A079364 -id:A079364 - cp's OEIS frontend

A221309 Numbers m such that no subset of {m-1, m, m+1} sums up to a prime number.

25, 77, 85, 92, 93, 94, 118, 122, 123, 124, 133, 143, 144, 145, 160, 161, 170, 171, 185, 188, 202, 203, 206, 207, 208, 213, 214, 218, 235, 236, 237, 247, 248, 253, 259, 265, 266, 267, 275, 287, 290, 291, 295, 298, 302, 305, 319, 325, 328, 333, 334, 335, 340

Offset: 1

Reinhard Zumkeller, Jan 10 2013

nonn

A117499(a(n)) = 0.

			a(1) = 25: there are 7 nonempty subsets of {25-1, 25, 25+1}: {24}, {25}, {26}, {24,25}, {24,26}, {25,26} and {24,25,26} with sums and factorizations: 24=3*2^3, 25=5^2, 26=13*2, 49=7^2, 50=5^2*2, 51=17*3 and 75=5^2*3.

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

Subsequence of A079364.

a221309 n = a221309_list !! (n-1)
a221309_list = map (+ 1) $ elemIndices 0 a117499_list

Flatten@Position[
  Plus @@ # & /@ (Rest@Subsets[# + {-1, 0, 1}]) & /@
Range@340, {?(! PrimeQ@# &) ..}] (* _Hans Rudolf Widmer, Oct 26 2024 *)

A167705 Composite numbers having four composite nearest neighbors.

Original entry on oeis.org

26, 34, 50, 56, 64, 76, 86, 92, 93, 94, 116, 117, 118, 119, 120, 121, 122, 123, 124, 134, 142, 143, 144, 145, 146, 154, 160, 170, 176, 184, 185, 186, 187, 188, 202, 203, 204, 205, 206, 207, 208, 214, 215, 216, 217, 218, 219, 220, 236, 244, 245, 246, 247, 248, 254, 260, 266, 274, 286, 287, 288, 289, 290, 296, 297, 298, 299, 300

Offset: 1

Juri-Stepan Gerasimov, Nov 10 2009

Almost all natural numbers are members of this sequence.

			a(1)=26 (24,25,27,28 are composite nearest neighbors), a(2)=34 (32,33,35,36 are composite nearest neighbors).

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A002808, A079364.

Select[Range[6!], !PrimeQ[#]&&!PrimeQ[#-1]&&!PrimeQ[#+1]&&!PrimeQ[#-2]&&!PrimeQ[#+2]&] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)
Flatten[Position[Partition[PrimeQ[Range[310]],5,1],?(Union[#]=={False}&),{1},Heads->False]]+2 (* _Harvey P. Dale, Nov 01 2013 *)

for(n=4,1e4,if(n-precprime(n)>2&&nextprime(n)-n>2,print1(n",")))

Comment and program from Charles R Greathouse IV, Nov 12 2009

Corrected terms, Clark Kimberling and Joerg Arndt, Jun 24 2011.

A210940 The prime numbers and their nonprime nearest-neighbors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 28, 29, 30, 31, 32, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 52, 53, 54, 58, 59, 60, 61, 62, 66, 67, 68, 70, 71, 72, 73, 74, 78, 79, 80, 82, 83, 84, 88, 89, 90, 96, 97, 98, 100

Offset: 1

Omar E. Pol, Apr 17 2012

nonn

The prime numbers and their nearest-neighbors without repetitions.

Union of A210939 and A000040. Complement of A079364.

Cf. A018252, A045718, A134928, A134930.

{#-1,#,#+1}&/@Prime[Range[30]]//Flatten//Union (* Harvey P. Dale, Jul 06 2019 *)

Corrected (74 added) by Harvey P. Dale, Jul 06 2019

A160666 Numbers whose distance to the closest prime number is a prime number.

Original entry on oeis.org

0, 9, 15, 21, 25, 26, 27, 33, 34, 35, 39, 45, 49, 50, 51, 55, 56, 57, 63, 64, 65, 69, 75, 76, 77, 81, 85, 86, 87, 91, 92, 94, 95, 99, 105, 111, 115, 116, 118, 120, 122, 124, 125, 129, 133, 134, 135, 141, 142, 144, 146, 147, 153, 154, 155, 159, 160, 161, 165, 169, 170

Offset: 1

Kyle Stern, May 22 2009

nonn

Terms n=2..31 are identical to terms n=1..30 of A079364.

K. Stern, Table of n, a(n) for n=1..10000

Cf. A000040, A051699.

isA160666 := proc(n) local ppl,pmi ; if isprime(n) then RETURN(false): elif n =0 then RETURN(true): elif n =1 then RETURN(false): fi; ppl := nextprime(n)-n ; pmi := n-prevprime(n) ; RETURN (isprime(min(ppl,pmi)) ) ; end: for n from 0 to 200 do if isA160666(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, May 25 2009

fQ[n_] := PrimeQ[ Min[ NextPrime[n] - n, n - NextPrime[n, -1]]]; Select[ Range[0, 174], !PrimeQ@ # && fQ@# &] (* Robert G. Wilson v, May 25 2009 *)

More terms from R. J. Mathar and Robert G. Wilson v, May 25 2009

A167776 Composite numbers having six composite nearest-neighbors.

Original entry on oeis.org

93, 117, 118, 119, 120, 121, 122, 123, 143, 144, 145, 185, 186, 187, 203, 204, 205, 206, 207, 215, 216, 217, 218, 219, 245, 246, 247, 287, 288, 289, 297, 298, 299, 300, 301, 302, 303, 321, 322, 323, 324, 325, 326, 327, 341, 342, 343, 363, 393, 405, 413, 414

Offset: 1

Juri-Stepan Gerasimov, Nov 11 2009

nonn

Terms lie between primes separated by a gap of at least 8 (see A083371). - David A. Corneth, Jun 24 2016

			a(1)=117 (114,115,116,118,119,120 are composite nearest-neighbors);
a(2)=118 (115,116,117,119,120,121 are composite nearest-neighbors).
There are no primes between primes 241 and 251 which gives a gap of 10 between them. Therefore, all numbers between (inclusive) 241 + 4 and 251 - 4 are terms. - _David A. Corneth_, Jun 24 2016

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

Cf. A002808, A079364, A083371, A167705.

Select[Range[6!],!PrimeQ[#] && !PrimeQ[#-1] && !PrimeQ[#+1] && !PrimeQ[#-2] && !PrimeQ[#+2] && !PrimeQ[#-3] && !PrimeQ[#+3]&] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)
Select[Range@ 414, Times @@ Boole@ Map[CompositeQ, Range[# - 3, # + 3]] == 1 &] (* Michael De Vlieger, Jun 24 2016 *)

lista(n) = {forprime(i=2,n+3,g=nextprime(i+1)-i;
for(j=i+4,i+g-4,print1(j", ")))}
a(n) = {forprime(i=88,,g=nextprime(i+1)-i;n-=max(0,g-7);
if(n<=0,return(i+g-4+n)))}
\\ gives the next term larger than n, whether n is a term or not.
nxt(n) = my(p=nextprime(n),g=0); if(p-n>4, n+1, while(1, q=nextprime(p+1); g=q-p; if(g>7, return(p+4), p=q))) \\ David A. Corneth, Jun 24 2016

Corrected (93, 144, 145 inserted) by R. J. Mathar, May 30 2010

A168141 a(n) = pi(n + 1) - pi(n - 2), where pi is the prime counting function.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0

Offset: 1

Juri-Stepan Gerasimov, Nov 19 2009

nonn

Conjecture: a(n) = 2 for infinitely many n. This is equivalent to the twin prime conjecture. - Andrew Slattery, Apr 26 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Twin Prime Conjecture
Wikipedia, Twin prime

Cf. A000720, A014574, A079364, A090406.

A168141 := proc(n) numtheory[pi](n+1)-numtheory[pi](n-2) ; end proc: seq(A168141(n),n=1..120) ; # R. J. Mathar, Nov 19 2009
# second Maple program:
a:= n-> add(`if`(isprime(n+i), 1, 0), i=-1..1):
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 28 2020

Table[PrimePi[n + 1] - PrimePi[n - 2], {n, 100}] (* Wesley Ivan Hurt, Apr 26 2020 *)

a(n) = primepi(n+1) - primepi(n-2); \\ Michel Marcus, Apr 27 2020

From Alois P. Heinz, Apr 28 2020: (Start)
a(n) = 2 <=> n in { 2,3 } union { A014574 }.
a(n) = 0 <=> n in A079364. (End)

cp's OEIS Frontend

A221309 Numbers m such that no subset of {m-1, m, m+1} sums up to a prime number.

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A167705 Composite numbers having four composite nearest neighbors.

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A210940 The prime numbers and their nonprime nearest-neighbors.

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A160666 Numbers whose distance to the closest prime number is a prime number.

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A167776 Composite numbers having six composite nearest-neighbors.

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A168141 a(n) = pi(n + 1) - pi(n - 2), where pi is the prime counting function.

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