A062298 -id:A062298 - cp's OEIS frontend

A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2a(n), a(not_an_oddprime(n)) = 2a(n-1).

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 9, 14, 11, 16, 20, 24, 13, 18, 15, 28, 22, 32, 17, 40, 48, 26, 36, 30, 21, 56, 25, 44, 64, 34, 80, 96, 19, 52, 72, 60, 29, 42, 23, 112, 50, 88, 33, 128, 68, 160, 192, 38, 41, 104, 144, 120, 58, 84, 49, 46, 27, 224, 100, 176, 66, 256, 37, 136, 320, 384, 31, 76, 57, 82, 208, 288, 240, 116, 45

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here oddprime(n) = n-th odd prime = A065091(n) = A000040(n+1), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).

			For n=2, which is the second natural number >= 1 that is not an odd prime [2 = A065090(2)], we compute 2*a(1) = 2 = a(2).
For n=4, which is A065090(3), we compute 2*a(3-1) = 2*2 = 4.
For n=5, and 5 is the second odd prime [5 = A065091(2)], thus a(5) = 1 + 2*a(2) = 5.
For n=9, which is the sixth natural number >= 1 not an odd prime (9 = A065090(6)), we compute 2*a(6-1) = 2*5 = 10.
For n=11, which is the fourth odd prime [11 = A065091(4)], we compute 1 + 2*a(4) = 1 + 2*4 = 9, thus a(11) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257728.
Cf. A000720, A010051, A062298, A065090, A065091.
Related or similar permutations: A246377, A246378, A257725, A257730, A257801.

a(1) = 1; a(2) = 2; and for n > 2, if A010051(n) = 1 [i.e., when n is a prime], then a(n) = 1 + 2*a(A000720(n)-1), otherwise a(n) = 2*a(A062298(n)).
As a composition of other permutations:
a(n) = A246377(A257730(n)).
a(n) = A257725(A257801(n)).

A337978 a(n) = n + pi(n) - pi(n + pi(n)), where pi(n) is the prime count of n (n>=1).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 7, 8, 10, 10, 11, 12, 13, 14, 15, 16, 18, 19, 19, 20, 21, 22, 23, 24, 25, 25, 27, 28, 29, 29, 30, 31, 32, 32, 34, 35, 36, 37, 38, 39, 41, 42, 42, 43, 44, 45, 46, 47, 48, 48, 50, 51, 51, 52, 52, 53, 55, 56, 57, 58, 59, 60, 60, 61, 63

Offset: 1

Ya-Ping Lu, Oct 06 2020

nonn

It seems that this is a nondecreasing sequence and a(n) < n for n >= 2.

Proofs of the above observations are provided in the Links below.

Robert Israel, Table of n, a(n) for n = 1..10000
Ya-Ping Lu, Proofs of the two observations in the Comments section

Cf. A000720, A062298, A095117, A337979.

f:= n -> n + numtheory:-pi(n) - numtheory:-pi(n + numtheory:-pi(n)):
map(f, [$1..100]); # Robert Israel, Feb 12 2024

pc[n_]:=With[{c=PrimePi[n]},n+c-PrimePi[n+c]]; Array[pc,70] (* Harvey P. Dale, Jan 18 2024 *)

a(n) = {my(x = n + primepi(n)); x - primepi(x); } \\ Michel Marcus, Oct 06 2020

from sympy import primepi
print(1)
n = 2
for n in range(2, 10001):
    n_f = n + primepi(n)
    a = n_f - primepi(n_f)
    print(a)

a(n) = n + pi(n) - pi(n + pi(n)).

A065857 The (10^n)-th composite number.

Original entry on oeis.org

4, 18, 133, 1197, 11374, 110487, 1084605, 10708555, 106091745, 1053422339, 10475688327, 104287176419, 1039019056246, 10358018863853, 103307491450820, 1030734020030318, 10287026204717358, 102692313540015924, 1025351434864118026, 10239531292310798956, 102270102190290407386

Offset: 0

Labos Elemer, Nov 26 2001

			The 100th composite number is C(100)=133, while the 100th prime is 541. In general: A000720(m) < A062298(m) < m < A002808(m) < A000040(m), for example pi(100)=25 < 75 < 100 < C(100)=133 < prime(100)=541.

A. E. Bojarincev, Asymptotic expressions for the n-th composite number. Univ. Mat. Zap. 6:21-43(1967). [in Russian]
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 133, p. 45, Ellipses, Paris 2008.

Chai Wah Wu, Table of n, a(n) for n = 0..22
Laurentiu Panaitopol, Some properties of the series of composed [sic] numbers, Journal of Inequalities in Pure and Applied Mathematics 2:3 (2001).
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp. 64-94.

Cf. A033844, A002808, A062298, A000720, A006988, A065855, A065856.

Composite[n_Integer] := Block[ {k = n + PrimePi[n] + 1 }, While[ k != n + PrimePi[k] + 1, k = n + PrimePi[k] + 1]; Return[k]];
Table[Composite[10^n], {n, 0, 9}]

a(n)=my(k=10^n);forcomposite(n=4,2*k+2,if(k--==0,return(n))) \\ Charles R Greathouse IV, May 30 2013

a(n) = A002808(A011557(n)).

a(n) = 10^(n + n/log n + 2n/log^2 + 4n/log^3 n + O(n/log^4 n)). See Bojarincev for an asymptotic expansion. - Charles R Greathouse IV, May 30 2013

More terms from Robert G. Wilson v, Nov 26 2001
a(14) from Lekraj Beedassy, Jul 14 2008
a(15)-a(19) from Chai Wah Wu, Apr 16 2018
a(20) from Chai Wah Wu, Aug 23 2018

A065863 Remainder when n-th prime is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 3, 5, 1, 2, 6, 3, 2, 3, 9, 6, 1, 11, 8, 9, 13, 14, 1, 16, 13, 12, 14, 13, 7, 5, 5, 1, 5, 1, 7, 7, 5, 5, 11, 7, 17, 13, 11, 7, 19, 25, 23, 19, 17, 17, 19, 23, 23, 23, 23, 19, 25, 23, 25, 29, 37, 35, 31, 29, 43, 43, 47, 43, 47, 47, 3, 2, 1, 53, 53, 55, 2, 3, 6, 1, 11, 6

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			For n=25, prime(25)=97, n - pi(n) = 25 - 9 = 16, a(25)=1 because 97 = 6*16 + 1.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A062298, A065858-A065864, A065134, A004648, A065133, A065134.

Table[Mod[Prime[n],n-PrimePi[n]],{n,90}] (* Harvey P. Dale, Aug 04 2015 *)

a(n) = { prime(n)%(n - primepi(n)) } \\ Harry J. Smith, Nov 02 2009

a(n) = prime(n) mod (n - pi(n)) = A000040(n) mod A062298(n).

A337979 Define a map f(n):= n-> n + pi(n) - pi(n + pi(n)), where pi(n) is the prime count of n (n>=1). a(n) is the number of steps for n to reach 1 under repeated iteration of f.

Original entry on oeis.org

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

Offset: 1

Ya-Ping Lu, Oct 05 2020

nonn

For any integer n > 1, pi(n + pi(n)) > pi(n) according to Lu and Deng (see Links). Thus, n + pi(n) - pi(n + pi(n)) < n, which means n is reduced by at least 1 every time map f is applied, eventually reaching 1 under repeated iteration of f.

It seems that the sequence contains all nonnegative integers.

			a(1) = 0 because f^0(1) = 1;
a(2) = 1 because f(2) = 2 + pi(2) - pi(2 + pi(2)) = 1;
a(4) = 3 because f^3(4) = f^2(f(4)) = f^2(3) = f(f(3)) = f(2) = 1.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.

Cf. A000720, A025003, A062298, A095117, A337978.

a:= proc(n) option remember; `if`(n=1, 0, 1+a((
      pi-> n+pi(n)-pi(n+pi(n)))(numtheory[pi])))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 24 2020

f[n_] := Module[{x = n + PrimePi[n]}, x - PrimePi[x]];
a[n_] := Module[{nb = 0, m = n}, While[m != 1, m = f[m]; nb++]; nb];
Array[a, 100] (* Jean-François Alcover, Oct 24 2020, after PARI code *)

f(n) = {my(x = n + primepi(n)); x - primepi(x);} \\ A337978
a(n) = {my(nb=0); while (n != 1, n = f(n); nb++); nb;} \\ Michel Marcus, Oct 06 2020

from sympy import primepi
print(0)
n = 2
for n in range (2, 10000001):
    ct = 0
    n_l = n
    pi_l = primepi(n)
    while ct >= 0:
        n_r = n_l + pi_l
        pi_r = primepi(n_r)
        n_l = n_r - pi_r
        pi_l = primepi(n_l)
        ct += 1
        if n_l == 1:
            print(ct)
            break

f^a(n) (n) = 1, where f = A062298(A095117) and m-fold iteration of f is denoted by f^m.

A092852 Number of composites <= A092802(n).

Original entry on oeis.org

2, 36, 412, 4371, 45118, 460161, 4663480, 47087659, 474329018, 4770493824, 47924729801, 481060418376, 4825817782189, 48387664042144, 484992123875142, 4859631205206357, 48681601698828085, 487571138851821274, 4882443976989269954, 48884842829781286250, 489393391263430721900

Offset: 1

Enoch Haga, Mar 07 2004

nonn

			Up to 10^1 there are 4 composites: 4 + 6 + 8 + 9 = 27. The rounded mean is A092802(1) = floor(27/4) = 7. There are 2 composites below 7: 4 and 6, so a(1) = 2.

Cf. A000720, A062298, A065855, A092802, A092853, A092854.

a(n) = A065855(A092802(n)) = A062298(A092802(n)) - 1 = A092802(n) - A000720(A092802(n)) - 1.

a(9)-a(15) from Max Alekseyev, Aug 14 2013

a(16)-a(21) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 05 2024

A157423 Triangle read by rows, T(n,k) = 0 if (n-k+1) is prime, else 1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 28 2009

Row sums = A062298: (1, 1, 1, 2, 2, 3, 3, 4, 5, 6,...). Eigensequence of the triangle = A052284: (1, 1, 1, 2, 3, 5, 7, 11, 17, 27,...).

			First few rows of the triangle =
1;
0, 1;
0, 0, 1;
1, 0, 0, 1;
0, 1, 0, 0, 1;
1, 0, 1, 0, 0, 1;
0, 1, 0, 1, 0, 0, 1;
1, 0, 1, 0, 1, 0, 0, 1;
1, 1, 0, 1, 0, 1, 0, 0, 1;
1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1;
...
Example: T(6,4) = 0 since (6 - 4 + 1) = 3, prime.

Cf. A005171, A062298, A052284, A157424

Table[If[PrimeQ[n-k+1],0,1],{n,15},{k,n}]//Flatten (* Harvey P. Dale, Jul 19 2016 *)

Triangle read by rows, T(n,k) = 0 if (n-k+1) is prime, else 1. By columns, A005171 in every column: where A005171(k) = 0 if k is prime.

A275306 Decimal expansion of 1/2 - Sum_{k>=1} 1/2^prime(k).

Original entry on oeis.org

0, 8, 5, 3, 1, 7, 4, 9, 0, 1, 4, 8, 8, 8, 8, 3, 3, 9, 7, 5, 1, 8, 9, 0, 3, 7, 7, 8, 4, 5, 6, 9, 2, 2, 9, 1, 6, 3, 4, 2, 2, 5, 7, 6, 1, 8, 6, 2, 0, 8, 3, 0, 2, 2, 1, 3, 1, 7, 5, 4, 5, 8, 5, 5, 1, 1, 3, 5, 9, 0, 3, 9, 3, 8, 0, 6, 4, 2, 6, 6, 5, 8, 0, 3, 7, 0, 9, 9, 5, 1, 5, 7, 1, 5, 2, 4, 2, 2, 2, 0, 6, 0, 3, 8, 3, 8, 4, 0, 6, 4, 7, 9, 1, 7, 0, 1, 4, 0, 4, 2, 1

Offset: 0

Ilya Gutkovskiy, Jul 22 2016

Composite constant: decimal value of A066247 interpreted as a binary number.
The characteristic function of composite numbers (A066247) has values 0, 0, 0, 1, 0, 1, 0, 1, 1, ... for n = 1, 2, 3, ... The constant obtained by concatenating these digits and interpreting them as a binary fraction is therefore C = 0.0001010111010... (base 2) = 0.0853174901...(base 10).
Continued fraction [0; 11, 1, 2, 1, 1, 2, 1, 1, 131, 2, 1, 1, 2, 6, 4, 2, 21, ...].

			0.0853174901... = (0.00010101110...)_2.
                        | | |||
                        4 6 8910

Simon Plouffe, Primes coded in binary to 1000 digits.
Eric Weisstein's World of Mathematics, Composite Number.
Eric Weisstein's World of Mathematics, Prime Constant.

Cf. A002808, A051006, A005171, A062298, A066247, A073718.

nn = 121; Take[#, nn] &@ PadLeft[First@ #, Abs@ Last@ # + Length@ First@ #] &@ RealDigits@ N[1/2 - Sum[ 1/2^Prime[k], {k, 10^4}], nn + 2] (* Michael De Vlieger, Jul 22 2016 *)

s=.5; forprime(p=2,bitprecision(s)+2, s-=1.>>p); s \\ Charles R Greathouse IV, Jul 22 2016

Equals Sum_{k>=1} 1/2^A002808(k).
From Amiram Eldar, Aug 11 2020: (Start)
Equals Sum_{k>=1} 1/A073718(k).
Equals Sum_{k>=1} A066247(k)/2^k.
Equals -(1/2) + Sum_{k>=1} A062298(k)/2^(k+1). (End)
Equals Sum_{k >= 1} ((-1)^A010051(k))/2^(k+1). - Antonio Graciá Llorente, Jan 13 2024

A377899 a(n) = number of composite numbers c_{2*k+1} <= n, where c_m = A002808(m) is the m-th composite number.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 14 2024

nonn

			c_1 = 4 and c_3 = 8 are <= 9, so a(9) = 2.

Cf. A000720, A002808, A062298, A099861.

from sympy import primepi
def A377899(n): return n-primepi(n)>>1 # Chai Wah Wu, Nov 14 2024

a(n) = floor((n-A000720(n))/2). - Chai Wah Wu, Nov 14 2024

A063084 a(n) = pi(n-1)n - pi(n)(n-1), where pi() = A000720().

Original entry on oeis.org

0, -1, -1, 2, -2, 3, -3, 4, 4, 4, -6, 5, -7, 6, 6, 6, -10, 7, -11, 8, 8, 8, -14, 9, 9, 9, 9, 9, -19, 10, -20, 11, 11, 11, 11, 11, -25, 12, 12, 12, -28, 13, -29, 14, 14, 14, -32, 15, 15, 15, 15, 15, -37, 16, 16, 16, 16, 16, -42, 17, -43, 18, 18, 18, 18, 18, -48, 19, 19, 19, -51, 20, -52, 21, 21, 21, 21, 21, -57, 22, 22, 22, -60, 23, 23

Offset: 1

Labos Elemer, Aug 06 2001

sign

To define as positive sequence let C(n)= A062298; f(a) = pi(a) if a is nonprime, f(a)= C(a) if a is prime. - Daniel Tisdale, Nov 07 2008

			The function is positive for composite and negative for prime numbers. It is zero at n=1.

G. A. Kudrevatow, (1970): Exercises in Number Theory. Problem 488; page 56; Prosveshenie, Moscow [in Russian].

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000720, A000027, A010051, A061397, A000040, A002808.

a(n)={if(n>1, primepi(n-1)*n - primepi(n)*(n-1), 0)} \\ Harry J. Smith, Aug 17 2009

cp's OEIS Frontend

A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2*a(n), a(not_an_oddprime(n)) = 2*a(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A337978 a(n) = n + pi(n) - pi(n + pi(n)), where pi(n) is the prime count of n (n>=1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A065857 The (10^n)-th composite number.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A065863 Remainder when n-th prime is divided by the number of nonprimes not exceeding n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A337979 Define a map f(n):= n-> n + pi(n) - pi(n + pi(n)), where pi(n) is the prime count of n (n>=1). a(n) is the number of steps for n to reach 1 under repeated iteration of f.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A092852 Number of composites <= A092802(n).

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A157423 Triangle read by rows, T(n,k) = 0 if (n-k+1) is prime, else 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A275306 Decimal expansion of 1/2 - Sum_{k>=1} 1/2^prime(k).

A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2a(n), a(not_an_oddprime(n)) = 2a(n-1).

A063084 a(n) = pi(n-1)n - pi(n)(n-1), where pi() = A000720().