A072731 -id:A072731 - cp's OEIS frontend

A097454 a(n) = (number of nonprimes <= n) - (number of primes <= n).

1, 0, -1, 0, -1, 0, -1, 0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 10, 9, 10, 9, 10, 11, 12, 13, 14, 13, 14, 15, 16, 15, 16, 15, 16, 17, 18, 17, 18, 19, 20, 21, 22, 21, 22, 23, 24, 25, 26, 25, 26, 25, 26, 27, 28, 29, 30, 29, 30, 31, 32, 31, 32, 31, 32, 33, 34, 35, 36, 35, 36, 37, 38, 37, 38, 39, 40, 41, 42, 41, 42, 43, 44, 45

Offset: 1

Cino Hilliard, Aug 23 2004

			a(7) = -1 because there are 3 nonprimes <= 7 (1,4 and 6) and 4 primes <= 7 (2,3,5 and 7).

Cf. A000720, A010051, A062298, A072731.

with(numtheory): seq(n-2*pi(n), n=1..93); # Emeric Deutsch, Apr 01 2006

qp=0;lst={};Do[If[PrimeQ[n],AppendTo[lst,qp-=1],AppendTo[lst,qp+=1]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 15 2010 *)
Accumulate[ -1 + 2 * Boole /@ Not /@ PrimeQ @ Range @ 100] (* Federico Provvedi, Oct 06 2013 *)

compsmprimes(n) = { for(x=1,n, y=composites(x) - pi(x); print1(y",") ) }
\\ The number of composite numbers less than or equal to n
composites(n) = { my(c,x); c=0; for(x=1,n, if(!isprime(x),c++); ); return(c) }
\\ pi(x) prime count function
pi(n) = { my(c,x); c=0;forprime(x=1,n,c++);return(c) }

a(n) = 1 + A072731(n).
a(n) = n - 2*pi(n) =  n - 2*A000720(n). - Wesley Ivan Hurt, Jun 16 2013
a(n) - a(n-1) = 1 - 2*A010051(n) for n > 1. - Wesley Ivan Hurt, Dec 18 2018
a(n) = A062298(n) - A000720(n). - Michel Marcus, Jan 31 2019

A118777 a(0) = 1; n > 0: a(n) = a(n-1) + d, d = +-1 if n is prime/nonprime.

Original entry on oeis.org

1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 10, 11, 10, 11, 10, 11, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 22, 23, 22, 23, 24, 25, 26, 27, 26, 27, 26, 27, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 32, 33, 34, 35, 36

Offset: 0

Zak Seidov, May 22 2006

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000720, A010051.
One more than A097454, two more than A072731.
For no apparent reason, the terms a(3)..a(24) are equal to the terms a(0)..a(21) of A276090.

 Table[n + 1 - 2*PrimePi[n], {n, 0, 100}]

;; With memoization-macro definec.
(definec (A118777 n) (if (zero? n) 1 (+ (A118777 (- n 1)) (expt -1 (A010051 n)))))
;; Antti Karttunen, Aug 19 2016

(define (A118777 n) (+ 1 (- n (* 2 (A000720 n))))) ;; After formula given by the original author.
;; Antti Karttunen, Aug 19 2016

a(n) = n + 1 - 2*primepi(n), n = 0, 1, 2, ..., where primepi(n) = A000720(n).
From Antti Karttunen, Aug 21 2016: (Start)
a(0) = 1, for n >= 1, a(n) = a(n-1) + (-1)^A010051(n). (from the definition).
For all n >= 1, a(n) = 1+A097454(n) = 2+A072731(n).
(End)

Offset and the name corrected by Antti Karttunen, Aug 19 2016

A383037 a(n) is the excess of composites over primes in the first n odd positive integers.

Original entry on oeis.org

0, -1, -2, -3, -2, -3, -4, -3, -4, -5, -4, -5, -4, -3, -4, -5, -4, -3, -4, -3, -4, -5, -4, -5, -4, -3, -4, -3, -2, -3, -4, -3, -2, -3, -2, -3, -4, -3, -2, -3, -2, -3, -2, -1, -2, -1, 0, 1, 0, 1, 0, -1, 0, -1, -2, -1, -2, -1, 0, 1, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 4

Offset: 1

Felix Huber, Apr 19 2025

			Of the first 5 odd positive integers (1, 3, 5, 7, 9), one (9) is a composite and three (3, 5, 7) are primes, so a(5) = 1 - 3 = -2.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Odd Number

Cf. A000720, A005408, A065091, A071904, A072731, A097454, A118777, A210469.

A383037:=n->n-NumberTheory:-pi(2*n)*2+1;seq(A383037(n),n=1..71);

a[n_]:=n - 2*PrimePi[2*n] + 1; Array[a,71] (* Stefano Spezia, Apr 20 2025 *)

a(n) = n - 2*pi(2*n) + 1.
a(n) = A210469(n) - pi(2*n) + 1 = A210469(n) - A000720(2*n) + 1 = for n > 1.
a(n) = A118777(2*n-1) - n + 1 for n > 1.
a(n) = A097454(2*n-1) - n + 2 for n > 1.
a(n) = A072731(2*n-1) - n + 3 for n > 1.

A361915 a(n) is the smallest prime p such that, for m >= nextprime(p), there are more composites than primes in the range [2, m], where multiples of primes prime(1) through prime(n) are excluded.

Original entry on oeis.org

13, 113, 1069, 5051, 18553, 44417, 99439, 190921, 356351, 603149, 933073, 1416223, 2044201, 2856559, 3957883, 5379287, 7093217, 9113263, 11693687, 14701529, 18345209, 22758829, 27879563, 33938257, 40808759, 48364003, 57099061, 67292237, 78919781, 92417891

Offset: 0

Ya-Ping Lu, Mar 29 2023

nonn

			The number of primes, N_p, and the number of composite, N_c, in the range [2, m] are listed in the table below, where N_p = N_c occurs at m = 9, 11 and 13. For m >= nextprime(13) = 17, N_c > N_p. So, a(0) = 13 is the case for n = 0, in which none of the multiples of primes is excluded from the integer list.
   m:   2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
   N_p: 1, 2, 2, 3, 3, 4, 4, 4,  4,  5,  5,  6,  6,  6,  6,  7, ...
   N_c: 0, 0, 1, 1, 2, 2, 3, 4,  5,  5,  6,  6,  7,  8,  9,  9, ...
If the multiples of prime(1) are excluded from the list, 113 is the smallest prime such that N_c > N_p for m >= nextprime(113) = 127 and, thus, a(1) = 113 (see below).
   m:   3, 5, 7, ..., 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, ...
   N_p: 1, 2, 3, ..., 23, 23, 24, 24,  25,  26,  26,  27,  28,  28,  29,  29, ...
   N_c: 0, 0, 0, ..., 23, 24, 24, 25,  25,  25,  26,  26,  26,  27,  27,  28, ...
If multiples of prime(1) and prime(2) are excluded, a(2) = 1069. If multiples of prime(1), prime(2) and prime(3) are excluded, a(3) = 5051.

Cf. A000040, A002808, A072731, A097454.

from sympy import isprime, prime
R = []; L = [x for x in range(2, 100000001)]
for n in range(30):
    np = 0; nc = 0; found = 0
    if n > 0: q = prime(n); L = [x for x in L if x%q != 0]
    for m in L:
        if isprime(m): np += 1; p = m
        else: nc += 1
        if np == nc: Lp = p; found = 1
    if found: R.append(Lp)
print(*R, sep = ", ")

cp's OEIS Frontend

A097454 a(n) = (number of nonprimes <= n) - (number of primes <= n).

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A118777 a(0) = 1; n > 0: a(n) = a(n-1) + d, d = +-1 if n is prime/nonprime.

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A383037 a(n) is the excess of composites over primes in the first n odd positive integers.

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Author

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Programs

Maple

Mathematica

Formula

A361915 a(n) is the smallest prime p such that, for m >= nextprime(p), there are more composites than primes in the range [2, m], where multiples of primes prime(1) through prime(n) are excluded.

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