A007510 -id:A007510 - cp's OEIS frontend

A117244 Single (or isolated or non-twin) primes (A007510) that are not Chen primes (A109611).

79, 97, 163, 173, 223, 277, 331, 367, 373, 383, 397, 439, 457, 547, 593, 607, 613, 673, 691, 709, 727, 733, 739, 757, 773, 853, 907, 929, 967, 997, 1013, 1069, 1087, 1103, 1123, 1129, 1171, 1181, 1213, 1223, 1237, 1249, 1307, 1373, 1423, 1433, 1447, 1493

Offset: 1

Jani Melik, Apr 22 2006

nonn

			79 is single prime, but not Chen prime, since 79 -2 = 77 = 7*11 is composite, and 79 + 2 = 81 = 3^4 is neither prime nor semiprime.

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

Cf. A007510, A109611.

isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc: isA109611 := proc(n) if isprime(n) then isprime(n+2) or isA001358(n+2) ; else false; end if; end proc: isA007510 := proc(n) if isprime(n) then not isprime(n-2) and not isprime(n+2) ; else false; end if ; end proc: isA117244 := proc(n) isA007510(n) and not isA109611(n) ; end proc: for n from 1 to 4000 do if isA117244(n) then printf("%d,",n) ; fi; end do ; # R. J. Mathar, Dec 09 2009

Select[Range[1500], PrimeQ[#] && !PrimeQ[#-2] && PrimeOmega[#+2] > 2 &] (* Amiram Eldar, Oct 19 2021 *)

isok(p) = isprime(p) && !isprime(p-2) && !isprime(p+2) && (bigomega(p+2) > 2); \\ Michel Marcus, Oct 19 2021

Terms beyond 397 from R. J. Mathar, Dec 09 2009

Offset corrected by Amiram Eldar, Oct 19 2021

A153478 Sum of first n isolated (or single) primes A007510.

Original entry on oeis.org

2, 25, 62, 109, 162, 229, 308, 391, 480, 577, 690, 817, 948, 1105, 1268, 1435, 1608, 1819, 2042, 2275, 2526, 2783, 3046, 3323, 3616, 3923, 4240, 4571, 4908, 5261, 5620, 5987, 6360, 6739, 7122, 7511, 7908, 8309, 8718, 9157, 9600

Offset: 1

Omar E. Pol, Dec 27 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A000040, A007504, A007510, A048598, A071148, A153474.

Accumulate[Select[Prime[Range[100]],!PrimeQ[#-2]&&!PrimeQ[#+2]&]]  (* Harvey P. Dale, Feb 08 2011 *)

A162308 Number of twin primes A001097 smaller than the non-twin prime A007510(n).

Original entry on oeis.org

0, 7, 9, 11, 11, 13, 15, 15, 15, 15, 19, 19, 19, 23, 23, 23, 23, 29, 29, 31, 33, 33, 33, 35, 37, 37, 39, 39, 39, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 45, 45, 45, 45, 47, 47, 47, 47, 47, 47, 47, 49, 49, 49, 49, 51, 51, 51, 53, 53, 55, 57, 57, 59, 59, 59, 59, 59, 59, 59

Offset: 1

Juri-Stepan Gerasimov, Jul 01 2009

nonn

			a(2)=7 counts the numbers 3, 5, 7, 11, 13, 17, 19 below 23=A007510(2).

Cf. A000040, A073425.

isA007510 := proc(n) RETURN(isprime(n) and not isprime(n-2) and not isprime(n+2)) ; end:
isA001097 := proc(n) RETURN(isprime(n) and (isprime(n-2) or isprime(n+2)) ) ; end:
A007510 := proc(n) local a; if n = 1 then 2; else for a from procname(n-1)+1 do if isA007510(a) then RETURN(a) ; fi; od: fi; end:
A162308 := proc(n) local a,k; a := 0 ; for k from 3 to A007510(n)-1 do if isA001097(k) then a := a+1; fi; od; a; end:
seq(A162308(n),n=1..120) ; # R. J. Mathar, Jul 02 2009

Edited by R. J. Mathar, Jul 02 2009

A064110 Let s(n) = n-th single prime (cf. A007510). Sequence is defined by recurrence a(n+1) = s(a(n)), n = 0,1,2,..., a(0)=1.

Original entry on oeis.org

1, 2, 23, 263, 2917, 38639, 603311, 11093633, 236524303, 5782539281

Offset: 0

Lubomir Alexandrov, Sep 07 2001

This is the "isolated prime Eratosthenes progression at base 1 (ipep(1))". The next ipep are: ipep(3) = 3, 37, 397, 4751, 64403, 1038629, 19661749,...; ipep(4) = 4, 47, 491, 5897, 81131, 1328167, 25467419,...; ipep(5) = 5, 53, 557, 6709, 93287, 1541191, 29778547,...; ...; ipep(22)= 22, 257, 2861, 37799, 589181, 10821757, 230452837,... ipep(24)= 24, 277, 3079, 40823, 640121, 11807167, 252480587,... and so on.

In the terminology of A007097 the name is "isolated_prime-th recurrence ..."

"Isolated Primes", by Richard L. Francis, J. Rec. Math., 11 (1978), 17-22.

Cf. A007097, A063502.

a(9) from Sean A. Irvine, Jun 12 2023

A133076 Successive digits of isolated primes A007510(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 10 2007

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

Cf. A033308, A007510.

With[{prs=Prime[Range[100]]},Flatten[IntegerDigits/@Complement[prs, Flatten[Select[Partition[prs,2,1],Last[#]-First[#]==2&]]]]] (* Harvey P. Dale, Sep 20 2011 *)

A167514 Index of prime(n) in A007510 or in A001097.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 2, 8, 9, 3, 10, 11, 4, 5, 12, 13, 6, 14, 15, 7, 8, 9, 10, 16, 17, 18, 19, 11, 12, 13, 20, 21, 22, 23, 14, 15, 16, 17, 24, 25, 26, 27, 28, 29, 18, 19, 30, 31, 20, 32, 33, 21, 22, 23, 34, 35, 24, 36, 37, 25, 26, 38, 39, 27, 28, 29, 40, 41, 30, 31, 32, 33, 34

Offset: 1

Juri-Stepan Gerasimov, Nov 05 2009

nonn

A007510 U A001097 = A000040.

Cf. A000040, A001097, A007510.

A007510(a(n))=n or A001097(a(n))=n.

Definition corrected by R. J. Mathar, May 30 2010

A374591 Even numbers that can be written as the sum of two isolated primes (A007510).

Original entry on oeis.org

4, 46, 60, 70, 74, 76, 84, 90, 94, 100, 102, 104, 106, 112, 114, 116, 120, 126, 130, 132, 134, 136, 142, 144, 146, 150, 154, 156, 158, 160, 162, 164, 166, 168, 172, 174, 176, 178, 180, 184, 186, 190, 192, 194, 196, 198, 200, 202, 204, 206, 210, 214, 216, 220

Offset: 1

Marc Groz, Jul 12 2024

nonn

			4 = 2 + 2 is a term, as 2 is the smallest isolated prime.
60 = 23 + 37 is the smallest term that is the sum of two distinct isolated primes.

Cf. A007510.

Lim=220;ip=Select[Prime[Range[Lim]], NoneTrue[#+{2, -2}, PrimeQ]&] ;ipp[a_]:={a,a};Select[Union[Total/@Join[ipp/@ip,Subsets[ip,{2}]]],EvenQ[#]&&#<=Lim&] (* James C. McMahon, Aug 10 2024 *)

A067774 Primes p such that p+2 is not a prime.

Original entry on oeis.org

2, 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383, 389

Offset: 1

Benoit Cloitre, Feb 06 2002

Primes n such that n!*B(n+1) is an integer where B(k) are the Bernoulli numbers.

All primes except for the lower members of twin primes - i.e. remove 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, ... - Gerard Schildberger, Feb 13 2005

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A049591.

Complement of A001359 in A000040, A025584, A007510.

[p: p in PrimesUpTo(400) | NextPrime(p)-p ne 2]; // Bruno Berselli, Apr 09 2013

f[n_]:=PrimeQ[n+2]; lst={}; Do[p=Prime[n]; If[ !f[p], AppendTo[lst,p]], {n,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 17 2009 *)

is(n)=isprime(n)&&!isprime(n+2) \\ Charles R Greathouse IV, Jul 01 2013

Except for a(1)=2, a(n+1)=A049591(n).

a(n) ~ n log n. - Charles R Greathouse IV, Jul 01 2013

Better description from Vladeta Jovovic, Dec 14 2002

A167706 The single or isolated numbers. The union of single (or isolated or non-twin) primes and single (or isolated or average of twin prime pairs) nonprimes.

Original entry on oeis.org

2, 4, 6, 12, 18, 23, 30, 37, 42, 47, 53, 60, 67, 72, 79, 83, 89, 97, 102, 108, 113, 127, 131, 138, 150, 157, 163, 167, 173, 180, 192, 198, 211, 223, 228, 233, 240, 251, 257, 263, 270, 277, 282, 293, 307, 312, 317, 331, 337, 348, 353, 359, 367, 373, 379, 383, 389

Offset: 1

Juri-Stepan Gerasimov, Nov 10 2009, Nov 14 2009

nonn

Equals A007510 U A014574.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A007510, A014574.

With[{nn = 78}, {2}~Join~Union[Transpose[Select[Partition[Prime@ Range@ nn, 3, 1], And[#[[2]] - #[[1]] != 2, #[[3]] - #[[2]] != 2] &]][[2]], Map[Mean, Select[Partition[Prime@ Range@ nn, 2, 1], Differences@ # == {2} &]]]] (* Michael De Vlieger, Feb 22 2017, after Harvey P. Dale at A007510 and A014574 *)

is(n)=if(n%6, (isprime(n) && !isprime(n-2) && !isprime(n+2)) || n==4, isprime(n-1) && isprime(n+1)) \\ Charles R Greathouse IV, Apr 29 2015

lista(pmax) = {my(p1 = 2, p2 = 3); print1(2, ", "); forprime(p3 = 5, pmax, if(p2 == p1 + 2, print1(p1 + 1, ", ")); if(p2 != p1 + 2 && p2 != p3 - 2, print1(p2, ", ")); p1 = p2; p2 = p3);} \\ Amiram Eldar, May 17 2024

a(n) ~ n log n. - Charles R Greathouse IV, Apr 29 2015

Corrected (97 inserted) by R. J. Mathar, Nov 16 2009

A132231 Primes congruent to 7 (mod 30).

Original entry on oeis.org

7, 37, 67, 97, 127, 157, 277, 307, 337, 367, 397, 457, 487, 547, 577, 607, 727, 757, 787, 877, 907, 937, 967, 997, 1087, 1117, 1237, 1297, 1327, 1447, 1567, 1597, 1627, 1657, 1747, 1777, 1867, 1987, 2017, 2137, 2287, 2347, 2377, 2437, 2467, 2557, 2617, 2647

Offset: 1

Omar E. Pol, Aug 15 2007

Primes ending in 7 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches, Feb 07 2009
Intersection of A030432 and A002476. - Ray Chandler, Apr 07 2009
Only from 4927 on, there are more composite numbers than primes in {7+30k}, see A227869. - M. F. Hasler, Nov 02 2013
Terms are non-twin primes A007510, except for 7. - Jonathan Sondow, Oct 27 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages
Omar E. Pol, Prime numbers in a sieve with 30 columns
Omar E. Pol, Sobre el patrón de los números primos

Cf. A007510, A007528, A010051, A068229, A117047, A129807, A039949, A132230-A132236, A214360.

a132231 n = a132231_list !! (n-1)
a132231_list = [x | k <- [0..], let x = 30 * k + 7, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

[p: p in PrimesUpTo(3000) | p mod 30 eq 7 ]; // Vincenzo Librandi, Aug 14 2012

Select[30*Range[0,100]+7,PrimeQ] (* Harvey P. Dale, Feb 01 2012 *)
Select[Prime[Range[1000]],MemberQ[{7},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)

forstep(p=7,1999,30,isprime(p)&&print1(p",")) \\ M. F. Hasler, Nov 02 2013

a(n) = A158573(n)*30 + 7. - Ray Chandler, Apr 07 2009

a(n) = A211890(4,n-1) for n <= 5. - Reinhard Zumkeller, Jul 13 2012

Extended by Ray Chandler, Apr 07 2009

cp's OEIS Frontend

A117244 Single (or isolated or non-twin) primes (A007510) that are not Chen primes (A109611).

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A153478 Sum of first n isolated (or single) primes A007510.

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A162308 Number of twin primes A001097 smaller than the non-twin prime A007510(n).

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A064110 Let s(n) = n-th single prime (cf. A007510). Sequence is defined by recurrence a(n+1) = s(a(n)), n = 0,1,2,..., a(0)=1.

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A133076 Successive digits of isolated primes A007510(n).

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A167514 Index of prime(n) in A007510 or in A001097.

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A374591 Even numbers that can be written as the sum of two isolated primes (A007510).

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Mathematica

A067774 Primes p such that p+2 is not a prime.

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Magma

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A167706 The single or isolated numbers. The union of single (or isolated or non-twin) primes and single (or isolated or average of twin prime pairs) nonprimes.

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