows.

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A089610 Number of primes between n^2 and (n+1/2)^2.

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A094189 Number of primes between n^2-n and n^2 (inclusive).

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A246826 Numbers n such that there is no prime of a prime twin pair between n^2 + n and n^2 + 3*n + 2.

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PARI