A062298 -id:A062298 - cp's OEIS frontend

A085604 T(n,k) = highest power of prime(k) dividing n!, read by rows.

0, 1, 0, 1, 1, 0, 3, 1, 0, 0, 3, 1, 1, 0, 0, 4, 2, 1, 0, 0, 0, 4, 2, 1, 1, 0, 0, 0, 7, 2, 1, 1, 0, 0, 0, 0, 7, 4, 1, 1, 0, 0, 0, 0, 0, 8, 4, 2, 1, 0, 0, 0, 0, 0, 0, 8, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 10, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 10, 5, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 11, 5, 2, 2, 1, 1, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Jul 07 2003

T(n,1) = A011371(n); T(n,2) = A054861(n) for n>1;
T(n,k) = number of occurrences of prime(k) as factor in numbers <= n (with repetitions);
Sum{T(n,k): 1<=k<=n} = A022559(n);
T(n, A000720(n)) = 1; T(n,k) = 0, A000720(n)T(n,k) = A115627(n,k) for n > 1 and k=1..A000720(n). - Reinhard Zumkeller, Nov 01 2013

			0;
1,0;
1,1,0;
3,1,0,0;
3,1,1,0,0;
4,2,1,0,0,0;
4,2,1,1,0,0,0;
7,2,1,1,0,0,0,0;
7,4,1,1,0,0,0,0,0;
8,4,2,1,0,0,0,0,0,0;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A141809, A115627, A000142.

a085604 n k = a085604_tabl !! (n-2) !! (k-1)
a085604_row 1 = [0]
a085604_row n = a115627_row n ++ (take $ a062298 $ fromIntegral n) [0,0..]
a085604_tabl = map a085604_row [1..]
-- Reinhard Zumkeller, Nov 01 2013

T[n_, k_] := Module[{p = Prime[k], jm}, jm = Floor[Log[p, n]]; Sum[Quotient[n, p^j], {j, 1, jm}]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

A101203 a(n) = sum of nonprimes <= n.

Original entry on oeis.org

0, 1, 1, 1, 5, 5, 11, 11, 19, 28, 38, 38, 50, 50, 64, 79, 95, 95, 113, 113, 133, 154, 176, 176, 200, 225, 251, 278, 306, 306, 336, 336, 368, 401, 435, 470, 506, 506, 544, 583, 623, 623, 665, 665, 709, 754, 800, 800, 848, 897, 947, 998, 1050, 1050, 1104, 1159, 1215

Offset: 0

Reinhard Zumkeller, Jan 23 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A062298, A018252.
Partial sums of A191558.
Cf. A000217, A034387, A101256.

a101203 n = a101203_list !! (n-1)
a101203_list = scanl (+) 0 $ zipWith (*) [1..] $ map (1 -) a010051_list
-- Reinhard Zumkeller, Oct 10 2013

Accumulate[Table[If[PrimeQ[n],0,n],{n,0,60}]] (* Harvey P. Dale, Oct 02 2020 *)

my(s=0); for(k=0,100,if(!isprime(k),s+=k);print1(s", ")); \\ Cino Hilliard, Feb 04 2006

a(n) = A000217(n) - A034387(n) = A101256(n) + 1.

A065856 The (2^n)-th composite number.

Original entry on oeis.org

4, 6, 9, 15, 26, 48, 88, 168, 323, 627, 1225, 2406, 4736, 9351, 18504, 36655, 72730, 144450, 287147, 571208, 1136971, 2264215, 4510963, 8990492, 17923944, 35743996, 71298762, 142249762, 283859985, 566537515, 1130886504, 2257704401, 4507834166, 9001524190

Offset: 0

Labos Elemer, Nov 26 2001

nonn

a(n) = A002808(A000079(n)).

			composite[1] = composite[2^0] = 4, composite[2] = composite[2^1] = 6, composite[1024] = composite[2^10] = 1225, composite[1073741824] = composite[2^30] = 1130886504.

Chai Wah Wu, Table of n, a(n) for n = 0..70

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

Composite[n_Integer] := Block[ {k = n + PrimePi[n] + 1 }, While[ k != n + PrimePi[k] + 1, k = n + PrimePi[k] + 1]; Return[ k ]]; Table[ Composite[2^n], {n, 0, 36} ]

a(n)-pi(a(n))-1 = 2^n.

More terms from Robert G. Wilson v, Nov 26 2001
Definition corrected by N. J. A. Sloane, Jun 07 2009
Further corrections from Reinhard Zumkeller, Jun 24 2009
a(32)-a(33) from Chai Wah Wu, Apr 16 2018

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

Original entry on oeis.org

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

A257730 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = prime(a(n)), a(not_an_oddprime(n)) = composite(a(n-1)).

Original entry on oeis.org

1, 4, 2, 9, 7, 6, 3, 16, 14, 12, 23, 8, 17, 26, 24, 21, 13, 35, 5, 15, 27, 39, 53, 36, 33, 22, 51, 10, 43, 25, 37, 40, 56, 75, 52, 49, 83, 34, 72, 18, 19, 62, 59, 38, 54, 57, 101, 78, 102, 74, 69, 114, 89, 50, 98, 28, 30, 86, 73, 82, 41, 55, 76, 80, 134, 106, 149, 135, 100, 94, 11, 150, 47, 120, 70, 130, 42, 45, 103, 117, 99, 112, 167, 58, 77

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here composite(n) = n-th composite = A002808(n), prime(n) = n-th prime = A000040(n), 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).

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

Inverse: A257729.
Cf. A000040, A000720, A002808, A010051, A062298, A065090, A065091.
Related or similar permutations: A246378, A257727, A257732, A257801, A236854.

a(1) = 1; if A000035(n) = 1 and A010051(n) = 1 [i.e., when n is an odd prime], then a(n) = A000040(a(A000720(n)-1)), otherwise a(n) = A002808(a(A062298(n))). [Here A062298(n) gives the index of n among numbers larger than 1 which are not odd primes, 1 for 2, 2 for 4, 3 for 6, etc.]
As a composition of other permutations:
a(n) = A246378(A257727(n)).
a(n) = A257732(A257801(n)).

A257801 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = lucky(1+a(n)), a(not_an_oddprime(n)) = unlucky(a(n-1)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 6, 11, 8, 13, 14, 25, 10, 17, 12, 15, 19, 33, 20, 35, 16, 21, 24, 18, 22, 27, 45, 43, 28, 31, 47, 23, 29, 34, 26, 51, 30, 38, 59, 63, 57, 115, 39, 42, 61, 37, 32, 40, 46, 36, 66, 73, 41, 52, 78, 83, 76, 49, 146, 67, 53, 56, 81, 50, 44, 79, 54, 60, 48, 163, 86, 87, 95, 55, 68, 101, 107, 171, 98, 64

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here lucky(n) = n-th lucky number = A000959(n), unlucky(n) = n-th unlucky number = A050505(n), oddprime(n) = n-th odd prime = A065091(n), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).

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

Inverse: A257802.
Cf. A000959, A000720, A010051, A050505, A062298, A065090, A065091.
Related or similar permutations: A257726, A257727, A257730, A257731.

a(1) = 1; a(2) = 2; if A010051(n) = 1 [i.e., when n is an (odd) prime] then a(n) = A000959(1+a(A000720(n)-1)), otherwise a(n) = A050505(a(A062298(n))).
As a composition of other permutations:
a(n) = A257726(A257727(n)).
a(n) = A257731(A257730(n)).

A316434 a(n) = a(pi(n)) + a(n-pi(n)) with a(1) = a(2) = 1.

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Jul 02 2018

nonn

This sequence hits every positive integer.

Robert Israel, Table of n, a(n) for n = 1..10000
Altug Alkan, A plot of a(n)/n
Altug Alkan and Orhan Ozgur Aybar, On a Family of Sequences Related to Prime Counting Function, International Journal of Statistics and Probability Vol. 7, No. 6; 2018.
Rémy Sigrist, C++ program for A316434

Cf. A000720, A062298.

f:= proc(n) option remember: local p;
     p:= numtheory:-pi(n);
     procname(p) + procname(n-p)
end proc:
f(1):= 1: f(2):= 1:
map(f, [$1..100]); # Robert Israel, Jul 03 2018

a[1]=a[2]=1; a[n_] := a[n] = a[PrimePi[n]] + a[n - PrimePi[n]]; Array[a, 75] (* Giovanni Resta, Nov 02 2018 *)

q=vector(75); for(n=1, 2, q[n] = 1); for(n=3, #q, q[n] = q[primepi(n)] + q[n-primepi(n)]); q
(C++) See Links section.

from sympy import primepi
def A316434(n):
    pp = primepi(n)
    return 1 if n == 1 or n == 2 else A316434(pp) + A316434(n-pp) # Chai Wah Wu, Nov 02 2018

a(n) = a(A000720(n)) + a(A062298(n)) with a(1) = a(2) = 1.

a(n+1) - a(n) = 0 or 1 for all n >= 1.

A025003 a(1) = 2; a(n+1) = a(n)-th nonprime, where nonprimes begin at 1.

Original entry on oeis.org

2, 4, 8, 14, 22, 33, 48, 66, 90, 120, 156, 202, 256, 322, 400, 494, 604, 734, 888, 1067, 1272, 1512, 1790, 2107, 2472, 2890, 3364, 3903, 4515, 5207, 5990, 6875, 7868, 8984, 10238, 11637, 13207, 14959, 16909, 19075, 21483, 24173, 27149, 30436, 34080, 38103

Offset: 1

David W. Wilson

nonn

Index of first occurrence of n in A090532.

Let b(n) (n >= 0) be the smallest integer k >= 1 that takes n steps to reach 1 iterating the map f: k -> k - pi(k). The sequence {b(n), n >= 0} begins 1, 2, 4, 8, 14, 22, 33, 48, 66, 90, 120, 156, ... and agrees with the present sequence except for b(0). - Ya-Ping Lu, Sep 07 2020

			From _Ya-Ping Lu_, Sep 07 2020: (Start)
a(1) = 2 because f(2) = 2 - pi(2) = 1 and m(2) = 1;
For the integer 3, since f(3) = 1. m(3) = 1, which is not bigger than m(1) or m(2). So, 3 is not a term in the sequence;
a(2) = 4 because f^2(4) = f(2) = 1 and m(4) = 2;
a(3) = 8 because f^3(8) = f^2(4) = 1 and m(8) = 3. (End)

Cf. A000040, A000720, A014688, A014689, A062298, A090532, A332086, A337334.

N:= 50: # to get a(0)..a(N)
V:= Array(0..N):
V[0]:= 1: V[1]:= 2:
m:= 2: p:= 3: g:= 1: n:= 1:
do
  if g+p-m-1 >= V[n] then
    m:= V[n]+m-g;
    n:= n+1;
    V[n]:= m;
    if n = N then break fi;
    g:= V[n-1];
  else
    g:= g+p-m;
    m:= p+1;
    p:= nextprime(m);
  fi;
od;
convert(V, list); # Robert Israel, Sep 08 2020

from sympy import prime, primepi
n_last = 0
pi_last = 0
ct_max = -1
for n in range(1, 100001):
    ct = 0
    pi = pi_last + primepi(n) - primepi(n_last)
    n_c = n
    pi_c = pi
    while n_c > 1:
        nc -= pi_c
        ct += 1
        pi_c -= primepi(n_c + pi_c) - primepi(n_c)
    if ct > ct_max:
        print(n)
        ct_max = ct
    n_last = n
    pi_last = pi # Ya-Ping Lu, Sep 07 2020

a(n) = min(k: f^n(k) = 1), where f = A062298 and n-fold iteration of f is denoted by f^n. - Ya-Ping Lu, Sep 07 2020

A161865 Numerators of ratio of nonprimes in a square interval to that of nonprimes in that interval and its successor.

Original entry on oeis.org

1, 3, 5, 2, 1, 3, 12, 13, 1, 16, 19, 10, 22, 1, 25, 13, 30, 31, 33, 17, 18, 38, 41, 40, 43, 46, 47, 16, 51, 1, 53, 56, 19, 60, 61, 32, 66, 65, 68, 23, 18, 76, 25, 1, 78, 83, 1, 82, 89, 45, 88, 89, 95, 24, 100, 101, 49, 104, 103, 21, 55, 27, 112, 1, 115, 59, 1, 20, 21, 15, 64, 1

Offset: 1

Daniel Tisdale, Jun 20 2009

			First few terms are 1/4, 3/8, 5/11, 2/5, 1/2, 3/7, 12/25, 13/29.
For n=1: there is 1 nonprime <= 1, 2 nonprimes <= 4, and 5 nonprimes <= 9. The ratio is (2 - 1)/(5 - 1) = 1/4.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A161621, A161622, A161867 (denominators for this sequence).

A062298 := proc(n) n-numtheory[pi](n) ; end: A078435 := proc(n) A062298(n^2) ; end: A161865 := proc(n) r := [ A078435(n),A078435(n+1),A078435(n+2)] ; (r[2]-r[1])/(r[3]-r[1]) ; numer(%) ; end: seq(A161865(n),n=1..120) ; # R. J. Mathar, Sep 27 2009

Numerator[Table[((2 n + 1) - (PrimePi[(n + 1)^2] - PrimePi[n^2]))/((4 n + 4) - (PrimePi[(n + 2)^2] - PrimePi[n^2])), {n, 1, 40}]] (* corrected by G. C. Greubel, Dec 20 2016 *)

The limit of this sequence is 1/2, as can be shown by setting an increasing lower bound on the ratio of composites in successive square intervals.

Extended beyond a(8) by R. J. Mathar, Sep 27 2009

A255573 a(n) = Number of terms of A205783 (including 1) in range 0 .. n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 14, 15, 15, 16, 17, 18, 18, 19, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 36, 37, 37, 38, 39, 40, 40, 41, 41, 42, 43, 44, 45, 46, 46, 47, 47, 48, 48, 49, 49, 50, 51, 52, 52

Offset: 0

Antti Karttunen, May 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Essentially one more than A255572 (after the initial zero).

Cf. A062298, A205783, A255574.

A255573_write_bfile(up_to_n) = { my(n,a_n=0); for(n=0, up_to_n, if(((n > 0) && !polisirreducible(Pol(binary(n)))),a_n++); write("b255573.txt", n, " ", a_n)); };
A255573_write_bfile(8192);

a(n) = n - A255574(n).
Other identities and observations. For all n >= 1:
a(n) = 1 + A255572(n).
a(n) <= A062298(n).

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cp's OEIS Frontend

A085604 T(n,k) = highest power of prime(k) dividing n!, read by rows.

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A101203 a(n) = sum of nonprimes <= n.

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A065856 The (2^n)-th composite number.

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

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A257730 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = prime(a(n)), a(not_an_oddprime(n)) = composite(a(n-1)).

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A257801 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = lucky(1+a(n)), a(not_an_oddprime(n)) = unlucky(a(n-1)).

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A316434 a(n) = a(pi(n)) + a(n-pi(n)) with a(1) = a(2) = 1.

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A025003 a(1) = 2; a(n+1) = a(n)-th nonprime, where nonprimes begin at 1.

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