A076368 -id:A076368 - cp's OEIS frontend

A119912 Scan A076368, discard any nonprimes.

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Aug 02 2006

Primes that are one greater than the difference between consecutive primes.

			The first 4 consecutive prime pairs are (2,3),(3,5),(5,7),(7,11). The differences + 1 are the primes 2,3,3,5, the first four entries in the sequence.

Cino Hilliard, Frequency of primes.

Cf. A076368.

P:=proc(n) local cont,i,j,k,w; for i from 1 by 1 to n do k:=ithprime(i); w:=ithprime(i+1); if isprime(w-k+1) then print(w-k+1); fi; od; end: P(10000);

Select[Differences[Prime[Range[200]]]+1,PrimeQ] (* Harvey P. Dale, Jul 02 2017 *)

diffp1p2(n) = { local(p1,p2,y); for(x=1,n, p1=prime(x); p2=prime(x+1); y=(p2-p1)+1; if(isprime(y), print1(y",") ) ) } \\ Cino Hilliard, May 23 2007

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

A106587 Sum of n-th prime squared and n-th perfect square.

Original entry on oeis.org

5, 13, 34, 65, 146, 205, 338, 425, 610, 941, 1082, 1513, 1850, 2045, 2434, 3065, 3770, 4045, 4850, 5441, 5770, 6725, 7418, 8497, 10034, 10877, 11338, 12233, 12722, 13669, 17090, 18185, 19858, 20477, 23426, 24097, 26018, 28013, 29410, 31529, 33722

Offset: 1

Alexandre Wajnberg, May 10 2005

			a(5)=146 because 121 (fifth prime^2) + 25 (fifth square) = 146.

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

Cf. A000290, A001248, A014688, A075526, A076368.

[NthPrime(n)^2 + n^2: n in [1..50]]; // G. C. Greubel, Sep 07 2021

Mathematica
```
Table[Prime[n]^2 + n^2, {n, 50}]
```

a(n) = n^2 + prime(n)^2; \\ Michel Marcus, Sep 08 2021

[nth_prime(n)^2 + n^2 for n in (1..50)] # G. C. Greubel, Sep 07 2021

a(n) = n^2 + prime(n)^2.

Extended by Ray Chandler, May 13 2005

A076367 Primes with subscripts from the Bonse sequence.

Original entry on oeis.org

2, 3, 3, 5, 5, 5, 7, 7, 7, 11, 11, 11, 11, 11, 13, 13, 13, 17, 17, 17, 17, 17, 19, 19, 19, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 37, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 41, 43, 43, 43, 47, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59

Offset: 1

Labos Elemer, Oct 14 2002

nonn

This and sequence A060646 was used to prove that 30 is the largest number whose RRS does not contain composite numbers. See A048597, A060646 and corresponding References.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A048597, A060646, A076368. See also A076366.

c[x_, j_] := x+1-(j+Prime[j]); c[x, 0]=x; a=1000; t=Table[0, {a}]; t1=Table[0, {a}]; Table[fl=1; (*Print["% ", u, " #"]; *)Do[s=c[u, n]; If[Equal[fl, 1]&&Equal[Sign[s], -1], Print[n]; t[[u]]=n; t1[[u]]=Prime[n]; fl=0], {n, 1, u}], {u, 1, a}] //t (*=A060646*)//t1 (* =A076367 *)

a(n) = prime(A060646(n)).

A106588 Difference between n-th prime squared and n-th perfect square.

Original entry on oeis.org

3, 5, 16, 33, 96, 133, 240, 297, 448, 741, 840, 1225, 1512, 1653, 1984, 2553, 3192, 3397, 4128, 4641, 4888, 5757, 6360, 7345, 8784, 9525, 9880, 10665, 11040, 11869, 15168, 16137, 17680, 18165, 20976, 21505, 23280, 25125, 26368, 28329, 30360

Offset: 1

Alexandre Wajnberg, May 10 2005

			a(5) = 96 because 121 (fifth prime^2) - 25 (fifth square) = 96.

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

Cf. A000290, A001248, A014688, A075526, A076368.

[NthPrime(n)^2 - n^2: n in [1..50]]; // G. C. Greubel, Sep 07 2021

Mathematica
```
Table[Prime[n]^2 - n^2, {n, 50}]
```

a(n) = prime(n)^2 - n^2; \\ Michel Marcus, Sep 08 2021

[nth_prime(n)^2 - n^2 for n in (1..50)] # G. C. Greubel, Sep 07 2021

a(n) = prime(n)^2 - n^2.

Extended by Ray Chandler, May 13 2005

A226534 a(n) = (p(n+1) + p(n) - 1) mod (p(n+1) - p(n) + 1) where p(n) is the n-th prime.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 2, 1, 2, 2, 4, 2, 2, 4, 1, 6, 2, 1, 2, 2, 4, 1, 3, 5, 2, 2, 4, 2, 1, 14, 2, 1, 2, 1, 2, 6, 4, 4, 3, 1, 2, 8, 2, 4, 2, 6, 4, 4, 2, 1, 2, 2, 7, 3, 1, 6, 2, 1, 2, 2, 3, 14, 2, 2, 4, 2, 2, 1, 2, 1, 4, 5, 4, 2, 1, 1, 2, 2, 8, 2, 2, 4, 2, 3, 1, 2, 5, 2, 2, 4, 9, 2, 2, 8, 1, 3, 2

Offset: 1

Juri-Stepan Gerasimov, Aug 31 2013

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

Table[Mod[Prime[p + 1] + Prime[p] - 1, Prime[p + 1] - Prime[p] + 1], {p, 100}] (* Alonso del Arte, Jan 18 2014 *)
Mod[Total[#]-1,#[[2]]-#[[1]]+1]&/@Partition[Prime[Range[100]],2,1] (* Harvey P. Dale, Mar 16 2023 *)

a(n)=lift(Mod(prime(n+1)+prime(n)-1,prime(n+1)-prime(n)+1)) /* Ralf Stephan, Sep 03 2013 */

a(n) = A076273(n+1) mod A076368(n+1).

A140785 a(n) = the single integer k, where p(n) <= k <= p(n+1), that is divisible by (p(n+1)-p(n)+1), where p(n) is the n-th prime.

Original entry on oeis.org

2, 3, 6, 10, 12, 15, 18, 20, 28, 30, 35, 40, 42, 45, 49, 56, 60, 63, 70, 72, 77, 80, 84, 90, 100, 102, 105, 108, 110, 120, 130, 133, 138, 143, 150, 154, 161, 165, 168, 175, 180, 187, 192, 195, 198, 208, 221, 225, 228, 230, 238, 240, 242, 252, 259, 266, 270, 273

Offset: 1

Leroy Quet, Jul 13 2008

nonn

Diana Mecum, Table of n, a(n) for n = 1..1000

Cf. A076368, A113709.

(#[[2]]-#[[1]]+1)Floor[#[[2]]/(#[[2]]-#[[1]]+1)]&/@Partition[ Prime[ Range[ 60]],2,1] (* Harvey P. Dale, Apr 07 2018 *)

a(n) = (p(n+1)-p(n)+1) * floor(p(n+1)/(p(n+1)-p(n)+1)), where p(n) is the n-th prime.

More terms from Diana L. Mecum, Jul 21 2008

A224934 Primes p for which there exists no prime q, different from p, such that p+q-1 is the next prime after p.

Original entry on oeis.org

2, 3, 89, 113, 293, 317, 359, 389, 401, 449, 479, 491, 683, 701, 719, 743, 761, 773, 839, 863, 887, 911, 929, 953, 983, 1109, 1163, 1193, 1327, 1373, 1409, 1439, 1523, 1559, 1571, 1583, 1637, 1669, 1733, 1823, 1847, 1979, 2003, 2039, 2153, 2179, 2213, 2243

Offset: 1

Jayanta Basu, Apr 20 2013

nonn

If we relax the restriction on q, where q is different from p, 2 and 3 fail to be members of this sequence.

Primes p = prime(k) for which A076368(k+1) = p or A076368(k+1) is composite. - Robert Israel, Nov 21 2016

			89 is in the list because there exists no prime q such that 89 + q - 1 = 97.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A076368, A224748.

N:= 10^4: # to get all terms p for which the next prime <= N
P:= select(isprime, [2,seq(i,i=3..N,2)]):
G:= P[2..-1]-P[1..-2]:
P[select(t -> G[t]=P[t]-1 or not isprime(G[t]+1), [$1..nops(G)])]; # Robert Israel, Nov 21 2016

t = {}; Do[p = Prime[n]; If[FreeQ[Table[k = p + Prime[i] - 1, {i, n - 1}], Prime[n + 1]], AppendTo[t, p]], {n, 335}]; t

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.

A230847 a(n) = 1 + A054541(n).

Original entry on oeis.org

3, 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 A014688.

			On the first quadrant of the square grid consider a diagram in which the n-th horizontal bar contains A000040(n) cells and in which the number of cells in the vertical bars gives 0 together with 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) = A054541(n) + 1. The total length of the boundary segments from [0, 0] after n-th stage is A014688(n).
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
31  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
29  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
23  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
19  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
17  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
13  |_ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
11  |_ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
7   |_ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
5   |_ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
3   |_ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
2   |_ _|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
.    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
.

Cf. A000040, A014688, A054541, A076368.

a(n) = A230846(n) = A076368(n), n>1. - R. J. Mathar, May 16 2023

cp's OEIS Frontend

A119912 Scan A076368, discard any nonprimes.

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A106587 Sum of n-th prime squared and n-th perfect square.

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Magma

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A076367 Primes with subscripts from the Bonse sequence.

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A106588 Difference between n-th prime squared and n-th perfect square.

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Magma

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A226534 a(n) = (p(n+1) + p(n) - 1) mod (p(n+1) - p(n) + 1) where p(n) is the n-th prime.

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A140785 a(n) = the single integer k, where p(n) <= k <= p(n+1), that is divisible by (p(n+1)-p(n)+1), where p(n) is the n-th prime.

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A224934 Primes p for which there exists no prime q, different from p, such that p+q-1 is the next prime after p.

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