A175944 -id:A175944 - cp's OEIS frontend

A018252 The nonprime numbers: 1 together with the composite numbers, A002808.

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88

Offset: 1

N. J. A. Sloane

d(a(n)) != 2 (cf. A000005). - Juri-Stepan Gerasimov, Oct 17 2009
Number of prime divisors of a(n) (counted with multiplicity) != 1. - Juri-Stepan Gerasimov, Oct 30 2009
Largest nonprime < n-th composite. - Juri-Stepan Gerasimov, Oct 29 2009
The nonnegative nonprimes A141468 without zero; the natural nonprimes; the whole nonprimes; the counting nonprimes. If the nonprime numbers A141468 which are also the nonnegative integers A001477, then the nonprimes A141468 also called the nonnegative nonprimes. If the nonprime numbers A018252 which are also the natural (or whole or counting) numbers A000027, then the nonprimes A018252 also called the natural nonprimes, the whole nonprimes and the counting nonprimes. - Juri-Stepan Gerasimov, Nov 22 2009
Smallest nonprime > n-th nonnegative nonprime. - Juri-Stepan Gerasimov, Dec 04 2009
a(n) = A175944(A014284(n)) = A175944(A175965(n)). - Reinhard Zumkeller, Mar 18 2011

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

N. J. A. Sloane, List of nonprimes up to 20000: Table of n, a(n) for n = 1..17738
Eric Weisstein's World of Mathematics, Monica Set
Eric Weisstein's World of Mathematics, Suzanne Set
Index entries for "core" sequences

Cf. A000040 (complement), A002808.
Cf. A000005, A001222, A141468.
Boustrophedon transforms: A230955, A230954.
Cf. A010051, A239968.

A018252 := Difference([1..10^5], Filtered([1..10^5], IsPrime)); # Muniru A Asiru, Oct 21 2017

a018252 n = a018252_list !! (n-1)
a018252_list = filter ((== 0) . a010051) [1..]
-- Reinhard Zumkeller, Mar 31 2014

Magma
```
[n : n in [1..100] | not IsPrime(n) ];
```

with(numtheory); sort(convert(convert([ seq(i,i=1..541) ],set) minus convert([ seq(ithprime(i),i=1..100) ],set),list));
seq(`if`(not isprime(n),n,NULL),n=1..88); # Peter Luschny, Jul 29 2009
A018252 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do; end if; end proc: # R. J. Mathar, Oct 22 2010

nonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n + PrimePi@ n]; Array[ nonPrime, 75] (* Robert G. Wilson v, Jan 29 2015, based on the algorithm by Labos Elemer in A006508 *)
max = 90; Complement[Range[max], Prime[Range[PrimePi[max]]]] (* Harvey P. Dale, Aug 12 2011 *)
Join[{1}, Select[Range[100], CompositeQ]] (* Jean-François Alcover, Nov 07 2021 *)

isA018252(n) = !isprime(n)
A018252(n) = {local(a,b);b=n;a=1;while(a!=b,a=b;b=n+primepi(a));b} \\ Michael B. Porter, Nov 06 2009

a(n) = my(k=0); while(-n+n-=k-k=primepi(n), ); n; \\ Ruud H.G. van Tol, Jul 15 2024 (after code in A002808)

from sympy import isprime
def ok(n): return not isprime(n)
print([k for k in range(1, 89) if ok(k)]) # Michael S. Branicky, Nov 10 2022

from sympy import composite
def A018252(n): return 1 if n == 1 else composite(n-1) # Chai Wah Wu, Nov 15 2022

def A018252_list(n) :
    return [k for k in (1..n) if not k.is_prime()]
A018252_list(88)  # Peter Luschny, Feb 03 2012

Let b(0) = n + pi(n) and b(n+1) = n + pi(b(n)), with pi(n) = A000720(n); then a(n) is the limit value of b(n). - Floor van Lamoen, Oct 08 2001
a(n) = A137621(A137624(n)). - Reinhard Zumkeller, Jan 30 2008
A010051(a(n)) = 0. - Reinhard Zumkeller, Mar 31 2014
A239968(a(n)) = n. - Reinhard Zumkeller, Dec 02 2014

A014284 Partial sums of primes, if 1 is regarded as a prime (as it was until quite recently, see A008578).

Original entry on oeis.org

1, 3, 6, 11, 18, 29, 42, 59, 78, 101, 130, 161, 198, 239, 282, 329, 382, 441, 502, 569, 640, 713, 792, 875, 964, 1061, 1162, 1265, 1372, 1481, 1594, 1721, 1852, 1989, 2128, 2277, 2428, 2585, 2748, 2915, 3088, 3267, 3448, 3639, 3832, 4029

Offset: 1

Deepan Majmudar (dmajmuda(AT)esq.com)

Lexicographically earliest sequence whose first differences are an increasing sequence of primes. Complement of A175969. - Jaroslav Krizek, Oct 31 2010
A175944(a(n)) = A018252(n). - Reinhard Zumkeller, Mar 18 2011
Partial sums of noncomposite numbers (A008578). - Omar E. Pol, Aug 09 2012

			a(7) = 42 because the first six primes (2, 3, 5, 7, 11, 13) add up to 41, and 1 + 41 = 42.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007504.
Equals A036439(n) - 1.
Cf. A175965, A175966, A175967, A175968, A175969, A051349, A175970. - Jaroslav Krizek, Oct 31 2010
Cf. A008578.

a014284 n = a014284_list !! n
a014284_list = scanl1 (+) a008578_list
-- Reinhard Zumkeller, Mar 26 2015

[1] cat [1 + (&+[NthPrime(j): j in [1..n]]): n in [1..50]]; // G. C. Greubel, Jun 18 2019

A014284 := proc(n)
    add(A008578(i),i=1..n) ;
end proc:
seq(A014284(n),n=1..60) ; # R. J. Mathar, Mar 05 2017

Join[{1}, Table[1+Sum[Prime[j], {j,1,n}], {n,1,50}]] (* Vladimir Joseph Stephan Orlovsky, Sep 25 2009, modified by G. C. Greubel, Jun 18 2019 *)
Accumulate[Join[{1}, Prime[Range[45]]]] (* Alonso del Arte, Oct 09 2012 *)

concat([1], vector(50, n, 1 + sum(j=1,n, prime(j)) )) \\ G. C. Greubel, Jun 18 2019

[1]+[1 + sum(nth_prime(j) for j in (1..n)) for n in (1..50)] # G. C. Greubel, Jun 18 2019

a(n) = Sum_{k <= n} [(A158611(k + 1)) * (A000012(n - k + 1))] = Sum_{k <= n} [(A158611(k + 1)) * (A000012(k))] = Sum_{k <= n} [(A008578(k)) * (A000012(n - k + 1))] = Sum_{k <= n} [(A008578(k)) * (A000012(k))] for n, k >= 1. - Jaroslav Krizek, Aug 05 2009
a(n + 1) = A007504(n) + 1. a(n + 1) - a(n) = A000040(n) = n-th primes. - Jaroslav Krizek, Aug 19 2009
a(n) = a(n-1) + prime(n-1), with a(1)=1. - Vincenzo Librandi, Jul 27 2013
G.f: (x*(1+b(x)))/(1-x) = c(x)/(1-x), where b(x) and c(x) are respectively the g.f. of A000040 and A008578. - Mario C. Enriquez, Dec 10 2016

Correction for Aug 2009 change of offset in A158611 and A008578 by Jaroslav Krizek, Jan 27 2010

A175965 Lexicographically earliest sequence with first differences as increasing sequence of noncomposites A008578.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 43, 60, 79, 102, 131, 162, 199, 240, 283, 330, 383, 442, 503, 570, 641, 714, 793, 876, 965, 1062, 1163, 1266, 1373, 1482, 1595, 1722, 1853, 1990, 2129, 2278, 2429, 2586, 2749, 2916, 3089, 3268, 3449, 3640, 3833, 4030, 4229, 4440, 4663

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Complement of A175966.

A175944(a(n)) = A018252(n). - Reinhard Zumkeller, Mar 18 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008578, A014284, A018252, A036439, A051349.

Cf. A175944, A175966, A175967, A175968, A175969, A175970.

a175965 n = a175965_list !! n
a175965_list = scanl (+) 1 a008578_list
-- Reinhard Zumkeller, Mar 26 2015

FoldList[Plus,1,Join[{1},Prime[Range[50]]]] (* or *) Accumulate[ Join[ {1,1},Prime[Range[50]]]] (* Harvey P. Dale, Sep 28 2016 *)

a(n) = A036439(n-1) for n > 1.
a(n) - a(n-1) = A008578(n-1) for n > 1.
a(n) = A014284(n-1) + 1 for n > 1.

A005145 n copies of n-th prime.

Original entry on oeis.org

2, 3, 3, 5, 5, 5, 7, 7, 7, 7, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31

Offset: 1

Russ Cox

Seen as a triangle read by rows: T(n,k) = A000040(n), 1 <= k <= n; row sums = A033286; central terms = A031368. - Reinhard Zumkeller, Aug 05 2009

Seen as a square array read by antidiagonals, a subtable of the binary operation multiplication tables A297845, A306697 and A329329. - Peter Munn, Jan 15 2020

			Triangle begins:
  2;
  3, 3;
  5, 5, 5;
  7, 7, 7, 7;
  ...

Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought", Basic Books, 1995.

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

Sequences with similar definitions: A002024, A175944.
Cf. A000040 (range of values), A003961, A031368 (main diagonal), A033286 (row sums), A097906.
Subtable of A297845, A306697, A329329.

a005145 n k = a005145_tabl !! (n-1) !! (k-1)
a005145_row n = a005145_tabl !! (n-1)
a005145_tabl = zipWith ($) (map replicate [1..]) a000040_list
a005145_list = concat a005145_tabl
-- Reinhard Zumkeller, Jul 12 2014, Mar 18 2011, Oct 17 2010

[NthPrime(Round(Sqrt(2*n))): n in [1..60]]; // Vincenzo Librandi, Jan 18 2020

Table[Prime[Floor[1/2 + Sqrt[2*n]]], {n, 1, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
Flatten[Table[Table[Prime[n], {n}], {n, 12}]] (* Alonso del Arte, Jan 18 2012 *)
Table[PadRight[{},n,Prime[n]],{n,15}]//Flatten (* Harvey P. Dale, Feb 29 2024 *)

a(n) = prime(round(sqrt(2*n))) \\ Charles R Greathouse IV, Oct 23 2015

from sympy import primerange
a = []; [a.extend([pn]*n) for n, pn in enumerate(primerange(1, 32), 1)]
print(a) # Michael S. Branicky, Jul 13 2022

from math import isqrt
from sympy import prime
def A005145(n): return prime(isqrt(n<<3)+1>>1) # Chai Wah Wu, Jun 08 2025

From Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006: (Start)
a(n) = prime(floor(1/2 + sqrt(2*n))).
a(n) = A000040(A002024(n)). (End)
From Peter Munn, Jan 15 2020: (Start)
When viewed as a square array A(n,k), the following hold for n >= 1, k >= 1:
A(n,k) = prime(n+k-1).
A(n,1) = A(1,n) = prime(n), where prime(n) = A000040(n).
A(n+1,k) = A(n,k+1) = A003961(A(n,k)).
A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)) = A329329(A(n,1), A(1,k)).
(End)
Sum_{n>=1} 1/a(n)^2 = A097906. - Amiram Eldar, Aug 16 2022

A063905 Each prime p appears p times.

Original entry on oeis.org

2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19

Offset: 1

Reinhard Zumkeller, Aug 30 2001

a(A007504(k)) = prime(k) and a(A007504(k) + 1) = prime(k + 1) for k > 0, where prime = A000040.

			a(A007504(4)) = prime(4) = 7 = a(17) and a(18) = a(A007504(4)+1) = prime(4+1) = prime(5) = 11.

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

Cf. A007504, A002024, A000040, A175944.

a063905 n = a063905_list !! (n-1)
a063905_list =
   concat $ zipWith ($) (map replicate a000040_list) a000040_list
-- Reinhard Zumkeller, Mar 18 2011

Flatten[Table[Prime[n], {n, 8}, {Prime[n]}]] (* Alonso del Arte, Sep 08 2011 based on Robert G. Wilson v's program for A002024 *)
Table[PadRight[{},p,p],{p,Prime[Range[8]]}]//Flatten (* Harvey P. Dale, Mar 13 2023 *)

{ n=0; p=1; for (m=1, 10^9, p=nextprime(p+1); for (i=1, p, write("b063905.txt", n++, " ", p); if (n==1000, break)); if (n==1000, break) ) } \\ Harry J. Smith, Sep 02 2009

A306366 For any sequence s of positive integers without infinitely many consecutive equal terms, let T(s) be the sequence obtained by replacing each run, say of k consecutive t's, with a run of t consecutive k's; this sequence corresponds to T(K) (where K denotes the Kolakoski sequence A000002).

Original entry on oeis.org

1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1

Offset: 1

Rémy Sigrist, Feb 10 2019

If s is finite, then s and T(s) have the same sum.
Fixed points of T correspond to sequences where each run, say of t's, has t elements; A001650, A001670, A002024, A130196, A167817, A175944 and A213083 are fixed points of T.
When s has no consecutive equal terms, then T(s) is all 1's (A000012).
Apparently, T^4(K) = T^2(K) (where T^i denotes the i-th iterate of K).

			The first terms of the Kolakoski sequence are:
       +-----+     +--+  +-----+  +-----+     +--
       |     |     |  |  |     |  |     |     |
    +--+     +-----+  +--+     +--+     +-----+
    |#1|#2   |#3   |#4|#5|#6   |#7|#8   |#9   |#10 ...
    +--+-----+-----+--+--+-----+--+-----+-----+--
      1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ...
.
The first terms of this sequence are:
       +-----+--+        +-----+  +-----+--
       |     .  |        |     |  |     .
    +--+     .  +-----+--+     +--+     .
    |#1|#2   .#3|#4   .#5|#6   |#7|#8   .#9  ...
    +--+-----+--+-----+--+-----+--+-----+--
      1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, ...

Rémy Sigrist, PARI program for A306366

Cf. A000002, A000012, A001650, A001670, A002024, A130196, A167817, A175944, A213083.

PARI
```
See Links section.
```

a(n) = A000002(ceiling(2*n/3)) (conjectured). - Jon Maiga, Jan 24 2021

cp's OEIS Frontend

A018252 The nonprime numbers: 1 together with the composite numbers, A002808.

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A014284 Partial sums of primes, if 1 is regarded as a prime (as it was until quite recently, see A008578).

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A175965 Lexicographically earliest sequence with first differences as increasing sequence of noncomposites A008578.

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A005145 n copies of n-th prime.

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A063905 Each prime p appears p times.

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Haskell

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PARI

A306366 For any sequence s of positive integers without infinitely many consecutive equal terms, let T(s) be the sequence obtained by replacing each run, say of k consecutive t's, with a run of t consecutive k's; this sequence corresponds to T(K) (where K denotes the Kolakoski sequence A000002).

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