A158611 -id:A158611 - cp's OEIS frontend

A182986 Zero together with the prime numbers (A000040).

0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Charles R Greathouse IV, Feb 05 2011

These numbers are the possible characteristics of a field.
First differences are in A054541. - Omar E. Pol, Oct 31 2013
Also A158611 without its second term. - Omar E. Pol, Nov 01 2013
The ideals generated by a(n) form Spec(Z), the set of prime ideals of the ring of integers. Due to its importance in algebraic geometry, algebraic geometers often consider 0 to be an honorary prime. - Keith J. Bauer, Jan 09 2024

Cf. A141468.

Complement of A018252. - Arkadiusz Wesolowski, Sep 15 2011

Prepend[Table[Prime[n], {n, 60}], 0] (* Arkadiusz Wesolowski, Sep 15 2011 *)
NestList[ NextPrime, 0, 57] (* Robert G. Wilson v, Jul 21 2015 *)

a(n)=if(n>1,prime(n-1),0) \\ Charles R Greathouse IV, Jul 15 2013

A073169 a(n)=A002808(n)-n, difference between n-th composite and n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 19 2002

nonn

a(n) = the number of numbers of set {1, prime} (A008578(n)) less than n-th composite numbers (A002808(n)). a(n) = inverse (frequency distribution) sequence of A162177(n), i.e. number of terms of sequence A162177(n) less than n for n >= 1. a(n) = A002808(n) + A162177(n) - A158611(n+1) for n >= 1. a(n) = A002808(n) + A162177(n) - A008578(n) for n >= 1. [From Jaroslav Krizek, Jul 23 2009]

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007821, A022449, A000040, A002808, A073168.

c[n_Integer] := FixedPoint[n+PrimePi[ # ]+1&, n] Table[c[w]-w, {w, 1, 128}]
With[{c=Select[Range[100],CompositeQ]},#[[1]]-#[[2]]&/@Thread[ {c,Range[ Length[ c]]}]] (* Harvey P. Dale, Feb 03 2015 *)

a(n)=1+A073425(n). [From R. J. Mathar, Jul 31 2009]

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

A075527 a(n) = A008578(n+3) - A008578(n+1).

Original entry on oeis.org

2, 3, 4, 6, 6, 6, 6, 6, 10, 8, 8, 10, 6, 6, 10, 12, 8, 8, 10, 6, 8, 10, 10, 14, 12, 6, 6, 6, 6, 18, 18, 10, 8, 12, 12, 8, 12, 10, 10, 12, 8, 12, 12, 6, 6, 14, 24, 16, 6, 6, 10, 8, 12, 16, 12, 12, 8, 8, 10, 6, 12, 24, 18, 6, 6, 18, 20, 16, 12, 6, 10, 14, 14, 12, 10, 10, 14, 12, 12, 18, 12

Offset: 0

Reinhard Zumkeller, Sep 22 2002

nonn

For n>0: a(n) = A031131(n) and a(n) - a(n-1) = A075526(n).

Cf. A000040, A076272, A076273.

Cf. A008578, A031131, A075526.

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

A162177 a(n) is the number of composite numbers that are smaller than A008578(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 9, 10, 13, 18, 19, 24, 27, 28, 31, 36, 41, 42, 47, 50, 51, 56, 59, 64, 71, 74, 75, 78, 79, 82, 95, 98, 103, 104, 113, 114, 119, 124, 127, 132, 137, 138, 147, 148, 151, 152, 163, 174, 177, 178, 181, 186, 187, 196, 201, 206, 211, 212, 217, 220, 221

Offset: 1

Jaroslav Krizek, Jun 27 2009

Essentially the same as A065890.

a(n) = number of terms of A073169(n) less than n.

			A008578(6) = 11, and there are 5 composites smaller than 11, viz. 4, 6, 8, 9, 10, hence a(6) = 5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002808 (composites), A008578 (1 and the primes), A065890, A073169.

T:=[0,1] cat PrimesUpTo(300); [ T[n+1]-n: n in [1..#T-1] ]; // Klaus Brockhaus, Sep 08 2009

Join[{0},Module[{nn=300,cmps},cmps=Accumulate[Table[If[CompositeQ[n],1,0],{n,nn}]];Table[cmps[[p]],{p,Prime[ Range[ PrimePi[ nn]]]}]]] (* Harvey P. Dale, Nov 11 2024 *)

a(n) = A008578(n) - n = A158611(n+1) -n.

a(n) = A065890(n-1) for n > 1.

Edited and extended by Klaus Brockhaus, Sep 09 2009

A369641 Composite numbers k such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

Original entry on oeis.org

9, 10, 14, 15, 16, 28, 30, 45, 58, 62, 74, 87, 108, 112, 136, 155, 161, 189, 198, 203, 209, 210, 212, 217, 221, 225, 236, 244, 246, 247, 282, 290, 299, 323, 361, 374, 399, 422, 435, 478, 482, 507, 717, 1055, 1205, 1477, 1480, 1631, 1673, 1687, 1940, 2132, 2189, 2212, 2308, 2356, 2519, 2524, 2561, 2587, 2655, 2766

Offset: 1

Antti Karttunen, Jan 31 2024

nonn

Composite terms of A341518, i.e., composite numbers k such that A327859(k) = A276086(A003415(k)) is squarefree number, or equally, k' is in A276156.

Antti Karttunen, Table of n, a(n) for n = 1..20054 (terms less than 2^30)

Setwise difference A341518 \ A158611.
Cf. A003415, A276086, A276156, A327859, A341517, A369640 (characteristic function).
Cf. A327978, A328243, A369642 (subsequences).

PARI
```
\\ See A369640
```

A159461 Numbers of previous and following composites of n-th prime.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 4, 8, 6, 6, 8, 4, 4, 8, 10, 6, 6, 8, 4, 6, 8, 8, 12, 10, 4, 4, 4, 4, 16, 16, 8, 6, 10, 10, 6, 10, 8, 8, 10, 6, 10, 10, 4, 4, 12, 22, 14, 4, 4, 8, 6, 10, 14, 10, 10, 6, 6, 8, 4, 10, 22, 16, 4, 4, 16, 18, 14, 10, 4, 8, 12, 12, 10, 8, 8, 12

Offset: 1

Jaroslav Krizek, Apr 12 2009

nonn

Essentially the same as A046930. - R. J. Mathar, Apr 16 2009

For twin primes this is the gap before or after the twins, e.g., a(17) = 6 = 59 - 53 = prime(17) - prime(16) for the twin (59, 61) with a(18) = 6 = 67 - 61 = prime(19) - prime(18). - Frank Ellermann, Mar 17 2020

			For a(16) = 10 = 59 - 47 - 2 = prime(16+1) - prime(16-1) - 2 is the sum of the prime gaps minus two ending and starting at prime(16) = 53.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for gaps between primes.

Cf. A000040, A001223, A046933, A008578, A158611.

Join[{0},Total[Differences[#]-1]&/@Partition[Prime[Range[60]],3,1]] (* Harvey P. Dale, Nov 27 2011 *)

For n >= 2, we have
a(n) = A001223(n) + A001223(n-1) - 2;
a(n) = A046933(n) + A046933(n-1);
a(n) = A008578(n+2) - A008578(n) - 2;
a(n) = A158611(n+3) - A158611(n+1) - 2.

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

A164653 a(1) = 1, for n>=2: a(n) = sum of two consecutive noncomposite numbers A008578.

Original entry on oeis.org

1, 3, 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, 138, 144, 152, 162, 172, 186, 198, 204, 210, 216, 222, 240, 258, 268, 276, 288, 300, 308, 320, 330, 340, 352, 360, 372, 384, 390, 396, 410, 434, 450, 456, 462, 472, 480, 492, 508

Offset: 1

Jaroslav Krizek, Aug 19 2009

nonn

Basically these are the sums of two successive primes. - N. J. A. Sloane, Nov 16 2018

Essentially the same as A001043, A011974 and A069102.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A008578, A011974, A069102, A076273, A158611.

ListConvolve[{1,1},Join[{0,1},Prime[Range[100]]]] (* Paolo Xausa, Nov 02 2023 *)

a(n) = A158611(n) + A158611(n+1).
a(n) = A008578(n-1) + A008578(n) for n >= 2.
a(n) = A076273(n-1) + 1 for n >= 2.
a(n) = A000040(n-1) + A008578(n-1) for n >= 2. - Jaroslav Krizek, Dec 13 2009

Edited by R. J. Mathar, Aug 21 2009
Correction for change of offset in A158611 and A008578 in Aug 2009 by Jaroslav Krizek, Jan 27 2010
Formulas edited by Paolo Xausa, Nov 04 2023

A023538 Convolution of natural numbers with (1, p(1), p(2), ... ), where p(k) is the k-th prime.

Original entry on oeis.org

1, 4, 10, 21, 39, 68, 110, 169, 247, 348, 478, 639, 837, 1076, 1358, 1687, 2069, 2510, 3012, 3581, 4221, 4934, 5726, 6601, 7565, 8626, 9788, 11053, 12425, 13906, 15500, 17221, 19073, 21062, 23190, 25467, 27895, 30480, 33228, 36143, 39231, 42498, 45946, 49585

Offset: 1

Clark Kimberling

nonn

F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.

Cf. A007504, A014148, A014150, A014284, A023538, A178138.

Nest[Accumulate,Join[{1},Select[Range@200,PrimeQ]],2] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2012 *)

a(n) = Sum_{k<=n} [(A158611(k+1)) * (A000027(n-k+1))] = Sum_{k<=n} [(A008578(k)) * (A000027(n-k+1))]. [Jaroslav Krizek, Aug 05 2009; Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010]

A159081 Let d be the largest element of A008578 which divides n, then a(n) is the position of d in A008578.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Apr 05 2009

Let p be the largest prime factor of n; if p = prime(k) then set a(n) = k + 1. a(n) = A061395(n) + 1.

			For n=30, the largest element of the set {1,2,3,5} (1 and prime divisors of 30) is 5, and 5 is a(n)=4th term of A008578, the extended set of primes.

Cf.: A061395, A049084, A006530, A008578.

a(n) = A049084(A006530(n)) + 1. A008578(a(n)) = A006530(n);

Edited by R. J. Mathar, Apr 06 2009

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

A160656 The odd prime numbers together with 0: p - (-1)^p - 1 where p = n-th prime.

Original entry on oeis.org

0, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Juri-Stepan Gerasimov, May 23 2009

Cf. A000040, A006005, A158611.

seq(ithprime(n) mod n^2, n=1..59) # Gary Detlefs, Jan 14 2012

a(n)=if(n>1,prime(n)) \\ Charles R Greathouse IV, Aug 26 2011

a(n) = p - (-1)^p - 1 where p is the n-th prime.

a(n) = p mod n^2, where p is the n-th prime. - Gary Detlefs, Jan 14 2012

cp's OEIS Frontend

A182986 Zero together with the prime numbers (A000040).

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A073169 a(n)=A002808(n)-n, difference between n-th composite and n.

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A075527 a(n) = A008578(n+3) - A008578(n+1).

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A162177 a(n) is the number of composite numbers that are smaller than A008578(n).

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A369641 Composite numbers k such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

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A159461 Numbers of previous and following composites of n-th prime.

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A164653 a(1) = 1, for n>=2: a(n) = sum of two consecutive noncomposite numbers A008578.

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A023538 Convolution of natural numbers with (1, p(1), p(2), ... ), where p(k) is the k-th prime.

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