A007510 -id:A007510 - cp's OEIS frontend

A134797 Odd isolated primes.

23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563, 577

Offset: 1

Omar E. Pol, Nov 14 2007

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Flatten[Select[Split[Prime[Range[106]], #2 - #1 < 4 &], Length[#] == 1 &]] (* Jayanta Basu, Jun 07 2013 *)

select(p->!isprime(p-2)&&!isprime(p+2),primes(100)[2..100]) \\ Charles R Greathouse IV, Jun 07 2013

a(n) = A007510(n+1).

a(n) ~ n log n. - Charles R Greathouse IV, Jun 07 2013

A062505 Numbers k such that if p is a prime that divides k, then either p + 2 or p - 2 is also prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 49, 51, 55, 57, 59, 61, 63, 65, 71, 73, 75, 77, 81, 85, 87, 91, 93, 95, 99, 101, 103, 105, 107, 109, 117, 119, 121, 123, 125, 129, 133, 135, 137, 139, 143, 145, 147, 149, 151, 153, 155, 165

Offset: 1

Leroy Quet, Jul 09 2001

nonn

Multiplicative closure of twin primes (A001097).

			35 is included because 35 = 5*7 and both (5+2) and (7-2) are primes.
65 = 5*13 where the factors are members of twin prime pairs: (3,5) and (11,13), therefore a(29) = 65 is a term; but 69 is not because 69 = 3*23 and 23 = A007510(2) is a single prime.

Stephan Ramon Garcia and Steven J. Miller, 100 Years of Math Milestones: The Pi Mu Epsilon Centennial Collection, American Mathematical Society, 2019, pp. 35-37.

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

Range of A072963.

Cf. A001097, A074480.

[k:k in [1..170] | forall{p:p in PrimeDivisors(k)| IsPrime(p+2) or IsPrime(p-2)}]; // Marius A. Burtea, Dec 30 2019

nmax = 15 (* corresponding to last twin prime pair (197,199) *); tp[1] = 3; tp[n_] := tp[n] = (p = NextPrime[tp[n-1]]; While[ !PrimeQ[p+2], p = NextPrime[p]]; p); twins = Flatten[ Table[ {tp[n], tp[n]+2}, {n, 1, nmax}]]; max = Last[twins]; mult[twins_] := Select[ Union[ twins, Apply[ Times, Tuples[twins, {2}], {1}]], # <= max & ]; A062505 = Join[{1}, FixedPoint[mult, twins] ] (* Jean-François Alcover, Feb 23 2012 *)

A072027 Swap (2,3) and all twin prime pairs >(3,5) in prime factorization of n.

Original entry on oeis.org

1, 3, 2, 9, 7, 6, 5, 27, 4, 21, 13, 18, 11, 15, 14, 81, 19, 12, 17, 63, 10, 39, 23, 54, 49, 33, 8, 45, 31, 42, 29, 243, 26, 57, 35, 36, 37, 51, 22, 189, 43, 30, 41, 117, 28, 69, 47, 162, 25, 147, 38, 99, 53, 24, 91, 135, 34, 93, 61, 126, 59, 87, 20, 729, 77

Offset: 1

Reinhard Zumkeller, Jun 07 2002

			a(143) = a(11*13) = a(11)*a(13) = 13*11 = 143.
a(77) = a(7*11) = a(7)*a(11) = 5*13 = 65.

Cf. A001359, A006512, A007510, A037074, A061898, A064505, A072026, A072028, A072029.

f[p_, e_] := If[p < 5, 5 - p, If[PrimeQ[p + 2], p + 2, If[PrimeQ[p - 2], p - 2, p]]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 26 2024 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; if(p < 5, 5-p, if(isprime(p+2), p+2, if(isprime(p-2), p-2, p)))^f[i,2]);} \\ Amiram Eldar, Feb 26 2024

Multiplicative with a(p) = (if p<=3 then 5-p else (if p+2 is prime then p+2 else (if p-2 is prime then p-2 else p))), p prime.
a(a(n)) = n, self-inverse permutation of natural numbers.
a(n) = n for single primes (A007510) and products of twin prime pairs (A037074).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{(p < q) swapped pair} ((p^2-p)*(q^2-q)/((p^2-q)*(q^2-p))) = 1.832194438922717... . - Amiram Eldar, Feb 26 2024

A102723 Smallest prime a(n) such that a(n)-x and a(n)+x, for x=1 to n, are all composite.

Original entry on oeis.org

5, 23, 23, 53, 53, 211, 211, 211, 211, 211, 211, 1847, 1847, 2179, 2179, 2179, 2179, 3967, 3967, 16033, 16033, 16033, 16033, 24281, 24281, 24281, 24281, 24281, 24281, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 58831

Offset: 1

Ray G. Opao, Feb 06 2005

nonn

a(2n+1)=a(2n). - Robert G. Wilson v, Feb 22 2005

Using Dirichlet's theorem, Sierpiński (1948) proved that a(n) exists for all n > 0. He noted that a(n) is a non-twin prime (A007510), except for a(1) = 5. - Jonathan Sondow, Oct 27 2017

David A. Corneth, Table of n, a(n) for n = 1..479 (first 97 terms from Harvey P. Dale)
W. Sierpiński, Remarque sur la répartition des nombres premiers, Colloq. Math., 1 (1948), 193-194.

Cf. A007510, A023186.

f[n_] := Block[{k = 1}, While[ Union[ PrimeQ /@ Sort[ Flatten[ Table[{Prime[k] - i, Prime[k] + i}, {i, n}]]]] != {False}, k++ ]; Prime[k]]; Table[ f[n], {n, 40}] (* Robert G. Wilson v, Feb 22 2005 *)
cmpgap[n_]:=Module[{p=Prime[n]},Min[p-NextPrime[p,-1],NextPrime[p]-p]]; Module[{nn=10000,prs},prs=Table[{Prime[n],cmpgap[n]},{n,nn}];Table[ SelectFirst[ prs,#[[2]]>=k&],{k,2,50}]][[All,1]] (* Harvey P. Dale, Oct 15 2021 *)

a(12)-a(40) from Robert G. Wilson v, Feb 22 2005

A132237 Primes congruent to {7, 23} mod 30.

Original entry on oeis.org

7, 23, 37, 53, 67, 83, 97, 113, 127, 157, 173, 233, 263, 277, 293, 307, 337, 353, 367, 383, 397, 443, 457, 487, 503, 547, 563, 577, 593, 607, 653, 683, 727, 743, 757, 773, 787, 863, 877, 907, 937, 953, 967, 983, 997, 1013, 1087, 1103

Offset: 1

Omar E. Pol, Aug 15 2007

Up to 4913, there are more primes of this form than composites. See also A132231 and A227869 (congruent to 7 only). - M. F. Hasler, Nov 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000040, A007510, A039949, A068229, A129807.

[ p: p in PrimesUpTo(1300) | p mod 30 in [7, 23] ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{7,23},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

is_A132237(n)=setsearch([7,23],n%30)&&isprime(n) \\ - M. F. Hasler, Nov 02 2013

A167511 The count of isolated primes between n-th non-isolated nonprime and n-th isolated nonprime.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 4, 5, 9, 9, 12, 11, 15, 15, 15, 17, 18, 21, 22, 24, 27, 36, 36, 40, 47, 51, 54, 55, 56, 58, 76, 76, 75, 77, 79, 96, 96, 97, 97, 99, 105, 114, 116, 117, 118, 119, 127, 130, 132, 132, 146, 147, 151, 151, 152, 159, 166, 166, 169, 169, 173, 176, 180, 180, 181

Offset: 1

Juri-Stepan Gerasimov, Nov 05 2009

nonn

			a(1)=1 (0<2<4); a(2)=1 (1<2<6); a(3)=0 (8<no<12); a(4)=0 (9<no<18); a(5)=1 ( 10<23<30); a(5)=2 (14<23&37<42); a(5)=4 (15<23&37&47&53<60).

Cf. A007510, A014574, A164276.

Contribution from R. J. Mathar, Mar 18 2010: (Start)
isA007510 := proc(n) isprime(n) and not isprime(n+2) and not isprime(n-2) ; end proc:
isA001359 := proc(n) isprime(n) and isprime(n+2) ; end proc:
A001359 := proc(n) if n = 1 then 3 ; else for a from procname(n-1)+2 do if isA001359(a) then return a; end if; end do: end if: end proc:
isA164276 := proc(n) not isprime(n) and (not isprime(n-1) or not isprime(n+1)) ; end proc:
A164276 := proc(n) if n = 1 then 0; else for a from procname(n-1)+1 do if isA164276(a) then return a; end if; end do: end if: end proc:
A014574 := proc(n) A001359(n)+1 ; end proc:
A167511 := proc(n) a := 0 ; for i from A164276(n)+1 to A014574(n)-1 do if isA007510(i) then a :=a +1 ; end if; end do; a ; end proc:
seq(A167511(n),n=1..80) ; (End)

a(n) = #{ A007510(i): A164276(n) < A007510(i) < A014574(n)}. [From R. J. Mathar, Mar 18 2010]

a(n) = SUM{A010051(k)*(1-A164292(k)): A164276(n)<=k<=A014574(n)}. [From Reinhard Zumkeller, Apr 02 2010]

a(12), a(31) and a(32) corrected by R. J. Mathar, Mar 18 2010

A167771 Twice-isolated primes: primes p such that neither p+-2 nor p+-4 is prime.

Original entry on oeis.org

2, 53, 89, 157, 173, 211, 251, 257, 263, 293, 331, 337, 359, 367, 373, 389, 409, 449, 479, 509, 541, 547, 557, 563, 577, 587, 593, 607, 631, 653, 683, 691, 701, 709, 719, 727, 733, 751, 787, 797, 839, 919, 929, 947, 953, 977, 983, 991, 997, 1039, 1069, 1103

Offset: 1

Juri-Stepan Gerasimov, Nov 11 2009

nonn

2 together with primes p with property that (p-previous prime)>=6 and (next prime-p)>=6.

By the finitude of the generalized Brun constants, this sequence includes almost all primes.

			a(1)=2 (-2,0,4,6 are nonprimes); a(2)=53 (49,51,55,57 are nonprimes).

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

Cf. A007510, A137869.

Join[{2},Select[Prime[Range[200]],NoneTrue[#+{4,2,-2,-4},PrimeQ]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 21 2016 *)

Comment from Charles R Greathouse IV, Nov 12 2009

A069456 Non-twin primes that are at least doubly lonely.

Original entry on oeis.org

1039, 2099, 4253, 91121, 386401, 626617, 754973, 873553, 908857, 972137, 1619353, 1749067, 1841681, 2007899, 2169007, 2241353, 2420633, 2484931, 2594971, 3075323, 3129601, 3151843, 3837451, 3843247, 3919229, 4038709, 4545683, 5502449, 5530529, 5921869

Offset: 1

Neil Fernandez, Mar 23 2002

nonn

			These are non-twin primes sandwiched between at least 2 pairs of twins on each side. The first number in the sequence is 1039 (sandwiched between 1019,1021,1031,1033 and 1049,1051,1061,1063).

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Cf. A007510, A069453.

Primes:= select(isprime,[seq(i,i=3..6*10^6,2)]):
good:= select(t -> Primes[t-3]-Primes[t-4]=2 and Primes[t-1]-Primes[t-2]=2 and Primes[t+2]-Primes[t+1]=2 and Primes[t+4]-Primes[t+3]=2, [$5..nops(Primes)-4]):
Primes[good]; # Robert Israel, May 13 2016

dltpQ[{a_,b_,c_,d_,e_,f_,g_,h_,i_}]:=b-a==d-c==g-f==i-h==2; Transpose[ Select[ Partition[Prime[Range[410000]],9,1],dltpQ]][[5]] (* Harvey P. Dale, May 14 2013 *)

More terms from Arkadiusz Wesolowski, May 08 2012

A074038 If n is the k-th single (i.e., non-twin) prime then a(n) = k, otherwise a(n) = 0.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 13 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007510, A049084, A074039.

upto = 100;
A074038list[upto_] := Module[{k = 0, v = Table[0, {upto}], n},
    For[n = 1, n <= upto, n++, If[PrimeQ[n] && (2 == n ||
    ((!PrimeQ[n-2]) && (!PrimeQ[n+2]))), k++; v[[n]] = k,
    v[[n]] = 0]]; v];
v074038 = A074038list[upto];
a[n_] := v074038[[n]];
Table[a[n], {n, 1, upto}] (* Jean-François Alcover, Dec 03 2021, after PARI code *)

up_to = 65537;
A074038list(up_to) = { my(k=0,v=vector(up_to)); for(n=1,up_to,if(isprime(n)&&(2==n||((!isprime(n-2))&&(!isprime(n+2)))),k++;v[n]=k, v[n] = 0)); (v); };
v074038 = A074038list(up_to);
A074038(n) = v074038[n];

A007510(a(n)) = n.

A121762 Single (or isolated or non-twin) primes of form 6n-1.

Original entry on oeis.org

23, 47, 53, 83, 89, 113, 131, 167, 173, 233, 251, 257, 263, 293, 317, 353, 359, 383, 389, 401, 443, 449, 467, 479, 491, 503, 509, 557, 563, 587, 593, 647, 653, 677, 683, 701, 719, 743, 761, 773, 797, 839, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013

Offset: 1

Lekraj Beedassy, Aug 20 2006

nonn

Subsequence of A007528. - Michel Marcus, Apr 26 2015

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A007510, A121763, A007528, A121764.

[n: n in [1..1050] | (n mod 6 eq 5) and not IsPrime(n+2) and  IsPrime(n)]; // G. C. Greubel, Feb 26 2019

Select[Table[6n - 1, {n, 200}], PrimeQ[ # ] && ! PrimeQ[ # + 2] &] (* Ray Chandler, Aug 22 2006 *)

is(n)=n%6==5 && isprime(n) && !isprime(n+2) \\ Charles R Greathouse IV, Apr 04 2016

[n for n in (1..1050) if mod(n,6)==5 and not is_prime(n+2) and  is_prime(n)] # G. C. Greubel, Feb 26 2019

Extended by Ray Chandler, Aug 22 2006

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A134797 Odd isolated primes.

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A062505 Numbers k such that if p is a prime that divides k, then either p + 2 or p - 2 is also prime.

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A072027 Swap (2,3) and all twin prime pairs >(3,5) in prime factorization of n.

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A102723 Smallest prime a(n) such that a(n)-x and a(n)+x, for x=1 to n, are all composite.

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A132237 Primes congruent to {7, 23} mod 30.

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A167511 The count of isolated primes between n-th non-isolated nonprime and n-th isolated nonprime.

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A167771 Twice-isolated primes: primes p such that neither p+-2 nor p+-4 is prime.

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A069456 Non-twin primes that are at least doubly lonely.

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