A031131 -id:A031131 - cp's OEIS frontend

A113784 Difference between semiprime(n) and semiprime(n+2).

5, 4, 5, 5, 7, 7, 4, 4, 8, 8, 2, 4, 4, 8, 10, 5, 6, 6, 3, 5, 7, 7, 9, 8, 8, 8, 4, 2, 5, 6, 3, 2, 12, 16, 9, 7, 4, 3, 3, 2, 7, 10, 5, 8, 8, 2, 3, 3, 10, 12, 4, 3, 7, 8, 11, 9, 6, 7, 4, 9, 14, 8, 2, 3, 3, 4, 7, 5, 2, 3, 3, 2, 3, 7, 14, 11, 12, 12, 6, 5, 6, 8, 6, 5, 9, 11, 13, 11, 4, 6, 7, 4, 3, 3, 2, 3, 6, 9

Offset: 1

Jonathan Vos Post, Jan 20 2006

Semiprime analog of A031131 "Difference between n-th prime and (n+2)nd prime."

			a(1) = 5 because 3rd semiprime - first semiprime = 9 - 4 = 5.
a(2) = 4 because semiprime(4) - semiprime(2) = 10 - 6 = 4.
a(3) = 5 because semiprime(5) - semiprime(3) = 14 - 9 = 5.
a(4) = 5 because semiprime(6) - semiprime(4) = 15 - 10 = 5.

Cf. A001358, A056809.

t = Select[ Range@320, Plus @@ Last /@ FactorInteger@# == 2 &]; Drop[t, 2] - Drop[t, -2] (* Robert G. Wilson v *)

a(n) = A001358(n+2) - A001358(n).

More terms from Robert G. Wilson v, Jan 21 2006

A115401 Record differences between prime(n+3) and prime(n). Records in A031165.

Original entry on oeis.org

5, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 40, 42, 46, 50, 54, 58, 60, 62, 64, 68, 78, 84, 112, 116, 118, 120, 126, 128, 142, 152, 170, 178, 184, 192, 194, 198, 208, 210, 216, 220, 222, 252, 258, 270, 300, 318, 336, 348, 354, 370, 408

Offset: 1

Jonathan Vos Post, Jan 22 2006

This is the k=3 case of the set of sequences "records in a(k,n) = prime(n+k) - prime(n)." The k=1 case is given by A005250 (ncreasing gaps between primes), A000101 [increasing gaps between primes (upper end)] and A002386, which gives lower ends of these gaps. The k=2 case is A031132. The merits of these records are (prime(n+3)-prime(n))/log (prime(n)). The first record merit is 5/log 2 = 16.6096405. The second record merit is 8/log 3 = 16.7672262.

			a(1) = A031165(1) = prime(4) - prime(1) = 7 - 2 = 5, which is the only odd element of this sequence.
a(2) = A031165(2) = prime(5) - prime(2) = 11 - 3 = 8.
a(3) = A031165(4) = prime(7) - prime(4) = 17 - 7 = 10.
a(4) = A031165(7) = prime(10) - prime(7) = 29 - 17 = 12.
a(5) = A031165(9) = prime(12) - prime(9) = 37 - 23 = 14.

Cf. A000040, A000101, A001223, A002386, A005250, A007530, A031131, A031132, A031165.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; d = 0; p = 1; q = 2; r = 3; s = 5; lst = {}; Do[{p, q, r, s} = {q, r, s, NextPrim[s]}; If[s > d + p, d = s - p; AppendTo[lst, d]; Print[d]], {n, 10^8}] (* Robert G. Wilson v *)

Corrected and extended by Robert G. Wilson v, Jan 23 2006

A161782 a(n) = sum of all numbers from and including (prime(n+1)-prime(n)) to and including (prime(n+2)-prime(n).)

Original entry on oeis.org

6, 9, 20, 15, 20, 15, 20, 49, 21, 35, 40, 15, 20, 49, 63, 21, 35, 40, 15, 35, 40, 49, 90, 50, 15, 20, 15, 20, 165, 80, 49, 21, 77, 33, 35, 63, 40, 49, 63, 21, 77, 33, 20, 15, 104, 234, 70, 15, 20, 49, 21, 77, 91, 63, 63, 21, 35, 40, 15, 77, 255, 80, 15, 20, 165, 119, 121, 33

Offset: 1

Juri-Stepan Gerasimov, Jun 20 2009

nonn

			n = 1: prime(1) = 2, prime(2) = 3, prime(3) = 5. Sum of all numbers from prime(2)-prime(1) = 1 to prime(3)-prime(1) = 3 is 1+2+3, hence a(1) = 6.
n = 11: prime(11) = 31, prime(12) = 37, prime(13) = 41. Sum of all numbers from prime(12)-prime(11) = 6 to prime(13)-prime(11) = 10 is 6+7+8+9+10, hence a(11) = 40.

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

Cf. A001223 (differences between consecutive primes), A031131 (difference between n-th prime and (n+2)nd prime).

[ &+[(NthPrime(n+1)-NthPrime(n))..(NthPrime(n+2)-NthPrime(n))]: n in [1..68] ];

Total[Range[#[[2]]-#[[1]],#[[3]]-#[[1]]]]&/@Partition[Prime[Range[70]],3,1] (* Harvey P. Dale, Oct 18 2021 *)

a(n) = Sum_{x=prime(n+1)-prime(n)..prime(n+2)-prime(n)} x = Sum_{x=A001223(n)..A031131(n)} x.

Edited and extended beyond a(33) by Klaus Brockhaus, Jun 23 2009

Definition clarified by Harvey P. Dale, Oct 18 2021

A161811 Difference between nonprime(n+2) and nonprime(n).

Original entry on oeis.org

4, 5, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 3, 2, 2, 2, 3, 4, 3, 2, 2, 2, 3, 3, 2, 3, 4, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 4, 3, 2, 2, 2, 3, 3, 2, 3, 4, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 4

Offset: 1

Juri-Stepan Gerasimov, Jun 20 2009

nonn

"nonprime(n)" is used for "n-th nonprime". Here the nonprimes start at 0 (see A141468), so nonprime(1) to nonprime(15) are 0, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22.

			nonprime(1+2)-nonprime(1) = 4-0; so a(1) = 4.
nonprime(5+2)-nonprime(5) = 10-8; so a(5) = 2.
nonprime(11+2)-nonprime(11) = 20-16; so a(11) = 4.

Cf. A031131, A141468.

#[[3]]-#[[1]]&/@Partition[Select[Range[0,150],!PrimeQ[#]&],3,1] (* Harvey P. Dale, Jul 08 2022 *)

a(n) = A141468(n+2)-A141468(n).

Edited, corrected (a(11)=2 replaced by 4) and extended by Klaus Brockhaus, Jun 24 2009

A261518 a(n+1) = prime(n + a(n)) - prime(n), a(1) = 1.

Original entry on oeis.org

1, 1, 2, 6, 22, 92, 508, 3674, 34452, 408104, 5925564, 103023888, 2102941162, 49588317960, 1332831700026, 40376512041704, 1365483356241318, 51130344360226830, 2104788801045148866, 94659739599219674872, 4625501078986781603540, 244380566194237434434094

Offset: 1

Reinhard Zumkeller, Aug 23 2015

nonn

Lucas A. Brown, Python program.

Cf. A000040, A001223, A031131, A031165, A031166, A031167, A031168, A031169, A031170, A031171, A031172.

a261518 n = a261518_list !! (n - 1)
a261518_list = 1 : zipWith (-)
               (map a000040 (zipWith (+) a261518_list [1..])) a000040_list

[1] cat [n le 1 select 1 else  NthPrime(n + Self(n-1)) - NthPrime(n): n in [1..12]]; // Vincenzo Librandi, Aug 24 2015

FoldList[(Prime[#2+#1]-Prime[#2])&,1,Range@15] (* Ivan N. Ianakiev, Aug 23 2015 *)
RecurrenceTable[{a[n+1] == Prime[n+a[n]] - Prime[n], a[1]==1}, a, {n, 1,16}] (* G. C. Greubel, Aug 24 2015 *)
nxt[{n_,a_}]:={n+1,Prime[n+a]-Prime[n]}; NestList[nxt,{1,1},16][[All,2]] (* Harvey P. Dale, May 15 2018 *)

a(n) = if(n==1, 1, prime(n-1 + a(n-1)) - prime(n-1));
vector(12, n, a(n)) \\ Altug Alkan, Oct 05 2015

Python
```
# see LINKS
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a(13)-a(16) from Ivan N. Ianakiev, Aug 23 2015
a(17)-a(19) from Jinyuan Wang, Jun 27 2020
a(20)-a(22) from Lucas A. Brown, Mar 18 2024

A338425 Numbers k such that the points [prime(k), prime(k+1)], [prime(k+2), prime(k+3)] and [prime(k+4), prime(k+5)] are collinear.

Original entry on oeis.org

3, 4, 25, 27, 41, 54, 103, 124, 140, 147, 149, 151, 186, 247, 271, 306, 345, 347, 354, 377, 398, 430, 464, 473, 504, 577, 578, 670, 682, 709, 767, 771, 787, 821, 823, 825, 827, 870, 1037, 1086, 1124, 1157, 1165, 1167, 1276, 1319, 1388, 1401, 1557, 1600, 1602, 1607, 1722, 1724, 1740, 1828, 1830

Offset: 1

Robert Israel, Oct 25 2020

nonn

Numbers k such that A031131(k)*A031131(k+3)=A031131(k+1)*A031131(k+2).

			a(3)=25 is in the sequence because the six primes starting with prime(25)=97 are 97, 101, 103, 107, 109, 113, and the points (97,101), (103,107) and (109,113) are collinear, all being on the line y=x+4.

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

Cf. A031131.

P:= [seq(ithprime(i), i=1..2005)]:
select(n -> (P[n+2]-P[n])*(P[n+5]-P[n+1]) = (P[n+3] - P[n+1])*(P[n+4]-P[n]), [$1..2000]);

A084856 Prime(n+2)^2-prime(n)^2.

Original entry on oeis.org

21, 40, 96, 120, 168, 192, 240, 480, 432, 528, 720, 480, 528, 960, 1272, 912, 1008, 1320, 840, 1200, 1560, 1680, 2520, 2280, 1200, 1248, 1272, 1320, 4248, 4392, 2640, 2160, 3432, 3480, 2448, 3768, 3240, 3360, 4152, 2832, 4440, 4488, 2328, 2352, 5712

Offset: 1

Meir Avshalomi (amomy(AT)yahoo.com), Jul 13 2003

nonn

			a(3)=prime(5)^2-prime(3)^2= 11^2-5^2=96

Table[ Prime[n + 2]^2 - Prime[n]^2, {n, 1, 45}]

A031131(n)*A048448(n).

Edited and extended by Robert G. Wilson v, Jul 19 2003

A110934 Difference between 3-almostprime(n) and 3-almostprime(n+2).

Original entry on oeis.org

10, 8, 9, 8, 3, 14, 14, 3, 6, 7, 13, 14, 5, 4, 7, 6, 3, 16, 20, 7, 4, 6, 8, 9, 6, 3, 8, 8, 6, 13, 17, 10, 6, 6, 11, 11, 6, 6, 2, 3, 3, 8, 11, 6, 4, 7, 17, 17, 15, 18, 9, 6, 7, 6, 6, 3, 2, 10, 12, 6, 8, 7, 7, 7, 6, 7, 5, 3, 2, 5, 6, 20, 24, 8, 6, 7, 10, 8, 6, 10, 7

Offset: 1

Jonathan Vos Post, Jan 21 2006

This is the 3-almost prime analog of what A113784 is for semiprimes and what A031131 is for primes. The minimum values in the sequence are 2 because we have, for example, the 3 consecutive 3-almost primes 170, 171, 172, so a(39) = A014612(41) - A014612(39) = 172 - 170 = 2. Equivalently, there are 2 consecutive 1 values of A114403 (3-almost prime gaps; first differences of A014612). This happens for elements of A113789 (numbers n such that n, n+1 and n+2 are 3-almost primes).

			a(1) = 10 because the difference between the first and third 3-almost primes is A014612(3) - A014612(1) = 18 - 8 = 10.
a(2) = A014612(4) - A014612(2) = 20 - 12 = 8.
a(3) = A014612(5) - A014612(3) = 27 - 18 = 9.

Cf. A014612, A031131, A067813, A113784, A113789, A114403.

a(n) = A014612(n+2) - A014612(n).

a(28) corrected by R. J. Mathar, Dec 22 2010

A214612 prime(n^3) - prime(n).

Original entry on oeis.org

0, 16, 98, 304, 680, 1308, 2292, 3652, 5496, 7890, 10926, 14716, 19362, 24766, 31272, 38820, 47598, 57498, 68964, 81728, 96064, 112212, 129990, 149628, 171432, 194942, 220758, 248744, 279322, 312470, 347580, 385962, 427032, 470794, 517404, 567720, 620374

Offset: 1

Jonathan Vos Post, Mar 06 2013

This is to exponent 3 as A213926 is to exponent 2.

			a(1) = prime(1^3) - prime(1) = 2-2 = 0.
a(2) = prime(2^3) - prime(2) = 19-3 = 16.
a(3) = prime(3^3) - prime(3) = 103-5 = 98.

Cf. A000040, A001223, A031131, A031165-A031172, A072473, A092504, A213926.

Table[Prime[n^3] - Prime[n], {n, 50}] (* T. D. Noe, Mar 07 2013 *)

A261468 a(n) = prime(n+2) mod prime(n).

Original entry on oeis.org

1, 1, 1, 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

Offset: 1

Altug Alkan, Aug 20 2015

			a(3) = 11 mod 5 = 1.

Cf. A031131.

[NthPrime(n+2) mod NthPrime(n): n in [1..80]]; // Vincenzo Librandi, Aug 20 2015

a[n_]:=PowerMod[Prime[n + 2], 1, Prime[n]]; Table[a[n], {n, 80}] (* Vincenzo Librandi, Aug 20 2015 *)
Mod[#[[3]],#[[1]]]&/@Partition[Prime[Range[80]],3,1] (* Harvey P. Dale, Mar 14 2020 *)

first(m)=vector(m,i,prime(i+2)% prime(i)); \\ Anders Hellström, Aug 20 2015

a(n) = A031131(n) = A075527(n) for n>3. - Alois P. Heinz, Aug 20 2015

cp's OEIS Frontend

A113784 Difference between semiprime(n) and semiprime(n+2).

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A115401 Record differences between prime(n+3) and prime(n). Records in A031165.

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A161782 a(n) = sum of all numbers from and including (prime(n+1)-prime(n)) to and including (prime(n+2)-prime(n).)

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A161811 Difference between nonprime(n+2) and nonprime(n).

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A261518 a(n+1) = prime(n + a(n)) - prime(n), a(1) = 1.

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A338425 Numbers k such that the points [prime(k), prime(k+1)], [prime(k+2), prime(k+3)] and [prime(k+4), prime(k+5)] are collinear.

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A084856 Prime(n+2)^2-prime(n)^2.

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A110934 Difference between 3-almostprime(n) and 3-almostprime(n+2).

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