A023200 -id:A023200 - cp's OEIS frontend

A052351 Least prime in A023200 (lesser of 4-twins) such that the distance to the next 4-twin is 6*n.

7, 67, 19, 43, 163, 127, 397, 229, 769, 1489, 673, 9547, 1009, 1783, 1693, 2857, 11677, 23869, 499, 1093, 4003, 28657, 10459, 29383, 12487, 6043, 41647, 7039, 17029, 19207, 15073, 24247, 65839, 29629, 18583, 9883, 66697, 100699, 7243, 53923, 82237, 6217, 76249

Offset: 1

Labos Elemer, Mar 07 2000

nonn

a(n) is a "lesser of a 4-twin" prime whose distance to the next twin is 6n.
Both the smallest distance (A052380) and its increment for 4-twins is 6.
The prime a(n)=p is the first which determines a prime quadruple [p, p+4, p+6n, p+6n+4] and difference pattern of [4, 6n-4, 4].

			a(1) = 7 gives [7, 11,7+6 = 13, 17] with no primes between 11 and 13.
a(5) = 163 specifies [163, 167, 163+30 = 191, 193] with 4 primes between 167 and 193.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A023200, A052380, A052381, A053320.

Cf. A052350, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 4] // Flatten; pp = p[[i]]; dd = Differences[pp]/6; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* Amiram Eldar, Mar 04 2025 *)

list(len) = {my(s = vector(len), c = 0, p1 = 7, q1 = 0, q2, d); forprime(p2 = 11, , if(p2 == p1 + 4, q2 = p1; if(q1 > 0, d = (q2 - q1)/6; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 04 2025

Name corrected by Amiram Eldar, Mar 04 2025

A172112 Partial sums of A023200.

Original entry on oeis.org

3, 10, 23, 42, 79, 122, 189, 268, 365, 468, 577, 704, 867, 1060, 1283, 1512, 1789, 2096, 2409, 2758, 3137, 3534, 3973, 4430, 4893, 5380, 5879, 6492, 7135, 7808, 8547, 9304, 10073, 10896, 11749, 12608, 13485, 14368, 15275, 16212, 17179, 18188, 19275

Offset: 1

Jonathan Vos Post, Jan 25 2010

Primes in the partial sum begin: a(1) = 3, a(3) = 23, a(5) = 79, a(11) = 577, a(15) = 1283, a(17) = 1789, a(21) = 3137, a(27) = 5879. Of these, the smaller members of cousin prime pairs which appear among the partial sums of smaller member p of cousin prime pairs begin: 3, 79; which are the next in this subset?

			a(30) = 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 = 7808.

Cf. A000040, A023200, A046132.

Accumulate[Select[Prime[Range[250]],PrimeQ[#+4]&]] (* Harvey P. Dale, Oct 09 2023 *)

a(n) = SUM[i=i..n] A023200(i) = SUM[i=i..n] {Primes p such that p and p + 4 are both primes}.

More terms from Max Alekseyev, Jan 31 2010

A172483 a(n) is the number of cousin primes between p^2 and p*(p+4) where p is the n-th cousin prime A023200(n).

Original entry on oeis.org

2, 1, 1, 2, 5, 4, 4, 2, 6, 4, 7, 7, 5, 9, 12, 13, 14, 14, 9, 12, 10, 11, 13, 20, 16, 15, 16, 15, 23, 19, 22, 26, 27, 28, 26, 22, 20, 27, 25, 27, 28, 26, 35, 29, 29, 29, 30, 45, 30, 36, 22, 30, 39, 39, 40, 44, 44, 43, 34, 38, 36, 48, 54, 43, 38, 43, 49, 45, 47, 53, 38, 51, 51, 62, 56

Offset: 1

Jaspal Singh Cheema, Feb 04 2010

nonn

If you graph the order of the consecutive cousin primes along the x-axis (i.e., first pair of cousin primes, second, third,...) and the number of cousin primes in the sequence given above along the y-axis, a clear pattern emerges. As you go farther along the x-axis, greater are the number of consecutive cousin primes, on average, within the interval obtained. If one can prove that there's at least one consecutive cousin prime within each interval, this would imply that cousin primes are infinite. I suspect the number of consecutive primes within each interval will never be zero. Can you prove it?

			The 1st pair of cousin primes is (3, 7), between 3^2=9 and 3*7=21 there is 2 cousin primes: 13 and 19. So a(1) = 2.
The 2nd pair of cousin primes is (7, 11), between 7^2=49 and 7*11=77 there is 1 cousin prime: 67. So a(2) = 1.

C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
M. D. Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.

J. S. Cheema, Table of n, a(n) for n = 1..1104 (2 prepended by Michael De Vlieger)

Cf. A023200, A046132, A087679.

vcp(nn) = my(list=List(), p=3); listput(list, p); p=7; forprime(q=11, nn, if(q-p==4, listput(list, p)); p=q); Vec(list); \\ A023200
nbcp(p) = my(nb=0); forprime(q=p^2, p*(p+4), if (isprime(q+4), nb++)); nb;
lista(nn) = my(v=vcp(nn)); vector(#v, n, nbcp(v[n])); \\ Michel Marcus, Nov 02 2022

New name and a(1)=2 prepended by Michel Marcus, Nov 02 2022

A174046 Places n for which A001359(n) and A023200(n) is a twin prime pair.

Original entry on oeis.org

2, 3, 4, 6, 14, 16, 29, 356, 358, 359, 403, 446, 464, 485, 652, 655, 764, 861, 866, 1123, 1301, 1304, 1324, 1328, 1358, 1486, 1610, 2631, 2632, 3735, 3931, 3953, 3956, 3957, 4679, 4855, 4931, 5222, 5226, 5269, 5283, 5292, 5403, 5427, 5445

Offset: 1

Vladimir Shevelev, Mar 06 2010

nonn

			2 is in the sequence because A001359(2)=5 and A023200(2)=7 are twin primes.

Michel Marcus, Table of n, a(n) for n = 1..149

Cf. A023200, A001359, A046132, A006512, A174042

lista(nn)  = {vp = primes(nn); va = select(x->isprime(x+2), vp); vb = select(x->isprime(x+4), vp); for (n=1, min(#va, #vb), if (vb[n] == va[n]+2, print1(n, ", ")););} \\ Michel Marcus, Jul 22 2017

Terms beyond 29 from R. J. Mathar, Nov 03 2011

Edited by Michel Marcus, Jul 22 2017

A352171 a(n) is the start of a sequence of exactly n members of A023200 under the iteration p -> 3*p+4.

Original entry on oeis.org

7, 13, 3, 1547803

Offset: 1

J. M. Bergot and Robert Israel, Mar 07 2022

Let s(1) = a(n) and s(k+1) = 3*s(k)+4. Then s(1), ..., s(n) are in A023200 but s(n+1) is not in A023200, and a(n) is the least value of s(n) for which this is the case.

			a(3) = 3 because with s(1) = 3 we have s(2) = 3*3+4 = 13, s(3) = 3*13+4 = 43, s(4) = 3*43+4 = 133; 3, 13, and 43 are in A023200 because 3, 7, 13, 17, 42, 47 are prime, but 133 is not in A023200 because 133 is composite.

Cf. A023200.

f:= proc(p) option remember;
  if isprime(p) and isprime(p+4) then 1 + procname(3*p+4) else 0 fi
end proc:
V:= Vector(5): V[1]:= 7: V[3]:= 3: count:= 2:
for p from 13 by 30 while count < 5 do
v:= f(p);
if v > 0 and V[v] = 0 then count:= count+1; V[v]:= p; fi
od:
convert(V,list);

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

A007694 Numbers k such that phi(k) divides k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 128, 144, 162, 192, 216, 256, 288, 324, 384, 432, 486, 512, 576, 648, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2304, 2592, 2916, 3072, 3456, 3888, 4096, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8192, 8748, 9216

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

a(n) divides p^a(n) - 1 for all primes p >= 5. - Benoit Cloitre, Mar 22 2002
Also k such that Sum_{d divides k} mu(d)/d has numerator 1. - Benoit Cloitre, Apr 15 2002
k is here if and only if phi(k) also divides cototient(k). On the other hand, cototient(k) divides phi(k) if and only if k is a prime or power of a prime. - Labos Elemer, May 03 2002
It follows that k/phi(k) = 2 if k is a power of 2 and equal to 3 if k is of the form 6*A003586. - Gary Detlefs, Jun 28 2011
1 and even 3-smooth numbers, cf. A003586. - Reinhard Zumkeller, Jan 06 2014
Numbers k such that k = (1+omega(k))*phi(k). - Farideh Firoozbakht, Oct 02 2014
These are the integers whose largest squarefree divisor is 1, 2 or 6. As such, this sequence is equal to the set V_infinite, defined as the intersection of the V_k for k >= 1, where V_k(x) = {phi_k(n) <= x} and phi_k is the k-th iterate of phi, the Euler function; for instance, V_1 is given by A002202 (see Theorem 7 in Pomerance and Luca). - Michel Marcus, Nov 09 2015
This sequence is contained in A068997. The terms of A068997 not in this sequence have largest squarefree divisor other than 1, 2, or 6, beginning with 10. - Torlach Rush, Dec 07 2017

			12 is in the sequence because 12/phi(12) = 12/4 = 3, which is an integer.
16 is in the sequence because 16/phi(16) = 16/8 = 2, which is an integer.
20 is not in the sequence because 20/phi(20) = 20/8 = 5/2 = 2.5, which is not an integer.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 526 pp. 71; 256, Ellipses Paris 2004.
Sárközy A. and Suranyi J., Number Theory Problem Book (in Hungarian), Tankonyvkiado, Budapest, 1972.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Roohollah Ebrahimian, When Does phi(n) divide n? A Number Theory Math Competition Problem., YouTube video, 2023.
Michael W. Ecker and Scott J. Beslin, Problem E3037, Amer. Math. Monthly, Vol. 93, No. 8 (1986), pp. 656-657.
Florian Luca and Carl Pomerance, On the range of the iterated Euler function, Article 8, Integers: Electronic Journal of Combinatorial Number Theory, Proceedings of the Integers Conference 2007, Volume 9 Supplement (2009).
Mathematics Stack Exchange, Is there phi(n)/n = 6
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.

Cf. A000010, A049237, A007694, A007947, A003557, A023200, A003586, A001221, A033950, A235353 (subsequence), A068997 (subsequence).

a007694 n = a007694_list !! (n-1)
a007694_list = 1 : filter even a003586_list
-- Reinhard Zumkeller, Jan 06 2014

select(n -> n mod numtheory:-phi(n) = 0, [$1..5000]); # Robert Israel, Nov 03 2014

Select[ Range[5000], IntegerQ[ #/EulerPhi[ # ]] &]
m = 5000; Join[{1}, Sort @ Flatten @ Table[2^i*3^j, {i, 1, Log2[m]}, {j, 0, Log[3, m/2^i]}]] (* Amiram Eldar, Oct 29 2020 *)

for(n=1,10^6, if (n%eulerphi(n)==0,print1(n,", "))); \\ Joerg Arndt, Apr 04 2013

list(lim)=my(v=List([1]),t); for(i=1,logint(lim\1,2), listput(v,t=2^i); for(j=1,logint(lim\t,3), listput(v,t*=3))); Set(v) \\ Charles R Greathouse IV, Nov 10 2015

library(numbers); j=N=1
while(j<200) if(isNatural((N=N+1)/eulersPhi(N))) dtot[(j=j+1)]=N # Christian N. K. Anderson, Apr 04 2013

is_A007694 = lambda n: euler_phi(n).divides(n)
A007694_list = lambda len: filter(is_A007694, (1..len))
A007694_list(4100) # Peter Luschny, Oct 03 2014

k/phi(k) is an integer if and only if k = 1 or k = 2^w * 3^u for w > 0 and u >= 0.
k/phi(k) = 3 if and only if phi(k)|k and 3|k. - Thomas Ordowski, Nov 03 2014
a(n) is approximately exp(sqrt(2*log(2)*log(3)*n))/sqrt(3/2). - Charles R Greathouse IV, Nov 10 2015
From Amiram Eldar, Oct 29 2020: (Start)
a(n) = 2 * A003586(n) for n > 1.
Sum_{n>=1} 1/a(n) = 5/2. (End)

A029710 Primes such that next prime is 4 greater.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

Union with A124588 gives A124589. - Reinhard Zumkeller, Dec 23 2006
For any prime p > 3, if p + 4 is prime then necessarily it is the next prime. But there cannot be three consecutive primes with mutual distance 4: If p and p + 4 are prime, then p+8 is an odd multiple of 3 (cf. formula). - M. F. Hasler, Jan 15 2013
The smaller members p of cousin prime pairs (p,p+4) excluding p=3. - Marc Morgenegg, Apr 19 2016

			79 is a term as the next prime is 79 + 4 = 83. 3 is not a term even though 3 + 4 = 7 is prime, since it is not the next one.

Marius A. Burtea, Table of n, a(n) for n = 1..14741 ( first 1000 terms from R. Zumkeller )
Index entries for primes, gaps between

Essentially the same as A023200.

Cf. A001359, A029708, A031505, A124588, A124589, A157834.

p=primes(1700);m=1;
for u=1:length(p)-4
   if and(isprime(p(u)+4)==1,p(u+1)==p(u)+4);sol(m)=p(u);m=m+1;end
end
sol % Marius A. Burtea, Jan 24 2019

[p:p in PrimesUpTo(1700)| IsPrime(p+4) and NextPrime(p) eq p+4] // Marius A. Burtea, Jan 24 2019

for i from 1 to 226 do if ithprime(i+1) = ithprime(i) + 4 then print({ithprime(i)}); fi; od; # Zerinvary Lajos, Mar 19 2007

Select[Prime[Range[225]], NextPrime[#] == # + 4 &] (* Alonso del Arte, Jan 17 2013 *)
Transpose[Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==4&]] [[1]] (* Harvey P. Dale, Mar 28 2016 *)

forprime(p=1, 1e4, if(nextprime(p+1)-p==4, print1(p, ", "))) \\ Felix Fröhlich, Aug 16 2014

a(n) = A031505(n + 1) - 4 = A029708(n) - 2.

a(n) = 1 (mod 6) for all n; (a(n) + 2)/3 = A157834(n), i.e., a(n) = 3*A157834(n) - 2. - M. F. Hasler, Jan 15 2013

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

A060308 Largest prime <= 2n.

Original entry on oeis.org

2, 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 23, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 47, 47, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 79, 83, 83, 83, 89, 89, 89, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 113, 113, 113, 113, 127, 127, 131

Offset: 1

Labos Elemer, Mar 27 2001

a(n) is the smallest k such that C(2n,n) divides k!. - Benoit Cloitre, May 30 2002
a(n) is largest prime factor of C(2n,n) = (2n)!/(n!)^2. - Alexander Adamchuk, Jul 11 2006
a(n) is also the largest prime in the interval [n,2n]. - Peter Luschny, Mar 04 2011
Odd prime p repeats (q-p)/2 times, where q > p is the next prime. In particular, every lesser of twin primes (A001359) occurs 1 time,  every lesser more than 3 of cousin primes (A023200) occurs 2 times, etc. - Vladimir Shevelev, Mar 12 2012

			n=1, 2n=2, p(1) = 2 = a(1) is the largest prime not exceeding 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Apart from initial term, same as A060265.
Cf. A007917 (largest prime <= n), A005843 (2n).
Cf. A020482, A049653, A060266, A060267, A060268, A060264.

a060308 = a007917 . a005843 -- Reinhard Zumkeller, May 25 2013

[NthPrime(#PrimesUpTo(2*n)): n in [2..100]]; // Vincenzo Librandi, Nov 25 2015

seq (prevprime(2*i+1), i=1..256);
seq(max(op(select(isprime,[$n..2*n]))),n=1..66); # Peter Luschny, Mar 04 2011

Table[Max[FactorInteger[(2n)!/(n!)^2]],{n,1,100}] (* Alexander Adamchuk, Jul 11 2006 *)
NextPrime[2*Range[80]+1,-1] (* Harvey P. Dale, Apr 23 2017 *)

a(n)=precprime(2*n) \\ Charles R Greathouse IV, May 24 2013

a(n) = Max[FactorInteger[(2n)!/(n!)^2]]. - Alexander Adamchuk, Jul 11 2006
a(n) = A006530(A000142(2*n)) and a(n) = A006530(A056040(2*n)). - Peter Luschny, Mar 04 2011
a(n) ~ 2*n as n tends to infinity. - Vladimir Shevelev, Mar 12 2012
a(n) = A007917(A005843(n)) = A226078(n, A067434(n)). - Reinhard Zumkeller, May 25 2013

More terms from Alexander Adamchuk, Jul 11 2006

cp's OEIS Frontend

A052351 Least prime in A023200 (lesser of 4-twins) such that the distance to the next 4-twin is 6*n.

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A172112 Partial sums of A023200.

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A172483 a(n) is the number of cousin primes between p^2 and p*(p+4) where p is the n-th cousin prime A023200(n).

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A174046 Places n for which A001359(n) and A023200(n) is a twin prime pair.

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A352171 a(n) is the start of a sequence of exactly n members of A023200 under the iteration p -> 3*p+4.

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

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A007694 Numbers k such that phi(k) divides k.

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