A060213 -id:A060213 - cp's OEIS frontend

A065505 Let p(k) denote k-th prime; consider solutions (p(n),p(m)) of Diophantine equation p(p(n)+1)-6.p(p(m))=1 (*), where p(p(n)) belongs to A060213 and p(p(m))=(p(p(n))+1)/6; sequence gives values of p(n).

7, 2309, 2753, 2789, 26183, 46933, 53597, 58411, 61357, 69481, 87691, 111487, 124991, 134327, 140659, 144651, 147551, 236519, 247711, 164643, 270223, 291359

Offset: 0

Lubomir Alexandrov, Nov 25 2001

nonn

			p(n)=395581 and p(m)=75277 satisfy equation (*) at the primes p(p(n))=5730617 and p(p(m))=955103.

A060213

A065503 Indices k of primes p(k) such that p(k) is in A060213.

Original entry on oeis.org

5, 7, 10, 13, 26, 33, 60, 113, 116, 142, 265, 288, 313, 332, 353, 384, 408, 484, 498, 542, 625, 636, 719, 805, 864, 1064, 1070, 1194, 1328, 1456, 1477, 1528, 1538, 1571, 1623, 1627, 1651, 1660, 1867, 2003, 2216, 2244, 2309, 2311, 2418, 2438, 2469, 2616, 2753

Offset: 1

Lubomir Alexandrov, Nov 25 2001

nonn

Original name: Let p(k) denote k-th prime; consider solutions (x,y) of Diophantine equation p(x+1)-6p(y)=1 (*), where p(x) belongs to A060213 and p(m)=(p(n)+1)/6; sequence gives values of x.

			x=13084 and y=2612 satisfy equation (*) at the primes p(x)=140837 and p(y)=23473.

Cf. A060213, A065505.

Offset corrected, more terms, and title clarified by Sean A. Irvine, Sep 03 2023

A164289 Sequence of twin primes p where the middle term p+1 has 5 prime factors (here p+2 is the associated twin prime, not listed).

Original entry on oeis.org

71, 107, 179, 269, 311, 419, 659, 827, 881, 1019, 1031, 1049, 1091, 1427, 1451, 1607, 1931, 1949, 1997, 2027, 2141, 2309, 2549, 2711, 2729, 2789, 3329, 3467, 3539, 3767, 3821, 3851, 4019, 4091, 4229, 4259, 4481, 4649, 4931, 5417, 5651, 5741, 5867, 6089

Offset: 1

Carlos Alves, Aug 12 2009

nonn

This sequence is similar to: A060213, with 3 prime factors in the middle number; A102168, with 4 prime factors in the middle number.

These sequences are of the form (p,p+1,p+2) with (p,p+2) twin primes and Omega(p+1)=m with m>=3 (m=1 or m=2 is impossible).

			71 is a term since 71 and 73 are twin primes and Omega(71 + 1) = Omega(72) = Omega(2*2*2*3*3) = 5.

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

Cf. A060213, A102168, A164290.

Omega = If[ # == 1, 0, Apply[Plus, Transpose[FactorInteger[ # ]][[2]]]] &; Wmil = Map[Omega, Range[1, 30000]]; Asequence = Flatten@Position[Partition[Wmil, 3, 1], {1, 5, 1}]
Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==2&&PrimeOmega[ #[[1]]+1] ==5&][[All,1]] (* Harvey P. Dale, Apr 02 2022 *)

A164290 Sequence of twin prime p where the middle term p+1 has 6 prime factors (here p+2 is the associated twin prime, not listed).

Original entry on oeis.org

239, 599, 809, 1319, 1487, 2087, 2339, 2969, 3299, 4157, 4271, 4787, 5021, 5099, 5231, 5639, 5849, 6359, 6659, 7307, 7349, 9431, 9767, 10007, 10139, 10331, 10709, 10889, 11069, 11171, 11351, 11549, 11717, 11831, 11969, 12539, 13007, 13337

Offset: 1

Carlos Alves, Aug 12 2009

nonn

This sequence is similar to A060213, A102168, A164289 respectively with 3, 4 and 5 prime factors in the middle number.

These sequences are of the form (p,p+1,p+2) with (p,p+2) twin primes and Omega(p+1)=m with m>=3 (m=1 or m=2 is impossible). Here m=6.

			(239, 240, 241): Omega(240)=Omega(2*2*2*2*3*5)=6 and 239, 241 are twin primes.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A060213, A102168, A164289, A164291.

Omega = If[ # == 1, 0, Apply[Plus, Transpose[FactorInteger[ # ]][[2]]]] &; Wmil = Map[Omega, Range[1, 30000]]; Asequence = Flatten@Position[Partition[Wmil, 3, 1], {1, 6, 1}]
Transpose[Select[Partition[Prime[Range[1600]],2,1],#[[2]]-#[[1]]==2 && PrimeOmega[ #[[1]]+1]==6&]][[1]] (* Harvey P. Dale, May 15 2012 *)

A164291 a(n) = p is the first twin prime (p, p+2) for which p+1 has n prime factors (n>=2, multiplicity counted).

Original entry on oeis.org

3, 11, 59, 71, 239, 191, 2111, 1151, 14591, 26111, 15359, 139967, 138239, 675839, 2101247, 737279, 4866047, 786431, 22118399, 36175871, 194641919, 63700991, 138412031, 169869311, 1321205759, 11123294207, 16357785599, 4076863487, 25165823999, 10871635967

Offset: 2

Carlos Alves, Aug 12 2009

nonn

a(3)-a(6) are the first elements of A060213, A102168, A164289, A164290 respectively with n=3,4,5,6 (prime factors in the middle number).

This gives the first p with (p,p+2) twin primes and Omega(p+1)=n with n>=2 (n=1 is impossible).

			a(7)=191 because in (191, 192, 193) we have Omega(192)=Omega(2*2*2*2*2*2*3)=7 and 191, 193 are twin primes.
The sequence oscillates and here we see that a(7)<a(6)=239.

Donovan Johnson, Table of n, a(n) for n = 2..200

Cf. A060213, A102168, A164289, A164290.

Omega = If[ # == 1, 0, Apply[Plus, Transpose[FactorInteger[ # ]][[2]]]] &; Wmil = Map[Omega, Range[1, 10000000]]; Aseq=(Flatten@Position[Partition[Wmil, 3, 1], {1, #, 1}])[[1]] & /@ Range[3,19]

Definition and comments corrected, a(2) and a(20)-a(29) from Donovan Johnson, Aug 20 2009

A283532 Primes p such that (q^2 - p^2) / 24 is prime, where q is the next prime after p.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 67, 83, 101, 109, 127, 131, 137, 251, 271, 281, 307, 331, 379, 383, 443, 487, 499, 563, 617, 641, 769, 821, 877, 937, 971, 1009, 1123, 1223, 1231, 1283, 1291, 1297, 1543, 1567, 1697, 1877, 2063, 2081, 2237, 2269, 2371, 2381, 2383, 2389, 2551, 2657, 2659, 2801, 2851, 2857

Offset: 1

Thomas Ordowski and Altug Alkan, Mar 10 2017

nonn

This sequence is union of primes of the form:
6t-1 such that 6t+1 and t are both prime,
6t-1 such that 6t+5 and 3t+1 are both prime and 6t+1 is composite,
6t+1 such that 6t+5 and 2t+1 are both prime,
6t+1 such that 6t+7 and 3t+2 are both prime and 6t+5 is composite.

			7 is a term since 11 is the next prime and (11^2 - 7^2)/24 = 3 is prime.

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

A060213 is a subsequence.

Cf. A075888.

N:= 10000: # to get all terms <= N
Primes:= select(isprime, [seq(i,i=3..N,2)]):
f:= proc(p,q)
  local r;
  r:= (q^2-p^2)/24;
  if r::integer and isprime(r) then p fi
end proc:
seq(f(Primes[i],Primes[i+1]),i=1..nops(Primes)-1); # Robert Israel, Mar 10 2017

Select[Prime@ Range@ 415, PrimeQ[(NextPrime[#]^2 - #^2)/24] &] (* Michael De Vlieger, Mar 13 2017 *)

is(n) = n>3 && isprime(n) && isprime((nextprime(n+1)^2-n^2)/24);

A358202 Lower twin primes p such that 6p-1 and 6p+1 are twin primes and (p+1)/6 is prime.

Original entry on oeis.org

17, 137, 23537, 92957, 157217, 318677, 326657, 440177, 510617, 521537, 558497, 577937, 617717, 651017, 661097, 861437, 969257, 1093997, 1152077, 1168337, 1177157, 1260317, 1299917, 1356077, 1463177, 1514657, 1600097, 1617437, 1768757, 1773977, 1957937, 2065577, 2271497, 2335637, 2382557, 2450597

Offset: 1

J. M. Bergot and Robert Israel, Nov 03 2022

nonn

			a(2) = 137 is a term because 137 and 139 are twin primes, 6*137-1 = 821 and 6*137+1 = 823 are twin primes, and (137+1)/6 = 23 is a prime.

Robert Israel, Table of n, a(n) for n = 1..1000

Intersection of A060213 and A176131.

P:= select(isprime, {seq(i,i=5..2*10^7,2)}):
T:= P intersect map(`-`,P,2):
R:=T intersect map(t -> (t+1)/6, T):
sort(convert(select(t -> isprime((t+1)/6), R),list));

Select[Prime[Range[180000]], PrimeQ[# + 2] && PrimeQ[6*# - 1] && PrimeQ[6*# + 1] && PrimeQ[(# + 1)/6] &] (* Amiram Eldar, Nov 03 2022 *)
Select[Prime[Range[180000]],AllTrue[{#+2,6#+1,6#-1,(#+1)/6},PrimeQ]&] (* Harvey P. Dale, Jan 29 2023 *)

A065511 Let p(k) denote k-th prime; consider solutions (n,m) of the Diophantine system {p(p(n)+1)-p(p(n))=2, p(p(n))-6.p(p(m))=-1} (*); sequence gives values of m.

Original entry on oeis.org

1, 92, 105, 106, 689, 1138, 1280, 1373, 1432, 1600, 1960, 2416, 2683, 2846, 2968, 3042, 3091, 4694, 4884, 5191, 5284, 5642, 6905, 6949, 7074, 7095, 7213, 7274, 7418

Offset: 0

Lubomir Alexandrov, Nov 26 2001

nonn

( p(p(n)), p(p(n)+1) ) is twin prime pair with average 6.p(p(m)) (A060213).

			n = 402 and m = 105 satisfy system (*) at the primes p(p(n)) = 24917 and p(p(m)) = 4153; n = 33521 and m = 7418 satisfy system (*) at the primes p(p(n)) = 5730617 and p(p(m)) = 955103.

A060213

cp's OEIS Frontend

A065505 Let p(k) denote k-th prime; consider solutions (p(n),p(m)) of Diophantine equation p(p(n)+1)-6.p(p(m))=1 (*), where p(p(n)) belongs to A060213 and p(p(m))=(p(p(n))+1)/6; sequence gives values of p(n).

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A065503 Indices k of primes p(k) such that p(k) is in A060213.

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A164289 Sequence of twin primes p where the middle term p+1 has 5 prime factors (here p+2 is the associated twin prime, not listed).

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A164290 Sequence of twin prime p where the middle term p+1 has 6 prime factors (here p+2 is the associated twin prime, not listed).

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A164291 a(n) = p is the first twin prime (p, p+2) for which p+1 has n prime factors (n>=2, multiplicity counted).

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A283532 Primes p such that (q^2 - p^2) / 24 is prime, where q is the next prime after p.

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PARI

A358202 Lower twin primes p such that 6*p-1 and 6*p+1 are twin primes and (p+1)/6 is prime.

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Maple

Mathematica

A065511 Let p(k) denote k-th prime; consider solutions (n,m) of the Diophantine system {p(p(n)+1)-p(p(n))=2, p(p(n))-6.p(p(m))=-1} (*); sequence gives values of m.

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A358202 Lower twin primes p such that 6p-1 and 6p+1 are twin primes and (p+1)/6 is prime.