A077800 -id:A077800 - cp's OEIS frontend

A344147 Primes in A191746.

193, 53069, 58422233, 1348470667, 2847740783, 3237916229, 5029745827, 7643871979, 15107731019, 17902513283, 21052092827, 22187962591, 28412311451, 59363922119, 81459096899, 85780812149, 102742076659, 123894775231, 137692362377, 143889901511, 170038274723, 174648621811

Offset: 1

Hartmut F. W. Hoft, May 10 2021

nonn

			a(1)=193=A191746(3) is the first prime in A191746 and a(2)=53069=A191746(11) is the second.

Cf. A037074, A074040, A077800, A079164, A191746.

(* function a037074[ ] and support functions are defined in A074040 *)
a191746[n_] := Rest[FoldList[Plus, 0, a037074[n]]]
a344147x[n_] := Select[a191746[n], PrimeQ]
a344147[550]

A357059 Decimal expansion of sum of squares of reciprocals of primes whose distance to the next prime is equal to 4, Sum_{j>=1} 1/A029710(j)^2.

Original entry on oeis.org

0, 3, 1, 3, 2, 1, 6, 2, 0, 6, 4, 6

Offset: 0

Artur Jasinski, Sep 10 2022

Convergence table:
   k       A029710(k)      Sum_{j=1..k} 1/A029710(j)^2
10000000   3285441223 0.031321620645456519799598611681
20000000   7067090263 0.031321620645890982910821292996
30000000  11044597393 0.031321620646019474620358985896
40000000  15153534937 0.031321620646079307404248696076
50000000  19360462153 0.031321620646113421819579063642
60000000  23647877233 0.031321620646135276227114122713
70000000  28000392817 0.031321620646150384406674037099

			0.031321620646...

Cf. A006512, A023200, A029710, A046132, A065421, A077800, A078437, A085548, A096247, A160910, A194098, A209328, A209329, A242301, A242302, A242303, A242304, A306539, A342714, A356793.

aa = {}; Do[g1[2 n] = 0, {n, 1, 1000}]; Do[g2[2 n] = 0, {n, 1, 1000}]; Do[g3[2 n] = 0, {n, 1, 1000}]; Do[g4[2 n] = 0, {n, 1, 1000}]; Do[g[2 n] = 0, {n, 1, 1000}];
w1 = 3; n = 3; Monitor[While[n < 10^10, w2 = NextPrime[w1]; kk = w2 - w1;
  If[kk < 2000, g[kk] = g[kk] + 1; g1[kk] = g1[kk] + N[1/w1, 1000];g2[kk] = g2[kk] + N[1/w1^2, 1000];g3[kk] = g3[kk] + N[1/w1^3, 1000];g4[kk] = g4[kk] + N[1/w1^4, 1000];
If[IntegerQ[g[kk]/1000000], Print[{n, w1, kk, g[kk]}];If[kk == 4,AppendTo[aa, {n, w1, kk, g[kk], g1[kk], g2[kk], g3[kk], g4[kk]}]]]];w1 = w2; n++], n];aa

A384102 Least x in absolute value, such that there exists y, |x| >= |y| > 0, such that n = |6xy + x + y|, or 0 if no such x exists. Choose x > 0 if x and -x are both possible.

Original entry on oeis.org

0, 0, 0, -1, 0, 1, 0, 1, -2, 0, 2, 0, -2, -3, 2, 3, 0, 0, -4, -2, 4, 3, 0, 2, 0, 5, -4, 2, 4, 0, -3, 0, 0, -5, 3, 5, -3, 0, -8, 0, 3, -4, 6, -9, 0, 4, 0, -3, -10, -4, 10, 0, -5, 3, -8, 11, 5, 0, -12, 3, 12, -9, -5, -6, -4, 13, 5, 6, -10, 0, 4, 0, -4, -15, -7, -6, 0, 11, 4, 6, 16, -5, -12, -17, 12, -8, 0, -4, -7, 8, 18, -5, 7, -19, 0, 4, -9, 5, -6

Offset: 1

M. F. Hasler, Jun 20 2025

(6n-1, 6n+1) are twin primes iff a(n) = 0, that is, if there are no nonzero integers x, y such that n = |6xy + x + y|. (These n are listed in A002822, the complement is A067611.)

a(n) <= (6*n-1)/5, with equality if 6*n+1 is prime and 6*n-1 is 5 times a prime. - Robert Israel, Jul 21 2025

			For n = 1, 2 and 3, there are no nonzero x,y such that n = |6xy + x + y|, and (6n-1, 6n+1) = (5, 7), (11, 13) and (17, 19) are indeed twin primes.
For n = 4 we have x = y = -1 such that |6xy + x + y| = |6 - 1 - 1| = 4 and (23, 25) is indeed not a twin prime pair.

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

Cf. A384103 (the corresponding y-values).
Cf. A002822 (indices of zeros: n such that 6n-1 and 6n+1 are twin primes).
Cf. A077800 (list of twin primes), A060461, A171696 (none among 6n+-1 is prime), A067611 (n = 6xy +- x +- y: 6n-1 or 6n+1 is composite).

f:= proc(n) local V, C, t, m,v, r;
       V:= numtheory:-divisors(6*n+1) minus {1,6*n+1};
       C:= map(u -> `if`(u mod 6 = 1,  [(u-1)/6, ((6*n+1)/u - 1)/6], [(-u-1)/6, (-(6*n+1)/u - 1)/6]), V);
       V:= numtheory:-divisors(6*n-1) minus {1,6*n-1};
       C:= C union map(u -> `if`(u mod 6 = 1, [(u-1)/6, ((-6*n+1)/u - 1)/6], [(-u-1)/6, ((-6*n+1)/u - 1)/6]), V);
       C:= select(t -> abs(t[1]) >= abs(t[2]), C)[..,1];
       if C = {} then return 0 fi;
       m:= infinity;
       for t in C do
         if abs(t) < m then m:= abs(t); r:= t;
         elif abs(t) = m and t > 0 then r:= t
         fi
       od;
       r
 end proc:
map(f, [$1..100]); # Robert Israel, Jul 21 2025

{A384102(n)=for(x=1,n\/5, my(p=6*x+1, q=6*x-1, r=if((n-x)%p==0, (n-x)\p, (n+x)%p==0, (n+x)\p, (n-x)%q==0, (x-n)\q, (n+x)%q==0,-(n+x)\q)); r && abs(r) <= x && return(sign(r)*x))}

A384103 a(n) = y with minimum |x| >= |y| > 0, such that n = |6xy + x + y|, or 0 if no such x, y exist. If x and -x are solutions, choose x > 0 > y = -x.

Original entry on oeis.org

0, 0, 0, -1, 0, -1, 0, 1, -1, 0, -1, 0, 1, -1, 1, -1, 0, 0, -1, -2, -1, 1, 0, -2, 0, -1, 1, 2, 1, 0, -2, 0, 0, 1, -2, 1, 2, 0, -1, 0, 2, -2, 1, -1, 0, -2, 0, -3, -1, 2, -1, 0, -2, -3, 1, -1, -2, 0, -1, 3, -1, 1, 2, -2, -3, -1, 2, -2, 1, 0, -3, 0, 3, -1, -2, 2, 0, 1, 3, 2, -1, -3, 1, -1, 1, -2, 0, -4, 2, -2

Offset: 1

M. F. Hasler, Jun 20 2025

(6n-1, 6n+1) are twin primes iff a(n) = 0, that is, if there are no nonzero integers x, y such that n = |6xy + x + y|. These n are listed in A002822, the complement is A067611.

The corresponding x-values are listed in A384102.

			For n = 1, 2 and 3, there are no nonzero x,y such that n = |6xy + x + y|, and (6n-1, 6n+1) = (5, 7), (11, 13) and (17, 19) are indeed twin primes.
For n = 4 we have x = y = -1 such that |6xy + x + y| = |6 - 1 - 1| = 4 and (23, 25) is indeed not a twin prime pair.

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

Cf. A384102 (the corresponding x-values).
Cf. A002822 (indices of zeros: n such that 6n-1 and 6n+1 are twin primes).
Cf. A077800 (list of twin primes), A060461, A171696 (none among 6n+-1 is prime), A067611 (n = 6xy +- x +- y: 6n-1 or 6n+1 is composite).

f:= proc(n) local V, C, t, m, v, r;
       V:= numtheory:-divisors(6*n+1) minus {1, 6*n+1};
       C:= map(u -> `if`(u mod 6 = 1,  [(u-1)/6, ((6*n+1)/u - 1)/6], [(-u-1)/6, (-(6*n+1)/u - 1)/6]), V);
       V:= numtheory:-divisors(6*n-1) minus {1, 6*n-1};
       C:= C union map(u -> `if`(u mod 6 = 1, [(u-1)/6, ((-6*n+1)/u - 1)/6], [(-u-1)/6, ((6*n-1)/u - 1)/6]), V);
       C:= select(t -> abs(t[1]) >= abs(t[2]), C);
       if C = {} then return 0 fi;
       m:= infinity;
       for t in C do
         if abs(t[1]) < m then m:= abs(t[1]); r:= t[2];
         elif abs(t[1]) = m and t[1] > 0 then r:= t[2]
         fi
       od;
       r
 end proc:
map(f, [$1..100]); # Robert Israel, Jul 21 2025

apply( {A384103(n)=for(x=1,n\/5, my(p=6*x+1, q=6*x-1, y=if((n-x)%p==0, (n-x)\p, (n+x)%p==0, -(n+x)\p, (n-x)%q==0, (n-x)\q, (n+x)%q==0,-(n+x)\q)); y && abs(y) <= x && return(y))}, [1..90])

A100819 Composite numbers whose prime factors are twin primes.

Original entry on oeis.org

9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 117, 119, 121, 123, 125, 129, 133, 135, 143, 145, 147, 153, 155, 165, 169, 171, 175, 177, 183, 187, 189, 195, 203, 205, 209, 213, 215, 217, 219, 221

Offset: 1

Walter Carlini, Jan 06 2005; corrected Jan 22 2005

			221 = 13 * 17 (13 is the twin prime of 11 and 17 is the twin prime to 19).

Cf. A077800 (twin primes).

nn = 100; p = Prime[Range[nn]]; t = {}; Do[If[p[[n + 1]] - p[[n]] == 2 || p[[n]] - p[[n - 1]] == 2, AppendTo[t, p[[n]]]], {n, nn - 1}]; Select[Range[2, t[[-1]]], ! PrimeQ[#] && Complement[Transpose[FactorInteger[#]][[1]], t] == {} &] (* T. D. Noe, May 21 2013 *)

New name, typo in example fixed by Zak Seidov, May 20 2013

A114379 Sums of p-th to the q-th prime where p and q are twin primes.

Original entry on oeis.org

23, 41, 109, 187, 349, 551, 841, 1079, 1667, 1779, 2357, 2599, 3219, 3487, 3631, 4319, 4533, 5197, 5501, 6213, 7039, 8709, 9031, 9829, 11233, 12425, 13227, 13677, 14329, 14813, 18667, 18951, 19073, 19973, 20561, 24329, 24685, 25153, 25561, 26261

Offset: 1

Cino Hilliard, Feb 10 2006

Conjecture: The number of terms in this sequence is infinite.

			3 and 5 are the first twin prime pair: prime(3) = 5, prime(4) = 7, prime(5) = 11
and 5+7+11 = 23, the first entry in the table.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Plus @@ (Prime /@ Range[#, # + 2]) & /@ Select[Prime@ Range@ 200, PrimeQ[# + 2] &] (* Michael De Vlieger, Apr 01 2015 *)

sumprimes(m, n) = { local(x); return(sum(x=m, n, prime(x))) }
g(n)=forprime(x=3,n,if(isprime(x+2),print1(sumprimes(x,x+2)",")))

prime(k) = A000040(k) is the k-th prime number.

a(n) = Sum_{k=A077800(2n-1)..A077800(2n)} prime(k). - Danny Rorabaugh, Apr 01 2015

A144550 Largest non-twin prime < n-th and (n+1)-th twin primes.

Original entry on oeis.org

2, 23, 37, 53, 67, 97, 131, 173, 223, 233, 263, 277, 307, 337, 409, 457, 509, 563, 593, 613, 631, 653, 797, 853, 877, 1013, 1039, 1087, 1129, 1223, 1259, 1283, 1297, 1307, 1423, 1447, 1471, 1601, 1613, 1663, 1693, 1709, 1783, 1867, 1913, 1993, 2017, 2069

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2008

nonn

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

Cf. A077800, A088175.

isA077800 := proc(n) isprime(n) and ( isprime(n+2) or isprime(n-2)) ; end proc:
isA007510 := proc(n) isprime(n) and not isA077800(n) ; end proc:
isA144550 := proc(n) isA007510(n) and isA077800( nextprime(n)) ; end proc:
for n from 2 to 2500 do if isA144550(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, May 01 2010

Module[{nn=400,tps},tps=Union[Flatten[Select[Partition[Prime[Range[nn]],2,1], #[[2]]-#[[1]] == 2&]]];Complement[NextPrime[#,-1]&/@tps,tps]] (* Harvey P. Dale, Jun 23 2022 *)

{A007510(k): nextprime(A007510(k)) in A077800}. - R. J. Mathar, May 01 2010

Corrected (1993 inserted) by R. J. Mathar, May 01 2010

A153740 Smallest non-twin prime > n-th and (n+1)-th twin primes.

Original entry on oeis.org

23, 37, 47, 67, 79, 113, 157, 211, 233, 251, 277, 293, 317, 353, 439, 467, 541, 577, 607, 631, 647, 673, 839, 863, 887, 1039, 1069, 1097, 1163, 1237, 1283, 1297, 1307, 1327, 1433, 1459, 1493, 1613, 1627, 1693, 1709, 1733, 1801, 1889, 1973, 2003, 2039, 2099

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2008

nonn

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

Cf. A000040, A077800.

Select[NextPrime/@Transpose[Select[Partition[Prime[Range[500]],2,1], Last[#]-First[#] == 2&]][[2]],!PrimeQ[#-2]&&!PrimeQ[#+2]&] (* Harvey P. Dale, Feb 25 2012 *)

863 inserted by R. J. Mathar, Jan 03 2009

A158284 List of pairs p, p+2 where p is prime and p and p+2 contain the same number of prime digits.

Original entry on oeis.org

3, 5, 5, 7, 13, 15, 23, 25, 29, 31, 43, 45, 53, 55, 73, 75, 83, 85, 89, 91, 103, 105, 109, 111, 113, 115, 163, 165, 173, 175, 193, 195, 223, 225, 229, 231, 233, 235, 263, 265, 283, 285, 293, 295, 313, 315, 353, 355, 373, 375, 383, 385, 389, 391, 409, 411, 433

Offset: 1

Juri-Stepan Gerasimov, Mar 15 2009

Prime digits are 2, 3, 5 or 7.

Cf. A077800.

snpdQ[n_]:=Count[IntegerDigits[n],?PrimeQ]==Count[IntegerDigits[n+2], ?PrimeQ]; {#,#+2}&/@Select[Prime[Range[100]],snpdQ]//Flatten (* Harvey P. Dale, Dec 04 2017 *)

(13,15) inserted, (23,25) inserted, and all other numbers replaced by R. J. Mathar, May 19 2010

Definition changed to match the terms by N. J. A. Sloane, Dec 03 2017

A158342 List of twin primes p1 and p2 with odd sum of digits of p1 and even sum of digits of p2.

Original entry on oeis.org

29, 31, 179, 181, 269, 271, 809, 811, 1019, 1021, 1619, 1621, 1949, 1951, 2339, 2341, 2999, 3001, 3329, 3331, 3389, 3391, 3929, 3931, 4049, 4051, 4229, 4231, 4649, 4651, 5279, 5281, 5639, 5641, 5879, 5881, 6089, 6091, 6269, 6271, 6359, 6361, 6449, 6451

Offset: 1

Juri-Stepan Gerasimov, Mar 16 2009

Harvey P. Dale, Table of n, a(n) for n = 1..3000

Cf. A077800.

Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2&&OddQ[Total[IntegerDigits[#[[1]]]]]&&EvenQ[Total[IntegerDigits[#[[2]]]]]&]//Flatten (* Harvey P. Dale, Jan 19 2025 *)

Corrected (269,271 inserted, 1229,1231 removed, 3119,3121 removed...) by R. J. Mathar, May 19 2010

cp's OEIS Frontend

A344147 Primes in A191746.

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A357059 Decimal expansion of sum of squares of reciprocals of primes whose distance to the next prime is equal to 4, Sum_{j>=1} 1/A029710(j)^2.

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A384102 Least x in absolute value, such that there exists y, |x| >= |y| > 0, such that n = |6xy + x + y|, or 0 if no such x exists. Choose x > 0 if x and -x are both possible.

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A384103 a(n) = y with minimum |x| >= |y| > 0, such that n = |6xy + x + y|, or 0 if no such x, y exist. If x and -x are solutions, choose x > 0 > y = -x.

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Maple

PARI

A100819 Composite numbers whose prime factors are twin primes.

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Mathematica

Extensions

A114379 Sums of p-th to the q-th prime where p and q are twin primes.

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Mathematica

PARI

Formula

A144550 Largest non-twin prime < n-th and (n+1)-th twin primes.

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A153740 Smallest non-twin prime > n-th and (n+1)-th twin primes.

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Extensions