A014688 -id:A014688 - cp's OEIS frontend

A098387 Prime(n)+Log2(n), where Log2=A000523.

2, 4, 6, 9, 13, 15, 19, 22, 26, 32, 34, 40, 44, 46, 50, 57, 63, 65, 71, 75, 77, 83, 87, 93, 101, 105, 107, 111, 113, 117, 131, 136, 142, 144, 154, 156, 162, 168, 172, 178, 184, 186, 196, 198, 202, 204, 216, 228, 232, 234, 238, 244, 246, 256, 262, 268, 274, 276

Offset: 1

Reinhard Zumkeller, Sep 06 2004

nonn

			a(10) = A000040(10) + A000523(10) = 29 + 3 = 32.

Cf. A098386, A014688, A098390, A098393.

A139027 This is to A139026 as A139026 to A139025, see A139025 for details.

Original entry on oeis.org

1292, 3865, 4666, 8973, 13936, 50339, 57266, 67597, 72316, 85343, 110934, 132941, 147990, 220203, 226652, 270239, 272950, 313361, 366186, 375253, 392090, 409619, 412024, 415237, 469982, 511263, 556808, 635279, 640716, 654559, 711018, 721629

Offset: 1

Zak Seidov, Apr 07 2008

nonn

Notice that a(n)-n is always prime by definition, e.g.,

a(1) - 1 = 1291, a(2) - 2 = 3863, a(3) - 3 = 4663, a(4) - 4 = 8969, etc.

Cf. A000027, A014688, A061068, A139025, A139026, A139028, A139029.

A139028 This is to A139027 as A139027 to A139026, see A139025 for details.

Original entry on oeis.org

270240, 375255, 635282, 1000695, 2039428, 2602013, 3398274, 3748771, 4300120, 4889577, 5643252, 6595775, 8684760, 12489373, 12758734, 15186995, 15557178, 17151151, 17988320, 18564859, 19878764, 20317745, 21560274, 22466983

Offset: 1

Zak Seidov, Apr 07 2008

nonn

Notice that a(n)-n is always prime by definition, e.g.,

a(1) - 1 = 270239, a(2) - 2 = 375253, a(3) - 3 = 635279, a(4) - 4 = 1000691, etc.

Cf. A000027, A014688, A061068, A139025, A139026, A139027, A139029.

A139029 This is to A139028 as A139028 to A139027, see A139025 for details.

Original entry on oeis.org

43448724, 59672019, 102128690, 113904945, 145135734, 169755139

Offset: 1

Zak Seidov, Apr 07 2008

Notice that a(n)-n is always prime by definition, e.g.,

a(1) - 1 = 43448723, a(2) - 2 = 59672017, a(3) - 3 = 102128687, a(4) - 4 = 113904941, etc.

Cf. A000027, A014688, A061068, A139025, A139026, A139027, A139028.

A060646 Bonse sequence: a(n) = minimal j such that n-j+1 < prime(j).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18

Offset: 1

Frank Ellermann, Apr 17 2001

For 35.

			For n=5, j=3 gives 5-3+1 = 3 < prime(3) = 5, true; but if j=2 we get 5-2+1 = 4 which is not < prime(2) = 3; hence a(5) = 3.
a(75)=18 because 75-18+1=58 < 61=prime(18), but 75-17+1=59=prime(17).

R. Remak, Archiv d. Math. u. Physik (3) vol. 15 (1908) 186-193

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
H. Bonse, Über eine bekannte Eigenshaft der Zahl 30 und ihre Verallgemeinerung, Archiv d. Math. u. Physik (3) vol. 12 (1907) 292-295.
H. Rademacher and O. Toeplitz, Eine Eigenschaft der Zahl 30, (A property of the number 30), Von Zahlen und Figuren (1930, reprint Springer 1968), ch. 22.

Cf. A014688.

import Data.List (findIndex)
import Data.Maybe (fromJust)
a060646 n = (fromJust $ findIndex ((n+1) <) a014688_list) + 1
-- Reinhard Zumkeller, Sep 16 2011

Table[j=0; While[j++; n-j+1 >= Prime[j]]; j, {n, 1, 76}] (* Jean-François Alcover, Aug 30 2011 *)

from sympy import nextprime
from itertools import count, islice
def agen(): # generator of terms
    n, pj = 1, 2
    for j in count(1):
        while n - j + 1 < pj: yield j; n += 1
        pj = nextprime(pj)
print(list(islice(agen(), 76))) # Michael S. Branicky, Aug 09 2022

A082500 a(n) = ceiling(n/2) if n is odd, or prime(n/2) otherwise.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 4, 7, 5, 11, 6, 13, 7, 17, 8, 19, 9, 23, 10, 29, 11, 31, 12, 37, 13, 41, 14, 43, 15, 47, 16, 53, 17, 59, 18, 61, 19, 67, 20, 71, 21, 73, 22, 79, 23, 83, 24, 89, 25, 97, 26, 101, 27, 103, 28, 107, 29, 109, 30, 113, 31, 127, 32, 131, 33, 137, 34, 139, 35, 149, 36

Offset: 1

Reinhard Zumkeller, May 11 2003

nonn

Alternatively, list of pairs n, prime(n). - Zak Seidov, Feb 18 2005
a(n) = (n mod 2)*(n+1)/2 + (1 - n mod 2)*A000040(n/2);
k>0: a(2*k-1)=k, a(2*k)=A000040(k), A049084(a(2*k))=k; a(2*k-1)+a(2*k)=A014688(k).
Each prime occurs at two positions, the distances between them are: 1, 1, 3, 5, 11, 13, 19, 21, 27, 37, 39, 49, 55, 57, 63, 73, 83, 85, ... - Zak Seidov, Mar 06 2011
See A239636. - Reinhard Zumkeller, Mar 22 2014

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

A000027 and A000040 interleaved. - Omar E. Pol, Mar 13 2012

import Data.List (transpose)
a082500 n = a082500_list !! (n-1)
a082500_list = concat $ transpose [[1..], a000040_list]
-- Reinhard Zumkeller, Mar 22 2014, Mar 19 2011, Oct 14 2010

[IsOdd(n) select Ceiling(n/2) else NthPrime(n div 2): n in[1..71]];  // Bruno Berselli, Mar 07 2011

s=Range[500];Do[s=Insert[s,Prime[n],2n],{n,100}];s (* Zak Seidov Mar 05 2011 *)
Table[If[OddQ[n],Ceiling[n/2 ],Prime[n/2]],{n,80}] (* Harvey P. Dale, Feb 22 2025 *)

A093571 LeastCommonMultiple({k+prime(k): 1<=k<=n}).

Original entry on oeis.org

3, 15, 120, 1320, 2640, 50160, 50160, 451440, 902880, 11737440, 82162080, 575134560, 575134560, 575134560, 17829171360, 410070941280, 410070941280, 32395604361120, 1393010987528160, 1393010987528160, 65471516413823520

Offset: 1

Reinhard Zumkeller, Apr 01 2004

nonn

a(n) <= A093570(n).

Cf. A093572, A014688.

Table[LCM @@ Table[k + Prime[k], {k, n}], {n, 21}] (* Robert G. Wilson v, Apr 07 2004 *)

More terms from Robert G. Wilson v, Apr 07 2004

A098393 Prime(n)+Log2(Log2(prime(n))), where Log2=A000523.

Original entry on oeis.org

2, 3, 6, 8, 12, 14, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 260, 266, 272, 274, 280

Offset: 1

Reinhard Zumkeller, Sep 06 2004

nonn

a(n) = A000040(n) + A098391(n).

			a(10) = A000040(10) + A098391(10) = 29 + 2 = 31.

Cf. A098389, A014688, A098387, A098390.

#+Floor[Log[2,Floor[Log[2,#]]]]&/@Prime[Range[60]] (* Harvey P. Dale, May 07 2017 *)

A135681 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=1 if n is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 4, 1, 4, 11, 4, 13, 4, 1, 4, 17, 4, 19, 4, 1, 4, 23, 4, 1, 4, 1, 4, 29, 4, 31, 4, 1, 4, 1, 4, 37, 4, 1, 4, 41, 4, 43, 4, 1, 4, 47, 4, 1, 4, 1, 4, 53, 4, 1, 4, 1, 4, 59, 4, 61, 4, 1, 4, 1, 4, 67, 4, 1, 4, 71, 4, 73

Offset: 1

Mohammad K. Azarian, Dec 01 2007

nonn

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

Cf. A135679; A135580; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690.

a[n_] := If[PrimeQ[n] || n == 1, n, If[EvenQ[n], 4, 1] ]; Table[a[n], {n,1,25}] (* G. C. Greubel, Oct 26 2016 *)

A230980 Number of primes <= n, starting at n=0.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Nov 02 2013

nonn

Essentially identical to A000720, except that sequence, being an arithmetical sequence, starts at n = 1. - N. J. A. Sloane, Jun 21 2017
Also, on the first quadrant of the square grid, consider a diagram in which the number of cells in the horizontal bar of the k-th row is equal to the k-th prime, see example. The total length of the boundary segments between the structure formed by the first k horizontal bars and the structure formed by the vertical bars, from [0, 0], is equal to A014688(k). a(n) is the number of cells in the vertical bar of the n-th column.
Note that in a similar diagram for A000720 the lengths of the horizontal bars give A006093 (primes minus 1) not A000040 (the prime numbers) because A000720 has only one zero, not two.
Also, the number of distinct prime factors of the factorial number n!. - Torlach Rush, Jan 17 2014
The lengths of the boundary horizontal segments between the structure formed by the horizontal bars and the structure formed by the vertical bars of the diagram gives A054541. The zig-zag path formed by the boundary segments is in A230850. - Omar E. Pol, Jun 22 2017

			Illustration of initial terms:
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
31   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
29   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
23   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
19   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
17   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
13   |_ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
11   |_ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
7    |_ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
5    |_ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
3    |_ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
2    |_ _|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
n:    0 1 2 3 4 5 6 7 8 9...
a(n): 0 0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A000040, A000720, A001223, A006093, A007504, A014688, A054541, A141042, A182986, A230850.

Partial sums of A010051.

[0] cat [#PrimesUpTo(n-1): n in [2..200] ]; // Vincenzo Librandi, Jun 22 2017

Array[PrimePi@# &, 90, 0] (* Robert G. Wilson v, Jun 21 2017 *)
Accumulate[Table[If[PrimeQ[n],1,0],{n,0,100}]] (* Harvey P. Dale, Mar 16 2023 *)

a(n)=primepi(n) \\ Charles R Greathouse IV, Jun 23 2017

Offset and definition changed by N. J. A. Sloane, Jun 21 2017

cp's OEIS Frontend

A098387 Prime(n)+Log2(n), where Log2=A000523.

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A139027 This is to A139026 as A139026 to A139025, see A139025 for details.

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A139028 This is to A139027 as A139027 to A139026, see A139025 for details.

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A139029 This is to A139028 as A139028 to A139027, see A139025 for details.

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A060646 Bonse sequence: a(n) = minimal j such that n-j+1 < prime(j).

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A082500 a(n) = ceiling(n/2) if n is odd, or prime(n/2) otherwise.

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Magma

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A093571 LeastCommonMultiple({k+prime(k): 1<=k<=n}).

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A098393 Prime(n)+Log2(Log2(prime(n))), where Log2=A000523.

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Mathematica

A135681 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=1 if n is odd.

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Mathematica

A230980 Number of primes <= n, starting at n=0.

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