A007510 -id:A007510 - cp's OEIS frontend

A121764 Single (or isolated or non-twin) primes of form 6n + 1.

37, 67, 79, 97, 127, 157, 163, 211, 223, 277, 307, 331, 337, 367, 373, 379, 397, 409, 439, 457, 487, 499, 541, 547, 577, 607, 613, 631, 673, 691, 709, 727, 733, 739, 751, 757, 769, 787, 853, 877, 907, 919, 937, 967, 991, 997, 1009, 1039, 1069, 1087, 1117

Offset: 1

Lekraj Beedassy, Aug 20 2006

nonn

For the first 30000 terms a(n) > A121762(n), see plot A121764(n) - A121762(n). But is it so for all n? - Zak Seidov, Apr 25 2015

Subsequence of A002476. - Michel Marcus, Apr 26 2015

Zak Seidov, Table of n, a(n) for n = 1..10000
Zak Seidov, Plot of A121764(n) - A121762(n)

Cf. A007510, A121765, A002476, A121762, A038179, A007310, A038511.

[n: n in [1..1150] | (n mod 6 eq 1) 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 *)
Select[Prime[Range[200]],Mod[#,6]==1&&NoneTrue[#+{2,-2},PrimeQ]&] (* Harvey P. Dale, Jul 16 2021 *)

{is(n)=n%6==1 && isprime(n) && !isprime(n-2)}; \\ G. C. Greubel, Feb 26 2019

[n for n in (1..1150) if mod(n,6)==1 and not is_prime(n-2) and is_prime(n)] # G. C. Greubel, Feb 26 2019

Extended by Ray Chandler, Aug 22 2006

A132259 Isolated primes congruent to {1, 17, 19, 29} mod 30.

Original entry on oeis.org

47, 79, 89, 167, 211, 257, 317, 331, 359, 379, 389, 409, 439, 449, 467, 479, 499, 509, 541, 557, 587, 631, 647, 677, 691, 709, 719, 739, 751, 769, 797, 839, 887, 919, 929, 947, 977, 991, 1009, 1039, 1069, 1097, 1109, 1129, 1171, 1187

Offset: 1

Omar E. Pol, Aug 20 2007

Harvey P. Dale, 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. A007510, A074822, A078857.

Select[Prime[Range[200]],NoneTrue[#+{2,-2},PrimeQ]&&MemberQ[ {1,17,19,29},Mod[ #,30]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 04 2019 *)

A162734 An alternating sum of all numbers from the n-th up to the (n+1)st isolated prime.

Original entry on oeis.org

11, 30, 42, 50, 60, 73, 81, 86, 93, 105, 120, 129, 144, 160, 165, 170, 192, 217, 228, 242, 254, 260, 270, 285, 300, 312, 324, 334, 345, 356, 363, 370, 376, 381, 386, 393, 399, 405, 424, 441, 446, 453, 462, 473, 483, 489, 495, 501, 506, 525, 544, 552, 560

Offset: 1

Juri-Stepan Gerasimov, Jul 13 2009

11 followed by the average of each two consecutive non-twin primes. - Colin Barker, Jul 17 2014

			a(1) = -2+3-4+5-6+7-8+9-10+11-12+13-14+15-16+17-18+19-20+21-22+23 = 11.
a(2) = 23-24+25-26+27-28+29-30+31-32+33-34+35-36+37 = 30.

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

Cf. A007510.

N:= 1000: # to get all terms where the larger non-twin <= N
Primes:= select(isprime,{seq(2*i-1,i=1..floor((N+1)/2))}):
NonTwins:= Primes minus (map(t->t+2,Primes) union map(t->t-2,Primes)):
11, seq((NonTwins[i]+NonTwins[i+1])/2,i=1..nops(NonTwins)-1); # Robert Israel, Jul 21 2014

non_twin_primes(pmax) = my(s=[]); forprime(p=2, pmax, if(!isprime(p-2) && !isprime(p+2), s=concat(s, p))); s
a162734(maxp) = my(ntp=non_twin_primes(maxp)); vector(#ntp-1, n, sum(k=ntp[n], ntp[n+1], -k*(-1)^k))
a162734(500) \\ Colin Barker, Jul 17 2014

from sympy import isprime, primerange
def nontwins(N):
    return [p for p in primerange(1, N+1) if not (isprime(p-2) or isprime(p+2))]
def auptont(N): # all terms where the larger non-twin <= N
    nt = nontwins(N)
    return [sum((-1)**(j+1)*j for j in range(nt[i], nt[i+1]+1)) for i in range(len(nt)-1)]
print(auptont(565)) # Michael S. Branicky, Nov 30 2021

a(n) = sum_{j= A007510(n).. A007510(n+1)} (-1)^(j+1)*j = A001057(A007510(n+1))-A001057(A007510(n)-1).

Replaced 55 by 60 and 447 by 446 - R. J. Mathar, Sep 23 2009

A176312 Numbers that are the products of two single (or isolated or non-twin) primes.

Original entry on oeis.org

4, 46, 74, 94, 106, 134, 158, 166, 178, 194, 226, 254, 262, 314, 326, 334, 346, 422, 446, 466, 502, 514, 526, 529, 554, 586, 614, 634, 662, 674, 706, 718, 734, 746, 758, 766, 778, 794, 802, 818, 851, 878, 886, 898, 914, 934, 958, 974, 982, 998, 1006, 1018, 1081

Offset: 1

Juri-Stepan Gerasimov, Apr 15 2010

nonn

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

Cf. A007510.

isA007510 := proc(n) isprime(n) and not isprime(n+2) and not isprime(n-2) ; simplify(%) ; end proc:
isA176312 := proc(n) for d in numtheory[divisors](n) do if isA007510(d) and isA007510(n/d) then return true; end if; end do: return false; end proc:
for n from 1 to 1200 do if isA176312(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Apr 20 2010:

Select[Range[1000], PrimeOmega[#] == 2 && AllTrue[FactorInteger[#][[;;, 1]], ! PrimeQ[#1 - 2] && ! PrimeQ[#1 + 2] &] &] (* Amiram Eldar, Nov 30 2020 *)

Entries checked by R. J. Mathar, Apr 20 2010

A176656 The positions of single (or isolated or non-twin) primes in A000040.

Original entry on oeis.org

1, 9, 12, 15, 16, 19, 22, 23, 24, 25, 30, 31, 32, 37, 38, 39, 40, 47, 48, 51, 54, 55, 56, 59, 62, 63, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 97, 100, 101, 102, 103, 106, 107, 108, 111, 112, 115, 118, 119, 122, 123, 124

Offset: 1

Juri-Stepan Gerasimov, Apr 23 2010

nonn

A176656 := proc(n) numtheory[pi](A007510(n)) ; end proc: seq(A176656(n),n=1..120) ; # R. J. Mathar, Apr 26 2010

A000040(a(n))=A007510(k).

Corrected (32 inserted, one of the 94 removed, 103 inserted, 121 replaced by 122) by R. J. Mathar, Apr 26 2010

A268343 Hermit primes: primes which are not a nearest neighbor of another prime.

Original entry on oeis.org

23, 37, 53, 67, 89, 97, 113, 157, 173, 211, 233, 277, 293, 307, 317, 359, 389, 409, 449, 457, 467, 479, 509, 577, 607, 631, 653, 691, 719, 751, 839, 853, 863, 877, 887, 919, 929, 1039, 1069, 1087, 1201, 1223, 1237, 1283, 1297, 1307, 1327, 1381, 1423, 1439

Offset: 1

Karl W. Heuer, Feb 02 2016

If p is a balanced prime (A006562), with two nearest neighbors, then it eliminates both of those neighbors from being hermits.
Conjecture: the asymptotic probability of a prime being in this list is 1/4.
A subsequence of the isolated primes A007510. The sequence of lonely primes A087770 appears to be a subsequence, except for its first three terms (2, 3 and 7). (This would not be true if one of these were followed by two increasingly larger gaps.) - M. F. Hasler, Mar 15 2016

			53 is in the list because the previous prime, 47, is closer to 43 than to 53, and the following prime, 59, is closer to 61 than to 53.

Karl W. Heuer, Table of n, a(n) for n = 1..30000
Robert Israel, Table of n, a(n) for n = 1..2600035

Cf. A269734 (number of hermit primes <= prime(n)).

N:= 1000: # to get all terms <= N
pr:= select(isprime, [$2 .. nextprime(nextprime(N))]):
Np:= nops(pr):
ishermit:= Vector(Np,1):
d:= pr[3..Np] + pr[1..Np-2] - 2*pr[2..Np-1]:
for i from 1 to Np-2 do
  if d[i] > 0 then ishermit[i]:= 0
elif d[i] < 0 then ishermit[i+2]:= 0
else ishermit[i]:= 0; ishermit[i+2]:= 0
fi
od:
subs(0=NULL, zip(`*`, pr[1..Np-2],convert(ishermit,list))); # Robert Israel, Mar 09 2016

Select[Prime@ Range@ 228, Function[n, AllTrue[{Subtract @@ Take[#, 2], Subtract @@ Reverse@ Take[#, -2]} &@ Differences[NextPrime[n, #] & /@ {-2, -1, 0, 1, 2}], # < 0 &]]] (* Michael De Vlieger, Feb 02 2016, Version 10 *)

A268343_list(LIM=1500)={my(d=vector(4),i,o,L=List());forprime(p=1,LIM,(d[i++%4+1]=-o+o=p)d[(i-3)%4+1]&&listput(L,p-d[i%4+1]-d[(i-1)%4+1]));Vec(L)} \\ M. F. Hasler, Mar 15 2016

is_A268343(n,p=precprime(n-1))={n-p>p-precprime(p-1)&&(p=nextprime(n+1))-n>nextprime(p+1)-p&&isprime(n)} \\ M. F. Hasler, Mar 15 2016

Deleted my incorrect conjecture about asymptotic behavior. - N. J. A. Sloane, Mar 10 2016

A072963 In prime factorization of n replace all single (i.e., non-twin) primes with 1.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 1, 3, 25, 13, 27, 7, 29, 15, 31, 1, 33, 17, 35, 9, 1, 19, 39, 5, 41, 21, 43, 11, 45, 1, 1, 3, 49, 25, 51, 13, 1, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 1, 65, 33, 1, 17, 3, 35, 71, 9, 73, 1, 75, 19, 77, 39, 1, 5, 81

Offset: 1

Reinhard Zumkeller, Aug 20 2002

a(a(n)) = a(n).

			a(92) = a(2*2*23) = 1; a(93) = a(3*31) = 3*31 = 93.

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

Cf. A007510, A000040.

f[p_, e_] := If[!Or @@ PrimeQ[p + {-2, 2}], 1, p^e]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 81] (* Amiram Eldar, Jan 10 2020 *)

Multiplicative with a(p) = (if p+2 or p-2 is prime then p else 1), p prime.

A074239 Related to cumulative number of non-twin primes.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 17, 18, 19, 20, 21, 21, 22, 22, 23, 24, 25, 26, 27, 27, 28, 28, 29, 29, 30, 31, 32, 32, 33, 34, 34, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 43, 43, 44, 45, 46, 47, 47, 48, 49, 50

Offset: 0

Rudi Huysmans (rudi.huysmans(AT)pandora.be), Sep 18 2002

Robert Israel and Paul Nanninga, Table of n, a(n) for n = 0..10000 (terms up to a(100) from Paul Nanninga)

Cf. A065091 (odd primes), A007510 (non-twin primes).

N:= 100: # to get a(0) to a(N)
P:= [seq(ithprime(i),i=2..N+2)]:
ListTools:-PartialSums([0,seq(`if`(P[i]-P[i-1]=2,0,1),i=2..N+1)]); # Robert Israel, May 13 2016

Accumulate@ Table[If[Prime@ n - Prime[n - 1] == 2, 0, 1], {n, 2, 120}] - 1 (* Michael De Vlieger, May 13 2016, after Robert Israel *)

op(n) = prime(n+1);
lista(nn) = {my(x=0); for (n=1, nn, print1(x, ", "); if ((op(n+1) - op(n)) > 2, x++););} \\ Michel Marcus, May 13 2016

Take the sequence of odd primes op(n); set a(0) = 0; if op(n+1)-op(n)=2 a(n+1) = a(n), if op(n+1)-op(n) > 2 a(n+1) = a(n) + 1.

A130286 Strongly-single primes: primes p such that neither previousprime(p) nor nextprime(p) is a twin prime.

Original entry on oeis.org

83, 89, 127, 163, 167, 257, 331, 359, 367, 373, 379, 383, 389, 397, 401, 443, 449, 479, 487, 491, 499, 503, 547, 557, 587, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977

Offset: 1

Zak Seidov, Aug 06 2007

nonn

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

Cf. A000040 (primes), A001097 (twin primes), A007510 (single primes), A071904 (odd composite numbers).

f:=func; g:=func; [p: p in PrimesInInterval(5,1000)| not IsPrime(f(p)-2) and not IsPrime(g(p)+2) and not IsPrime(f(f(p))-2) and not IsPrime(g(g(p))+2)]; // Marius A. Burtea, Jan 02 2020

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k]; lst={};Do[p=Prime[n];If[ !PrimeQ[p-2]&&!PrimeQ[PrimePrev[p]-2]&&!PrimeQ[p+2]&&!PrimeQ[PrimeNext[p]+2],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)

Some terms corrected by Vladimir Joseph Stephan Orlovsky, Jul 21 2009

A132248 Isolated primes congruent to 1 (mod 30).

Original entry on oeis.org

211, 331, 541, 631, 691, 751, 991, 1171, 1201, 1381, 1471, 1531, 1741, 1801, 1831, 1861, 2011, 2161, 2221, 2251, 2281, 2371, 2521, 2671, 2851, 3061, 3181, 3271, 3511, 3571, 3631, 3691, 4111, 4201, 4441, 4561, 4591, 4621, 4831, 4861

Offset: 1

Omar E. Pol, Aug 20 2007

Harvey P. Dale, 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. A007510.

Select[Select[Partition[Prime[Range[700]],3,1],FreeQ[Differences[#],2]&][[All,2]],Mod[#,30]==1&] (* Harvey P. Dale, Nov 30 2019 *)

cp's OEIS Frontend

A121764 Single (or isolated or non-twin) primes of form 6n + 1.

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A132259 Isolated primes congruent to {1, 17, 19, 29} mod 30.

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A162734 An alternating sum of all numbers from the n-th up to the (n+1)st isolated prime.

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A176312 Numbers that are the products of two single (or isolated or non-twin) primes.

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Maple

Mathematica

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A176656 The positions of single (or isolated or non-twin) primes in A000040.

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Maple

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Extensions

A268343 Hermit primes: primes which are not a nearest neighbor of another prime.

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Maple

Mathematica

PARI

PARI

Extensions

A072963 In prime factorization of n replace all single (i.e., non-twin) primes with 1.

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Mathematica

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A074239 Related to cumulative number of non-twin primes.