A062298 -id:A062298 - cp's OEIS frontend

A065862 Remainder when n-th composite number is divided by the number of nonprimes not exceeding n.

0, 0, 0, 1, 0, 0, 2, 3, 1, 0, 2, 0, 1, 0, 7, 6, 7, 6, 8, 8, 7, 6, 7, 6, 6, 5, 4, 4, 6, 5, 6, 6, 5, 4, 3, 2, 4, 3, 2, 1, 2, 2, 4, 3, 2, 1, 2, 2, 1, 0, 0, 0, 1, 0, 38, 38, 39, 39, 40, 41, 42, 42, 42, 42, 43, 43, 44, 44, 44, 44, 45, 46, 47, 47, 48, 49, 49, 49, 51, 52, 52, 52, 54, 54, 54, 54, 54

Offset: 1

Labos Elemer, Nov 26 2001

nonn

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

Cf. A002808, A000720, A062298, A065858-A065864, A065134.

Module[{nn=150,cmps,len},cmps=Select[Range[nn],CompositeQ];len=Length[ cmps];Mod[#[[1]],#[[2]]-PrimePi[#[[2]]]]&/@Thread[{cmps,Range[len]}]] (* Harvey P. Dale, Feb 21 2020 *)

Composite(n) = { local(k); k=n + primepi(n) + 1; while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); return(k) } { for (n = 1, 1000, a=Composite(n)%(n - primepi(n)); write("b065862.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

a(n) = c(n) mod (n - pi(n)) = A002808(n) mod (n - A000720(n)) = A002808(n) mod A062298(n).

A168048 a(n) = C(n)*Pi(n) where C(n) = number of nonprimes <= n, Pi(n) = number of primes <= n.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 16, 20, 24, 30, 35, 42, 48, 54, 60, 70, 77, 88, 96, 104, 112, 126, 135, 144, 153, 162, 171, 190, 200, 220, 231, 242, 253, 264, 275, 300, 312, 324, 336, 364, 377, 406, 420, 434, 448, 480, 495, 510, 525, 540, 555, 592, 608, 624, 640, 656, 672

Offset: 1

Daniel Tisdale, Nov 17 2009

nonn

Cf. A000720, A062298.

A000720 := proc(n) numtheory[pi](n) ; end proc: A168048 := proc(n) pi := A000720(n) ; pi*(n-pi) ; end proc: seq(A168048(n),n=1..120) ; # R. J. Mathar, Nov 18 2009

a(n) = A062298(n)*A000720(n).

a(1) and terms after a(16) from R. J. Mathar, Nov 18 2009

Edited by Jon E. Schoenfield, May 10 2019

A279436 Number of nonprimes less than or equal to n that do not divide n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 3, 4, 5, 3, 6, 6, 7, 6, 9, 7, 10, 8, 11, 12, 13, 9, 14, 15, 15, 15, 18, 15, 19, 16, 20, 21, 22, 18, 24, 24, 25, 22, 27, 24, 28, 26, 27, 30, 31, 25, 32, 31, 34, 33, 36, 32, 37, 34, 39, 40, 41, 34, 42, 42, 41, 40, 45, 43, 47, 45, 48, 46, 50, 42, 51, 51, 50, 51, 54, 52, 56, 50, 55, 58, 59, 52, 60, 61, 62, 59, 64, 57, 65, 64, 67, 68, 69, 62, 71, 69, 70, 68

Offset: 1

Ilya Gutkovskiy, Dec 12 2016

nonn

			a(10) = 4 because 10 has 4 divisors {1,2,5,10} therefore 6 non-divisors {3,4,6,7,8,9} out of which 4 are nonprimes {4,6,8,9}.

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

Cf. A000005, A000040, A000720, A001221, A014689, A033273, A048865, A049820, A062298, A065890, A082514, A229109.

Table[n - PrimePi[n] - DivisorSigma[0, n] + PrimeNu[n], {n, 1, 100}]

for(n=1,50, print1(n - primepi(n) - numdiv(n) + omega(n), ", ")) \\ G. C. Greubel, May 22 2017

first(n)=my(v=vector(n),pp); forfactored(k=1,n, if(k[2][,2]==[1]~, pp++); v[k[1]]=k[1] - pp - numdiv(k) + omega(k)); v \\ Charles R Greathouse IV, May 23 2017

from sympy import primepi, divisor_count, primefactors
def a(n): return 0 if n==1 else n - primepi(n) - divisor_count(n) + len(primefactors(n)) # Indranil Ghosh, May 23 2017

G.f.: A(x) = B(x) + C(x) - D(x), where B(x) = Sum_{k>=1} x^(2*k+1)/((1 - x^k)*(1 - x^(k+1))), C(x) = Sum_{k>=1} x^prime(k)/(1 - x^prime(k)), D(x) = Sum_{k>=1} x^prime(k)/(1 - x).
a(n) = n - A000720(n) - A000005(n) + A001221(n).
a(n) = A062298(n) - A033273(n).
a(n) = A049820(n) - A048865(n).
a(n) = A229109(n) - A082514(n).
a(A000040(n)) = A065890(n).
a(A000040(n)) + 1 = A014689(n).
A000040(n) - a(A000040(n)) = n + 1.

A334614 a(n) = pi(prime(n) - n) + n, where pi is the prime counting function.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 11, 13, 15, 18, 19, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 45, 47, 48, 50, 51, 53, 55, 57, 60, 61, 65, 66, 67, 68, 70, 72, 74, 76, 77, 79, 81, 82, 85, 88, 89, 91, 93, 94, 95, 99, 101, 102, 104, 105, 106, 107, 108, 112, 116, 117

Offset: 1

Ya-Ping Lu, Sep 08 2020

nonn

It can be shown that a(n) > a(n-1) >= 1 and a(n) <= 2*n - 1 < 2*n (see proofs in the Links section).

Ya-Ping Lu, Proofs of the two observations in the Comments section

Cf. A000040, A000720, A014688, A014689, A062298, A065328, A097933.

Cf. A332086, A337334.

Table[PrimePi[Prime[n] - n] + n, {n, 1, 64}] (* Amiram Eldar, Sep 09 2020 *)

a(n) = n + primepi(prime(n) - n); \\ Michel Marcus, Sep 09 2020

from sympy import prime, primepi
for n in range(1, 100001):
    a_n = primepi(prime(n) - n) + n
    print(a_n)

a(n) = A000720(A014689(n)) + n.

a(n) = A065328(n) + n. - Michel Marcus, Sep 12 2020

A338260 a(n) is the number of nodes with depth of n in a binary tree defined as: root = 1 and a child (C) of a node (N) is such that A337978(C) = N. For nodes with two children, the smaller child is assigned as the left child and the bigger one as the right child. A child of a one-child node is assigned as the left child.

Original entry on oeis.org

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

Offset: 0

Ya-Ping Lu, Oct 19 2020

nonn

The binary tree, read from left to right in the order of increasing depth n, is the positive integer sequence A000027. The first 66 numbers are shown in the figure below.
                         1
                        2
                       3
                      4
                     5
                    6
                   7
                  8 \_(9)
                10
               11 \_12
              13   14
             15   16
           (17)  18
                19
               20 \_21
              22   23
             24   25
           (26)  27 \______28
                29        30
               31 \_32  (33)
              34   35 \______36
             37   38        39
           (40)  41        42
                43        44 \_45
               46        47   48
             (49)       50   51 \______52
                       53  (54)\_55   56 \______57
                      58        59   60        61
                    (62)       63   64 \_65  (66)
All right children are composite numbers and all prime numbers are left children.
a(n) in this sequence is the number of terms with value of n in A337979.

Cf. A000027, A062298, A095117, A337978, A337979.

from sympy import primepi
def depth(k):
    d = 0
    while k > 1:
        k += primepi(k)
        k -= primepi(k)
        d += 1
    return d
m = 1
for n in range (0, 101):
    a = 0
    while depth(m + a) == n:
        a += 1
    print(a)
    m += a

A338521 The number of primes between n-primepi(n) and n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 4, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Ya-Ping Lu, Nov 01 2020

nonn

There is at least one prime number between n-primepi(n) and n, or a(n) >= 1, for n >= 3 (see Corollary 3 in the paper by Ya_Ping Lu attached in the links).

Ya-Ping Lu, Lower Bounds for the Number of Primes in Some Integer Intervals

Cf. A000720, A062298, A337788.

Array[Subtract @@ Map[PrimePi, {#1 - 1, #1 - #2}] & @@ {#, PrimePi[#]} &, 105] (* Michael De Vlieger, Nov 05 2020 *)

a(n) = primepi(n - 1) - primepi(n - primepi(n)); \\ Michel Marcus, Nov 01 2020

from sympy import primepi
for n in range(1, 101):
    pi = primepi(n)
    pi_1 = primepi(n - 1)
    a = pi_1 - primepi(n - pi)
    print(a)

a(n) = primepi(n - 1) - primepi(n - primepi(n)).
a(n) = A000720(n - 1) - A000720(n - A000720(n)).
a(n) = A000720(n -1) - A000720(A062298(n)).

A349214 a(n) = Sum_{k=1..n} k^c(k), where c is the prime characteristic (A010051).

Original entry on oeis.org

1, 3, 6, 7, 12, 13, 20, 21, 22, 23, 34, 35, 48, 49, 50, 51, 68, 69, 88, 89, 90, 91, 114, 115, 116, 117, 118, 119, 148, 149, 180, 181, 182, 183, 184, 185, 222, 223, 224, 225, 266, 267, 310, 311, 312, 313, 360, 361, 362, 363, 364, 365, 418, 419, 420, 421, 422, 423, 482, 483, 544

Offset: 1

Wesley Ivan Hurt, Nov 10 2021

nonn

For k in 1 <= k <= n, add k if k is prime, otherwise add 1. For example a(6) = 1 + 2 + 3 + 1 + 5 + 1 = 13.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Partial sums of A089026.

Cf. A010051, A034387, A062298.

a[n_] := Sum[k^Boole[PrimeQ[k]], {k, 1, n}]; Array[a, 60] (* Amiram Eldar, Nov 11 2021 *)

a(n) = sum(k=1, n, if (isprime(k), k, 1)); \\ Michel Marcus, Nov 11 2021

from sympy import primerange
def A349214(n):
    p = list(primerange(2,n+1))
    return n-len(p)+sum(p) # Chai Wah Wu, Nov 11 2021

a(n) = A034387(n) + A062298(n). - Wesley Ivan Hurt, Nov 23 2021

A073725 a(n)-th composite number = phi(n-th composite number); a(1)=a(2)=0.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 2, 3, 3, 2, 3, 6, 5, 3, 11, 6, 10, 6, 3, 9, 11, 9, 14, 6, 10, 14, 9, 6, 11, 14, 13, 9, 28, 11, 20, 14, 10, 27, 14, 24, 18, 9, 19, 24, 20, 32, 11, 20, 29, 14, 14, 24, 27, 24, 42, 14, 20, 37, 27, 14, 45, 28, 39, 27, 14, 51, 29, 42, 31, 51, 20, 28, 42, 27, 20, 32, 32

Offset: 1

Labos Elemer, Aug 05 2002

nonn

			Phi of 25th composite number = 10th composite number: n=25: A002808(25)=38, phi(38) = 18 = A002808(10) so a(25)=10.

Cf. A000010, A002808, A062298.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x]; G[x_] := x-PrimePi[x]-1; a(n)=G[EulerPhi[c[n]]]

See program.

A102613 Numerator of the reduced fractions of the ratios of the number of primes less than n over the number of composites less than n.

Original entry on oeis.org

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

Offset: 1

Cino Hilliard, Jan 30 2005

Conjecture: The ratio pi(x)/(n-pi(x)) tends to 0 as n tends to infinity. This is evident from the fact that Li(x)/(n-Li(x)) -> 0 as n -> infinity but unfortunately not proof.

Cf. A000720, A062298.

pixovcmpx(n) = for(x=1,n,print1(numerator(pi(x)/(x-pi(x)))",")) pi(n) = \Number of primes less than or equal to n. { local(c,x); c=0;forprime(x=1,n,c++);return(c) }

a(n)=numerator(primepi(n)/(n-primepi(n))) \\ Jason Yuen, Aug 31 2024

a(n) = numerator(pi(n)/(n-pi(n))) = numerator(A000720(n)/A062298(n)). - Jason Yuen, Aug 31 2024

A131872 Set m = 0, n = 1. Find smallest k >= 2 such that pi(k) = (k-pi(k)) - (m-pi(m)); set a(n) = pi(k), m = k, n = n+1. Repeat.

Original entry on oeis.org

1, 4, 8, 11, 16, 23, 30, 39, 50, 62, 78, 97, 119, 141, 172, 205, 242, 284, 334, 393, 455, 531, 615, 704, 811, 928, 1059, 1213, 1373, 1560, 1761, 1988, 2239, 2524, 2833, 3180, 3557, 3983, 4448, 4942, 5503, 6126, 6791, 7522, 8331, 9228, 10188, 11228

Offset: 1

Manuel Valdivia, Oct 05 2007

nonn

For n>1, a(n)-a(n-1) is approximately pi(n)^2/n.

			m=0, n=1; pi(2) = (2-1)-(0) = 1 = number of nonprimes from 1 to 2, a(1) = 1 is a term. Now n=2, m=2.
pi(9) = (9-4)-(2-1) = 4 = number of nonprimes from 3 to 9, a(2) = 4 is a term. Now n=3, m=9.
pi(21) = (21-8)-(9-4) = 8 = number of nonprimes from 10 to 21, a(3) = 8 is a term.

A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Eric Weisstein's World of Mathematics, Prime Counting Function

Cf. A000720, A062298.

m=0; Do[If[PrimePi[n]==(n-PrimePi[n])-(m-PrimePi[m]), Print[PrimePi[n]]; m=n], {n, 1, 10^6, 1}]

lista(nn) = my(m=0, list = List()); for (n=1, nn, my(k=2); while(primepi(k) != (k-primepi(k)) - (m-primepi(m)), k++); listput(list, primepi(k)); m = k;); Vec(list); \\ Michel Marcus, Nov 13 2023

Edited by N. J. A. Sloane, Nov 05 2007

cp's OEIS Frontend

A065862 Remainder when n-th composite number is divided by the number of nonprimes not exceeding n.

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A168048 a(n) = C(n)*Pi(n) where C(n) = number of nonprimes <= n, Pi(n) = number of primes <= n.

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A279436 Number of nonprimes less than or equal to n that do not divide n.

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A334614 a(n) = pi(prime(n) - n) + n, where pi is the prime counting function.

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A338521 The number of primes between n-primepi(n) and n.

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A349214 a(n) = Sum_{k=1..n} k^c(k), where c is the prime characteristic (A010051).

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A073725 a(n)-th composite number = phi(n-th composite number); a(1)=a(2)=0.

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