A097454 -id:A097454 - cp's OEIS frontend

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

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.

A243106 a(n) = Sum_{k=1..n} (-1)^isprime(k)*10^k.

Original entry on oeis.org

10, -90, -1090, 8910, -91090, 908910, -9091090, 90908910, 1090908910, 11090908910, -88909091090, 911090908910, -9088909091090, 90911090908910, 1090911090908910, 11090911090908910, -88909088909091090, 911090911090908910, -9088909088909091090

Offset: 1

R. J. Cano, Aug 19 2014

Alternative definition: a(n,x)=T(x,1) for a dichromate or Tutte-Whitney polynomial in which the matrix t[i,j] is defined as t[i,j]=Delta(i,j)*((-1)^isprime(i)) and "Delta" is the Kronecker Delta function. - Michel Marcus, Aug 19 2014
If 10 is replaced by 1, then this becomes A097454. If it is replaced by 2, one gets A242002. Choosing powers of the base b=10, as done here, allows one to easily read off the equivalent for any other base b > 4, by simply replacing digits 8,9 with b-2,b-1 (when terms are written in base b). [Comment extended by M. F. Hasler, Aug 20 2014]
There are 2^n ways of taking the partial sum of the first n powers of b=10 if exponent zero is excluded and the signs can be assigned arbitrarily. Conjecture: When expressed in base b, the absolute value for any of these terms only contains digits belonging to {0,1,b-2,b-1}; here {0,1,8,9}.

			n=1 is not prime x^1 = (10)^1 = 10, therefore a(1)=10;
n=2 is prime and x^2 = (10)^2 = 100, taking it negative, a(2) = 10 - 100 = -90;
n=3 also is prime, x^3 = 1000, and we have a(3) = 10 - 100 - 1000 = -1090;
n=4 is not prime, so a(4) = 10 - 100 - 1000 + 10000 = 8910;
n=5 is prime, then a(5) = 10 - 100 - 1000 + 10000 - 100000 = -91090;
Examples of analysis for the concatenation patterns among the terms can be found at the "Additional Information" link.

R. J. Cano, Table of n, a(n) for n = 1..100
R. J. Cano, Additional information.
Eric Weisstein's World of Mathematics, Alternating Series
Eric Weisstein's World of Mathematics, Tutte Polynomial

Cf. A097454.
The same kind of base-independent behavior: A215940, A217626.
Partial sums of alternating series: A181482, A222739, A213203.

Table[Sum[ (-1)^Boole@ PrimeQ@ k*10^k, {k, n}], {n, 19}] (* Michael De Vlieger, Jan 03 2016 *)

ap(n, x)={my(s); forprime(p=1, n, s+=x^p); s}
a=(n, x=10)->(x^(n+1)-1)/(x-1)-2*ap(n, x)-1;

Delta=(i, j)->(i==j); /* Kronecker's Delta function */
t=n->matrix(n, n, i, j, Delta(i, j)*((-1)^isprime(i))); /* coeffs t[i, j] */
/* Tutte polynomial over n */
T(n, x, y)={my(t0=t(n)); sum(i=1, n, sum(j=1, n, t0[i, j]*(x^i)*(y^j)))};
a=(n, x=10)->T(n, x, 1);

A243106(n,b=10)=sum(k=1,n,(-1)^isprime(k)*b^k) \\ M. F. Hasler, Aug 20 2014

a(n,x) = Sum_{k=1..n} (-1)^isprime(k)*(x^k), for x=10 in decimal.

Definition simplified by N. J. A. Sloane, Aug 19 2014

Definition further simplified and more terms from M. F. Hasler, Aug 20 2014

A116568 Difference between n and the absolute value of the difference between number of nonprimes not exceeding n and number of primes not exceeding n.

Original entry on oeis.org

0, 2, 2, 4, 4, 6, 6, 8, 8, 8, 10, 10, 12, 12, 12, 12, 14, 14, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 20, 20, 22, 22, 22, 22, 22, 22, 24, 24, 24, 24, 26, 26, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 32, 32, 32, 32, 32, 32, 34, 34, 36, 36, 36, 36, 36, 36, 38, 38, 38, 38, 40, 40, 42

Offset: 1

Roger L. Bagula, Mar 18 2006

nonn

			a(11)=10 because the nonprimes not exceeding 11 are 1,4,6,8,9 and 10, the primes not exceeding 11 are 2,3,5,7 and 11 and 11-abs(6-5)=10.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A097454.

with(numtheory): seq(n-abs(n-2*pi(n)),n=1..73);

Table[n-Abs[2*PrimePi[n]-n],{n,80}] (* Harvey P. Dale, Oct 08 2015 *)

for(n=1,50, print1(n - abs(2*primepi(n) - n) , ", ")) \\ G. C. Greubel, Sep 20 2017

a(n) = n - Abs[A097454(n)].

Edited by N. J. A. Sloane, Apr 05 2006

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

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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A243106 a(n) = Sum_{k=1..n} (-1)^isprime(k)*10^k.

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A116568 Difference between n and the absolute value of the difference between number of nonprimes not exceeding n and number of primes not exceeding n.

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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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