A029710 -id:A029710 - cp's OEIS frontend

A078860 Least positive residues [mod 210] representing those residue classes which can be lesser of prime pairs from A029710.

7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 139, 163, 169, 187, 193

Offset: 1

Labos Elemer, Dec 13 2002

t=Flatten[Position[Table[GCD[w, 210], {w, 1, 210}], 1]] t2=Intersection[t, t+4]-4

Intersection[RRS(210), 4+RRS{210)]-4 and {7}. RRS[210]=reduced residue system of 210=first 48=phi[210] terms of A008364; additional term 7 is a singular cases; 210k+r generates complete A029710 with suitable k and r taken from these 15+1 numbers.

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

A078863 Smallest primes from A029710, each belonging to those different residue class of mod 210 which are listed in A078860. Arranged according to possible least positive residues mod 210.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 349, 163, 379, 397, 193

Offset: 1

Labos Elemer, Dec 13 2002

			Several terms are equal to corresponding ones in A078860, while others are larger like: 397=210.1+187 where r=187 is in A078860.

Cf. A031924, A008364, A078860, A078859-A078864.

f[x_] := Mod[Prime[x], 210] d[x_] := Prime[x+1]-Prime[x] t=Table[0, {210}]; Do[s=f[n]; If[Equal[d[n], 4]&&s<211&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 10000}]; t

A023200 Primes p such that p + 4 is also prime.

Original entry on oeis.org

3, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423, 1429, 1447, 1483

Offset: 1

David W. Wilson

nonn

Smaller member p of cousin prime pairs (p, p+4).
A015913 contains the composite number 305635357, so it is different from both the present sequence and A029710. (305635357 is the only composite member of A015913 < 10^9.) - Jud McCranie, Jan 07 2001
Apart from the first term, all terms are of the form 6n + 1.
Complement of A067775 (primes p such that p + 4 is composite) with respect to A000040 (primes). With prime 2 also primes p such that q^2 + p is prime for some prime q (q = 3 if p = 2, q = 2 if p > 2). Subsequence of A232012. - Jaroslav Krizek, Nov 23 2013
Conjecture: The sequence is infinite and for every n, a(n+1) < a(n)^(1+1/n). Namely a(n)^(1/n) is a strictly decreasing function of n. - Jahangeer Kholdi and Farideh Firoozbakht, Nov 24 2014
From Alonso del Arte, Jan 12 2019: (Start)
If p splits in Z[sqrt(-2)], p + 4 is an inert prime in that domain. Likewise, if p splits in Z[sqrt(2)], p + 4 is an inert prime in that domain.
The only way for p or p + 4 to split in both domains is if it is congruent to 1 modulo 24, in which case the other prime is inert in both domains.
For example, 3 = (1 - sqrt(-2))*(1 + sqrt(-2)) but is inert in Z[sqrt(2)], while 7 = (3 - sqrt(2))*(3 + sqrt(2)) but is inert in Z[sqrt(-2)]. And also 11 = (3 - sqrt(-2))*(3 + sqrt(-2)) but 15 is composite in Z or any quadratic integer ring.
And 97 = (5 - 6*sqrt(-2))*(5 + 6*sqrt(-2)) = (1 - 7*sqrt(2))*(1 + 7*sqrt(2)), but 101 is inert in both Z[sqrt(-2)] and Z[sqrt(2)]. (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
H. J. Weber, A Sieve for Cousin Primes, arXiv:1204.3795v1 [math.NT], 2012.
Eric Weisstein's World of Mathematics, Cousin Primes
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Cousin prime.
Index entries for primes, gaps between

Exactly the same as A029710 except for the exclusion of 3.

Cf. A000010, A003557, A007947, A046132, A098429, A000040, A010051.

a023200 n = a023200_list !! (n-1)
a023200_list = filter ((== 1) . a010051') $
               map (subtract 4) $ drop 2 a000040_list
-- Reinhard Zumkeller, Aug 01 2014

[p: p in PrimesUpTo(1500) | NextPrime(p)-p eq 4]; // Bruno Berselli, Apr 09 2013

A023200 := proc(n) option remember; if n = 1 then 3; else p := nextprime(procname(n-1)) ; while not isprime(p+4) do p := nextprime(p) ;  end do: p ; end if; end proc: # R. J. Mathar, Sep 03 2011

Select[Range[10^2], PrimeQ[#] && PrimeQ[# + 4] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[250]],PrimeQ[#+4]&] (* Harvey P. Dale, Oct 09 2023 *)

print1(3);p=7;forprime(q=11,1e3,if(q-p==4,print1(", "p)); p=q) \\ Charles R Greathouse IV, Mar 20 2013

a(n) = A046132(n) - 4 = A087679(n) - 2.

a(n) >> n log^2 n via the Selberg sieve. - Charles R Greathouse IV, Nov 20 2016

Definition modified by Vincenzo Librandi, Aug 02 2009

Definition revised by N. J. A. Sloane, Mar 05 2010

A046132 Larger member p+4 of cousin primes (p, p+4).

Original entry on oeis.org

7, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433

Offset: 1

Eric W. Weisstein

nonn

A pair of cousin primes are primes of the form p and p+4 (where p+2 may or may not be a prime). - N. J. A. Sloane, Mar 18 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cousin Primes
Wikipedia, Cousin prime.

Essentially the same as A031505. Cf. A023200, A029710, A098429.

Cf. A000040, A010051.

a046132 n = a046132_list !! (n-1)
a046132_list = filter ((== 1) . a010051') $ map (+ 4) a000040_list
-- Reinhard Zumkeller, Aug 01 2014

lst={};Do[p=Prime[n];If[PrimeQ[p4=p+4], (*Print[p4];*)AppendTo[lst, p4]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Prime[Range[300]],PrimeQ[#+4]&]+4 (* Harvey P. Dale, Dec 15 2017 *)

forprime(p=2,1e5,if(isprime(p-4),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A023200(n) + 4 = A087679(n) + 2.

a(n) = 3*A157834(n-1) + 2 = A029710(n-1) + 4 = 6*A056956(n-1) + 5 (thus a(n) mod 6 == 5), for all n>1. - M. F. Hasler, Jan 15 2013

A022005 Initial members of prime triples (p, p+4, p+6).

Original entry on oeis.org

7, 13, 37, 67, 97, 103, 193, 223, 277, 307, 457, 613, 823, 853, 877, 1087, 1297, 1423, 1447, 1483, 1663, 1693, 1783, 1867, 1873, 1993, 2083, 2137, 2377, 2683, 2707, 2797, 3163, 3253, 3457, 3463, 3847, 4153, 4513, 4783, 5227, 5413, 5437, 5647, 5653, 5737, 6547

Offset: 1

Warut Roonguthai

Subsequence of A029710. - R. J. Mathar, May 06 2017

All terms are congruent to 1 (modulo 6). - Matt C. Anderson, May 22 2015

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe).
Tony Forbes and Norman Luhn, Prime k-tuplets.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Prime Triplet.

Subsequence of A029710 and of A002476.
Subsequence of A007529.
Cf. A001359, A073649, A098413.

[p: p in PrimesUpTo(10000) | IsPrime(p+4) and IsPrime(p+6)]; // Vincenzo Librandi, Aug 23 2015

Select[Table[Prime[n], {n, 2000}], PrimeQ[# + 4] && PrimeQ[# + 6] &] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)

select(p->isprime(p+4) && isprime(p+6), primes(1000)) \\ Charles R Greathouse IV, Mar 17 2023

A029709 Numbers k such that k-th and (k+1)st primes differ by 4.

Original entry on oeis.org

4, 6, 8, 12, 14, 19, 22, 25, 27, 29, 31, 38, 44, 48, 50, 59, 63, 65, 70, 75, 78, 85, 88, 90, 93, 95, 112, 117, 122, 131, 134, 136, 143, 147, 149, 151, 153, 155, 159, 163, 169, 181, 183, 198, 207, 211, 213, 224, 226, 229, 235, 237, 244, 247, 249, 251

Offset: 1

N. J. A. Sloane

nonn

Positions of 4 in A001223. - Zak Seidov, Apr 28 2015

N. J. A. Sloane and K. D. Bajpai, Table of n, a(n) for n = 1..10213 (first 56 terms from N. J. A. Sloane)

Cf. A023200, A029710, A001223.

[n: n in [0..300] | NthPrime(n+1) - NthPrime(n) eq 4]; // Vincenzo Librandi, Apr 28 2015

Select[Range[2, 300], 4 == (Prime[# + 1] - Prime[#]) &] (* Vincenzo Librandi, Apr 28 2015 *)

A029710(n) = prime(a(n)). - R. J. Mathar, Apr 30 2024

A015913 Numbers k such that sigma(k) + 4 = sigma(k+4).

Original entry on oeis.org

3, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423

Offset: 1

Robert G. Wilson v

nonn

This sequence contains the composite number 305635357, so is different from A023200 and A029710 (305635357 is the only composite member of the present sequence below 10^9). - Jud McCranie, Jan 07 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A023200, A029710.

Select[Range[1500],DivisorSigma[1,#]+4==DivisorSigma[1,#+4]&] (* Harvey P. Dale, Nov 04 2011 *)

is(n)=sigma(n)+4==sigma(n+4) \\ Charles R Greathouse IV, Mar 09 2014

A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)A003961(n)) - (nsigma(A003961(n))).

Original entry on oeis.org

0, 1, 2, 11, 2, 36, 4, 85, 46, 58, 2, 324, 4, 120, 120, 575, 2, 693, 4, 566, 248, 172, 6, 2340, 94, 270, 788, 1176, 2, 1800, 6, 3661, 348, 358, 336, 5967, 4, 492, 548, 4210, 2, 3744, 4, 1820, 2490, 744, 6, 15372, 380, 2271, 720, 2826, 6, 11304, 392, 8760, 992, 946, 2, 15480, 6, 1232, 5164, 22631, 636, 5904, 4, 3866

Offset: 1

Antti Karttunen, Feb 22 2021

Cf. A000203, A000720, A001223, A003961, A003973, A286385, A341528, A341529, A341530.

Cf. Sequences A001359, A029710, A031924 give the positions of 2's, 4's and 6's in this sequence, or at least subsets of such positions.

Array[#2 DivisorSigma[1, #1] - #1 DivisorSigma[1, #2] & @@ {#, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 68] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341512(n) = (A341529(n)-A341528(n));

a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).
For all primes p, a(p) = (q*(p+1)) - (p*(q+1)) = (pq + q) - (pq + p) = q - p = A001223(A000720(p)), where q = nextprime(p) = A003961(p).
And in general, a(p^e) = (q^e * (p^(e+1)-1)/(p-1)) - ((p^e) * (q^(e+1)-1)/(q-1)), where q = A003961(p).
Thus, a(p^2) = (p + 1)*q^2 - p^2*q - p^2,
      a(p^3) = (p^2 + p + 1)*q^3 - p^3*q^2 - p^3*q - p^3,
      a(p^4) = (p^3 + p^2 + p + 1)*q^4 - p^4*q^3 - p^4*q^2 - p^4*q - p^4,
      etc.

A031505 Upper prime of a difference of 4 between consecutive primes.

Original entry on oeis.org

11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433, 1451

Offset: 1

Jeff Burch

nonn

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

Essentially the same as A046132.

for i from 1 to 232 do if ithprime(i+1) = ithprime(i) + 4 then print({ithprime(i+1)}); fi; od; # Zerinvary Lajos, Mar 19 2007

Select[Partition[Prime[Range[250]],2,1],#[[2]]-#[[1]]==4&][[All,2]] (* Harvey P. Dale, Feb 02 2023 *)

a(n) = A029710(n-1) + 4 = A029708(n) + 2.

Corrected and extended by Henry Bottomley, Jul 18 2000

Definition clarified by Harvey P. Dale, Feb 02 2023

cp's OEIS Frontend

A078860 Least positive residues [mod 210] representing those residue classes which can be lesser of prime pairs from A029710.

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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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A078863 Smallest primes from A029710, each belonging to those different residue class of mod 210 which are listed in A078860. Arranged according to possible least positive residues mod 210.

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A023200 Primes p such that p + 4 is also prime.

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A046132 Larger member p+4 of cousin primes (p, p+4).

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A022005 Initial members of prime triples (p, p+4, p+6).

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A029709 Numbers k such that k-th and (k+1)st primes differ by 4.

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A015913 Numbers k such that sigma(k) + 4 = sigma(k+4).

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A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)A003961(n)) - (nsigma(A003961(n))).