A054541 -id:A054541 - cp's OEIS frontend

A299763 a(n) = 1 + A182986(n).

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

Omar E. Pol, Mar 14 2018

Are these the indices of the rows of A299762 where there is a record?

G. C. Greubel, Table of n, a(n) for n = 1..10000

First differences are in A054541.
Cf. A008864, A028815, A182986, A299762.
Essentially the same as A008864, A028815, A055670, A135731, A175216.

A299763:= func< n | n eq 1 select 1 else NthPrime(n-1) + 1 >;
[A299763(n): n in [1..70]]; // G. C. Greubel, Aug 05 2024

A299763[n_]:= If[n==1, 1, Prime[n-1] +1];
Table[A299763[n], {n,70}] (* G. C. Greubel, Aug 05 2024 *)

def A299763(n): return 1 if n==1 else nth_prime(n-1) + 1
[A299763(n) for n in range(1,71)] # G. C. Greubel, Aug 05 2024

a(n) = A028815(n-1) - [n=1].

a(n) = A008864(n-1) for n >= 2, with a(1) = 1.

A181346 Absolute difference between (sum of previous terms) and prime(n) with a(0) = 1 and a(1) = 2.

Original entry on oeis.org

1, 2, 0, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12

Offset: 0

Giovanni Teofilatto, Oct 14 2010

Essentially a duplicate of A001223: equals A001223 with terms 2,0 inserted after the initial one.

Cf. A001223.

Cf. A054541. - Georg Fischer, Oct 12 2018

A181346 := proc(n) option remember; if n <= 1 then return n+1 ; end if; add( procname(i),i=0..n-1)-ithprime(n) ; abs(%) ;end proc:
seq(A181346(n),n=0..100) ; # R. J. Mathar, Oct 15 2010

Offset and indexing corrected by R. J. Mathar, Oct 15 2010

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.

A300951 a(n) = Product_{j=1..floor(n/2)} p(j) where p(j) = j if j is prime else 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 6, 6, 6, 6, 30, 30, 30, 30, 210, 210, 210, 210, 210, 210, 210, 210, 2310, 2310, 2310, 2310, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 510510, 510510, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 9699690, 9699690, 9699690

Offset: 0

Peter Luschny, Mar 16 2018

nonn

a(4*n+2)=a(4*n+3)=a(4*n+4)=a(4*n+5) for n >= 1. - Robert Israel, Mar 16 2018

The length of the n-th run is given by 2*A054541(n). - Michel Marcus, Mar 17 2018

Robert Israel, Table of n, a(n) for n = 0..4713

Cf. A002110, A054541, A056172, A089026.

a := n -> mul(`if`(isprime(j), j, 1), j=1..iquo(n,2)):
seq(a(n), n=0..44);
# Alternative:
f:= proc(n) option remember;
  if n::even and isprime(n/2) then procname(n-1)*n/2 else procname(n-1) fi
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Mar 16 2018

{#,#}&/@FoldList[Times,Table[If[PrimeQ[n],n,1],{n,0,30}]]//Flatten (* Harvey P. Dale, Dec 25 2019 *)

a(n) = prod(i=1, n\2, if(isprime(i), i, 1)); \\ Altug Alkan, Mar 16 2018

a(n) = A002110(A056172(n)). - Robert Israel, Mar 16 2018

A333471 a(n) = 2 * mu(n) + Sum_{d|n, d > 1} mu(n/d) * (prime(d) - prime(d-1)).

Original entry on oeis.org

2, -1, 0, 1, 2, 1, 2, 0, 2, 3, 0, 3, 2, -1, 0, 4, 4, -2, 4, -3, -2, 5, 2, 0, 4, 1, -2, 1, 0, -3, 12, -2, 4, -3, 4, -4, 4, 1, 0, 2, 4, 1, 8, -5, -2, -1, 10, 2, 0, -8, -2, 1, 0, 10, 2, 2, 0, 1, 4, -1, 0, -3, 10, 0, -4, -7, 12, 3, 6, -9, 2, 4, 6, 1, -2, -3, 2, 3, 2, -2

Offset: 1

Ilya Gutkovskiy, Mar 23 2020

sign

Moebius transform of A054541 (2 followed by prime gaps).

Cf. A000040, A001223, A007444, A008683, A030013, A054541, A182986, A333450 (partial sums).

a[n_] := 2 MoebiusMu[n] + Sum[If[d > 1, MoebiusMu[n/d] (Prime[d] - Prime[d - 1]), 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 80}]

a(n) = Sum_{d|n} mu(n/d) * A054541(d).

Sum_{k=1..n} floor(n/k) * a(k) = prime(n).

cp's OEIS Frontend

A299763 a(n) = 1 + A182986(n).

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A181346 Absolute difference between (sum of previous terms) and prime(n) with a(0) = 1 and a(1) = 2.

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A230846 1 + A075526(n).

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A300951 a(n) = Product_{j=1..floor(n/2)} p(j) where p(j) = j if j is prime else 1.

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A333471 a(n) = 2 * mu(n) + Sum_{d|n, d > 1} mu(n/d) * (prime(d) - prime(d-1)).

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