A037040 -id:A037040 - cp's OEIS frontend

A038377 Number of odd nonprimes <= (2n+1)^2.

1, 2, 5, 11, 20, 32, 47, 66, 85, 110, 137, 167, 200, 237, 276, 320, 365, 414, 467, 522, 579, 643, 708, 777, 845, 924, 997, 1080, 1169, 1255, 1343, 1437, 1536, 1637, 1741, 1847, 1961, 2075, 2187, 2311, 2435, 2560, 2691, 2826, 2962, 3104, 3249, 3393, 3543

Offset: 0

N. J. A. Sloane and Erich Friedman

nonn

			a(2) = 5 because there are 5 odd nonprimes that are not exceeding (2*2+1)^2 = 25: 1, 9, 15, 21 and 25.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000720, A037040, A051890.

nn=20001; With[{onps=Complement[Range[1,nn,2],Prime[Range[PrimePi[nn+1]]]]}, Table[Count[onps,?(#<=(2n+1)^2&)],{n,0,60}]]  (* _Harvey P. Dale, Apr 13 2011 *)
a[n_] := 2*n^2 + 2*n + 2 - PrimePi[(2*n + 1)^2]; a[0] = 1; Array[a, 61, 0] (* Amiram Eldar, Sep 06 2024 *)

a(n) = if(n == 0, 1, 2*n^2 + 2*n + 2 - primepi((2*n + 1)^2)); \\ Amiram Eldar, Sep 06 2024

a(n) = A037040(n) + 1.

For n>=1, a(n) = 2n^2 + 2n + 2 - PrimePi((2n+1)^2) = A051890(n+1) - A000720((2n+1)^2). - Zak Seidov, Mar 03 2008

Offset corrected by Amiram Eldar, Sep 06 2024

A287016 a(n) = smallest number k such that A071904(n) + k^2 is a perfect square.

Original entry on oeis.org

0, 1, 2, 0, 3, 4, 1, 5, 2, 0, 7, 3, 8, 1, 4, 10, 5, 2, 0, 6, 13, 3, 14, 7, 1, 4, 17, 9, 2, 5, 0, 19, 10, 20, 6, 3, 22, 1, 12, 7, 4, 13, 25, 8, 2, 0, 5, 9, 28, 29, 16, 3, 6, 1, 32, 11, 18, 7, 4, 34, 19, 12, 35, 2, 0, 5, 21, 38, 9, 14, 3, 40, 6, 1, 15, 10, 24

Offset: 1

Zhandos Mambetaliyev, Jean-François Alcover, May 18 2017

			The third odd composite number is A071904(3) = 21. and 21+2^2 = 25 = 5^2, so a(3) = 2.

Cf. A016754, A037040, A071904.

Subsequence of A068527.

q[n_] := SelectFirst[Range[0, (n-1)/2], IntegerQ@ Sqrt[#^2 + n] &]; q /@ Select[Range[1, 300, 2], CompositeQ] (* Giovanni Resta, May 18 2017 *)

from sympy import primepi, divisors
from sympy.ntheory.primetest import is_square
def A287016(n):
    if n == 1: return 0
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return 0 if is_square(int(m)) else -(d:=divisors(m))[l:=(len(d)>>1)-1]+d[l+1]>>1 # Chai Wah Wu, Aug 02 2024

a(m) = 0 for m>0 in A037040, the corresponding odd composites being in A016754\{1}. - Michel Marcus, May 19 2017

More terms from Giovanni Resta, May 18 2017

A378814 a(n) = round(n/(A000005(A071904(n))-2)).

Original entry on oeis.org

1, 1, 2, 4, 3, 3, 4, 4, 2, 10, 6, 6, 7, 4, 8, 8, 4, 9, 6, 10, 11, 11, 12, 12, 6, 4, 14, 14, 7, 15, 31, 16, 17, 17, 18, 6, 19, 19, 20, 10, 10, 21, 22, 22, 8, 46, 12, 12, 25, 25, 26, 26, 9, 9, 28, 28, 29, 15, 30, 30, 31, 31, 32, 32, 9, 11, 34, 34, 17, 18, 36

Offset: 1

Bill McEachen, Dec 08 2024

nonn

Nearly all of the data falls on lines discussed below. There are a few "outliers" visible on the graph. There are <120 such outliers in the first 20000 terms (about 0.6%). Many of the outlier indices belong to A037040. The lines are n, (n+0)/2, (n+2)/4, (n+4)/6, (n+6)/8,....

			Let n=14, A071904(n)=63, tau(63)=6 so a(14)=round(14/(6-2))=4.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Bill McEachen, Plot 20K terms
Bill McEachen, A037040 overlap (TEMP)

Cf. A000005, A037040, A071904.

MapIndexed[Floor[#2[[1]]/# + 1/2] &, DivisorSigma[0, Select[Range[9, 500, 2], CompositeQ]] - 2] (* Paolo Xausa, Dec 16 2024 *)

cp's OEIS Frontend

A038377 Number of odd nonprimes <= (2n+1)^2.

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A287016 a(n) = smallest number k such that A071904(n) + k^2 is a perfect square.

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A378814 a(n) = round(n/(A000005(A071904(n))-2)).

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