A095116 -id:A095116 - cp's OEIS frontend

A143350 Triangle read by rows, replace column 1 of triangle A143349 with A095116, 1<=k<=n.

2, 4, -1, 7, -1, -1, 10, -2, -1, 0, 15, -2, -1, 0, -1, 18, -3, -2, 0, -1, 1, 23, -3, -2, 0, -1, 1, -1, 26, -4, -2, 0, -1, 1, -1, 0, 31, -4, -3, 0, -1, 1, -1, 0, 0, 38, -5, -3, 0, -2, 1, -1, 0, 0, 1, 41, -5, -3, 0, -2, 1, -1, 0, 0, 1, -1, 48, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0, 53, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0, -1, 56, -7, -4, 0, -2, 2, -2, 0, 0

Offset: 1

Gary W. Adamson, Aug 10 2008

Triangle A143349 = a type of Mobius transform which converts sequences to triangles with row sums = the same sequence. In this case, we convert p(n) to triangle A143349 having row sums = p(n), the primes.

We begin with p(n), adding (n-1) = A095116: (2, 4, 7, 10, 15, 18, 23,...). We then replace column 1 of triangle A143349 with A095116 resulting in A143350 with row sums = p(n).

			First few rows of the triangle =
2;
4, -1;
7, -1, -1;
10, -2, -1, 0;
15, -2, -1, 0, -1;
18, -3, -2, 0, -1, 1;
23, -3, -2, 0, -1, 1, -1;
26, -4, -2, 0, -1, 1, -1, 0;
31, -4, -3, 0, -1, 1, -1, 0, 0;
38, -5, -3, 0, -2, 1, -1, 0, 0, 1;
...

Cf. A143349, A095116, A008683, A000040.

Triangle read by rows, replace column 1 of triangle A143349 with A095116, 1<=k<=n. A143349 = p(n)+(n-1) & A143349 = a type of Mobius transform.

A095117 a(n) = pi(n) + n, where pi(n) = A000720(n) is the number of primes <= n.

Original entry on oeis.org

0, 1, 3, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 91

Offset: 0

Dean Hickerson, following a suggestion of Leroy Quet, May 28 2004

Positions of first occurrences of n in A165634: A165634(a(n))=n for n>0. - Reinhard Zumkeller, Sep 23 2009

There exists at least one prime number p such that n < p <= a(n) for n >= 2. For example, 2 is in (2, 3], 5 in (3, 5], 5 in (4, 6], ..., and primes 73, 79, 83 and 89 are in (71, 91] (see Corollary 1 in the paper by Ya-Ping Lu attached in the links section). - Ya-Ping Lu, Feb 21 2021

Carmine Suriano, Table of n, a(n) for n = 0..9999
Ya-Ping Lu, Lower Bounds for the Number of Primes in Some Integer Intervals

Complement of A095116.

Cf. A000720, A064427.

a095117 n = a000720 n + toInteger n  -- Reinhard Zumkeller, Apr 17 2012

with(numtheory): seq(n+pi(n),n=1..90); # Emeric Deutsch, May 02 2007

Table[ PrimePi@n + n, {n, 0, 71}] (* Robert G. Wilson v, Apr 22 2007 *)

a(n) = n + primepi(n); \\ Michel Marcus, Feb 21 2021

from sympy import primepi
def a(n): return primepi(n) + n
print([a(n) for n in range(72)]) # Michael S. Branicky, Feb 21 2021

a(0) = 0; for n>0, a(n) = a(n-1) + (if n is prime then 2, else 1). - Robert G. Wilson v, Apr 22 2007; corrected by David James Sycamore, Aug 16 2018

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A014690 a(n) = n + prime(n+1).

Original entry on oeis.org

4, 7, 10, 15, 18, 23, 26, 31, 38, 41, 48, 53, 56, 61, 68, 75, 78, 85, 90, 93, 100, 105, 112, 121, 126, 129, 134, 137, 142, 157, 162, 169, 172, 183, 186, 193, 200, 205, 212, 219, 222, 233, 236, 241, 244, 257, 270, 275, 278, 283, 290, 293, 304, 311, 318, 325, 328

Offset: 1

Mohammad K. Azarian

In sequence of odd primes add 1 to first prime, add 2 to 2nd prime, add 3 to 3rd prime and so on.

Essentially the same sequence as A095116.

a[n_]:=Prime[n+1]+n (* Zak Seidov, May 05 2005 *)
With[{nn=60},Prime[Range[2,nn+1]]+Range[nn]] (* Harvey P. Dale, Oct 02 2015 *)

More terms from Erich Friedman

A165634 Start with x=1 and repeat: if x is a prime number then (append i and then x, with x=prime(i)) else (only append x), continue with x:=x+1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 5, 6, 4, 7, 8, 9, 10, 5, 11, 12, 6, 13, 14, 15, 16, 7, 17, 18, 8, 19, 20, 21, 22, 9, 23, 24, 25, 26, 27, 28, 10, 29, 30, 11, 31, 32, 33, 34, 35, 36, 12, 37, 38, 39, 40, 13, 41, 42, 14, 43, 44, 45, 46, 15, 47, 48, 49, 50, 51, 52, 16, 53, 54, 55, 56, 57, 58, 17, 59

Offset: 1

Reinhard Zumkeller, Sep 23 2009

nonn

All positive integers occur exactly twice: A095117 and A095116 give positions of first and second occurrences.

			1,(1,2),(2,3),4,(3,5),6,(4,7),8,9,10,(5,11),12, ... .

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

A049084, A000040.

a165634 n = a165634_list !! (fromInteger n - 1)
a165634_list = concatMap (\x ->
   if a010051 x == 1 then [a049084 x, x] else [x]) [1..]
-- Reinhard Zumkeller, Apr 17 2012

A340716 Lexicographically earliest sequence of positive integers with as many distinct values as possible such that for any n > 0, a(n + pi(n)) = a(n) (where pi(n) = A000720(n) corresponds to the number of prime numbers <= n).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Jan 17 2021

The condition "with as many distinct values as possible" means here that for any distinct m and n, provided the orbits of m and n under the map x -> x + pi(x) do not merge, then a(m) <> a(n).

This sequence has similarities with A003602 (A003602(2*n) = A003602(n)) and with A163491 (A163491(n+ceiling(n/2)) = A163491(n)).

			The first terms, alongside n + pi(n), are:
  n   a(n)  n + pi(n)
  --  ----  ---------
   1     1          1
   2     2          3
   3     2          5
   4     3          6
   5     2          8
   6     3          9
   7     4         11
   8     2         12
   9     3         13
  10     5         14
  11     4         16
  12     2         17

Rémy Sigrist, Table of n, a(n) for n = 1..10000

See A003602, A163491 and A340717 for similar sequences.

Cf. A000720, A095117.

u=0; for (n=1, #a=vector(80), if (a[n]==0, a[n]=u++); print1 (a[n]", "); m=n+primepi(n); if (m<=#a, a[m]=a[n]))

a(n) = 2 iff n belongs to A061535.

a(A095116(n)) = n + 1.

A123128 Add n to the n-th difference between consecutive primes.

Original entry on oeis.org

2, 4, 5, 8, 7, 10, 9, 12, 15, 12, 17, 16, 15, 18, 21, 22, 19, 24, 23, 22, 27, 26, 29, 32, 29, 28, 31, 30, 33, 44, 35, 38, 35, 44, 37, 42, 43, 42, 45, 46, 43, 52, 45, 48, 47, 58, 59, 52, 51, 54, 57, 54, 63, 60, 61, 62, 59, 64, 63, 62, 71, 76, 67, 66, 69, 80, 73, 78, 71, 74, 77

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 30 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
A. Frank & P. Jacqueroux, International Contest, 2001. Item 20.

Cf. A000040 (primes), A001223, A014689, A014690, A095116.

[n + NthPrime(n+1) - NthPrime(n): n in [1..80]]; // G. C. Greubel, Aug 03 2021

With[{nn=80},Total/@Thread[{Differences[Prime[Range[nn]]],Range[nn-1]}]] (* Harvey P. Dale, Jun 02 2014 *)

for(n=1,100,print1(prime(n+1)-prime(n)+n,","))

[n + nth_prime(n+1) - nth_prime(n) for n in (1..80)] # G. C. Greubel, Aug 03 2021

a(n) = n + (prime(n+1) - prime(n)) = n + A001223(n).
From G. C. Greubel, Aug 03 2021: (Start)
a(n) = A014690(n) - prime(n) = A095116(n+1) - prime(n).
a(n) = prime(n+1) - A014689(n). (End)

A123129 a(n) = ( n + prime(n+1) - prime(n) )^(n-1).

Original entry on oeis.org

1, 4, 25, 512, 2401, 100000, 531441, 35831808, 2562890625, 5159780352, 2015993900449, 17592186044416, 129746337890625, 20822964865671168, 3243919932521508681, 136880068015412051968, 288441413567621167681

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 30 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..350
Sergio Silva, Teste Numerico.

Cf. A000040, A001223, A014690, A095116.

Table[(n +Prime[n+1] -Prime[n])^(n-1), {n, 1, 30}] (* G. C. Greubel, Aug 04 2021 *)

for(n=1, 25, print1( (prime(n+1) -prime(n) +n)^(n-1), ","))

[(n + nth_prime(n+1) - nth_prime(n))^(n-1) for n in (1..30)] # G. C. Greubel, Aug 04 2021

a(n) = ( n + prime(n+1) - prime(n) )^(n-1) = ( n + A001223(n) )^(n-1).
From G. C. Greubel, Aug 04 2021: (Start)
a(n) = (A014690(n) - prime(n))^(n-1) = (A095116(n+1) - prime(n))^(n-1).
a(n) = (prime(n+1) - A014689(n))^(n-1). (End)

A230846 1 + A075526(n).

Original entry on oeis.org

2, 2, 3, 3, 5, 3, 5, 3, 5, 7, 3, 7, 5, 3, 5, 7, 7, 3, 7, 5, 3, 7, 5, 7, 9, 5, 3, 5, 3, 5, 15, 5, 7, 3, 11, 3, 7, 7, 5, 7, 7, 3, 11, 3, 5, 3, 13, 13, 5, 3, 5, 7, 3, 11, 7, 7, 7, 3, 7, 5, 3, 11, 15, 5, 3, 5, 15, 7, 11, 3, 5, 7, 9, 7, 7, 5, 7, 9, 5, 9, 11, 3, 11, 3, 7, 5, 7, 9, 5, 3, 5, 13, 9, 5, 9, 5, 7

Offset: 1

Omar E. Pol, Nov 01 2013

nonn

Partial sums give A095116.

			On the first quadrant of the square grid consider a diagram in which the n-th horizontal bar contains A006093(n) cells and in which the number of cells in the vertical bars gives A000720 as shown below. a(n) is the sum of the length of the n-th horizontal boundary segment and the length of the n-th vertical boundary segment between the structure formed by the horizontal bars and the structure formed by the vertical bars, hence a(n) = A075526(n) + 1. The total length of the boundary segments from [0, 0] after n-th stage is A095116(n).
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
30  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
28  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
22  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
18  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
16  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
12  |_ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
10  |_ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
6   |_ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
4   |_ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
2   |_ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
1   |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
.    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
.

Cf. A000720, A006093, A014688, A054541, A075526, A095116.

Essentially the same as A076368.

cp's OEIS Frontend

A143350 Triangle read by rows, replace column 1 of triangle A143349 with A095116, 1<=k<=n.

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A095117 a(n) = pi(n) + n, where pi(n) = A000720(n) is the number of primes <= n.

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A014690 a(n) = n + prime(n+1).

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A165634 Start with x=1 and repeat: if x is a prime number then (append i and then x, with x=prime(i)) else (only append x), continue with x:=x+1.

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Haskell

A340716 Lexicographically earliest sequence of positive integers with as many distinct values as possible such that for any n > 0, a(n + pi(n)) = a(n) (where pi(n) = A000720(n) corresponds to the number of prime numbers <= n).

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A123128 Add n to the n-th difference between consecutive primes.

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Magma

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A123129 a(n) = ( n + prime(n+1) - prime(n) )^(n-1).

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A230846 1 + A075526(n).

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