A232882 -id:A232882 - cp's OEIS frontend

A232880 Twin primes with digital root 2 or 4.

11, 13, 29, 31, 101, 103, 137, 139, 191, 193, 227, 229, 281, 283, 461, 463, 569, 571, 641, 643, 659, 661, 821, 823, 857, 859, 1019, 1021, 1091, 1093, 1289, 1291, 1451, 1453, 1487, 1489, 1667, 1669, 1721, 1723, 2027, 2029, 2081, 2083, 2549, 2551, 2657, 2659

Offset: 1

Gary Croft, Dec 01 2013

All twin primes except (3, 5) have one of 3 digital root pairings: {2, 4}, {5, 7} or {8, 1}: see A232881 for {5, 7} and A232882 for {8, 1}.

Or primes congruent to 11 or 13 mod 18 such that the number congruent to 13 or 11 mod 18 is also prime. - Alonso del Arte, Dec 02 2013

			11 and 13 are in the sequence because they form a twin prime pair in which 11 has a digital root of 2 and 13 has one of 4.
Likewise 29 and 31 form a twin prime pair with 29 has 2 for a digital root and 31 has 4.

Cf. A001097, A077800, A232881, A232882.

partialList = Select[18Range[100] - 7, PrimeQ[#] && PrimeQ[# + 2] &]; A232880 = Sort[Flatten[Join[partialList, partialList + 2]]] (* Alonso del Arte, Dec 02 2013 *)
dRoot[n_] := 1 + Mod[n - 1, 9]; tw = Select[Prime[Range[1000]], PrimeQ[# + 2] &]; Select[Union[tw, tw + 2], MemberQ[{2, 4}, dRoot[#]] &] (* T. D. Noe, Dec 10 2013 *)

p=5; forprime(q=7,1e4,if(q-p==2 && q%9==4, print1(p", "q", ")); p=q) \\ Charles R Greathouse IV, Aug 26 2014

A282321 Lesser of twin primes congruent to 11 (mod 30).

Original entry on oeis.org

11, 41, 71, 101, 191, 281, 311, 431, 461, 521, 641, 821, 881, 1031, 1061, 1091, 1151, 1301, 1451, 1481, 1721, 1871, 1931, 2081, 2111, 2141, 2381, 2591, 2711, 2801, 3251, 3371, 3461, 3581, 3671, 3821, 3851, 4001, 4091, 4241, 4271, 4421, 4481, 4721, 4931, 5021

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [this sequence and A282322] is A132241.
The union of [{3, 5}, this sequence, A282323 and A060229] is the lesser of twin primes sequence A001359.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Subset of A001097, A001359, A132232, A132241 and A132247.

Cf. A006512, A232880, A232881, A232882, A282322, A282323, A060229, A060229, A282326.

[p: p in PrimesUpTo(5000) | IsPrime(p+2) and p mod 30 eq 11 ]; // Vincenzo Librandi, Feb 12 2017

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)+2) and ithprime(i) mod 30 = 11 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;

list(lim)=my(v=List(), p=2); forprime(q=3, lim+2, if(q-p==2 && q%30==13, listput(v, p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A232881 Twin primes with digital root 5 or 7.

Original entry on oeis.org

5, 41, 59, 149, 239, 311, 347, 419, 599, 617, 1031, 1049, 1229, 1301, 1319, 1427, 1481, 1607, 1697, 1787, 1877, 1931, 1949, 2111, 2129, 2237, 2309, 2381, 2687, 3119, 3299, 3371, 3389, 3461, 3767, 3821, 3929, 4001, 4019, 4091, 4127, 4217, 4271, 4649, 4721

Offset: 1

Gary Croft, Dec 01 2013

All twin primes except (3,5) have one of 3 digital root pairings: {2,4}, {5,7} or {8,1}: see A232880 for {2,4} and A232882 for {8,1}.

			41 and 43 are in the sequence because they form a twin prime pair in which 41 has a digital root of 5 and 43 has a digital root of 7. Likewise 59 and 61 form a twin prime pair where 59 has a digital root of 5 and 61 has one of 7.

Cf. A001097, A077800, A232880, A232882.

dRoot[n_] := 1 + Mod[n - 1, 9]; tw = Select[Prime[Range[1000]], PrimeQ[# + 2] &]; Select[Union[tw, tw + 2], MemberQ[{5, 7}, dRoot[#]] &] (* T. D. Noe, Dec 10 2013 *)

A282323 Lesser of twin primes congruent to 17 (mod 30).

Original entry on oeis.org

17, 107, 137, 197, 227, 347, 617, 827, 857, 1277, 1427, 1487, 1607, 1667, 1697, 1787, 1877, 1997, 2027, 2087, 2237, 2267, 2657, 2687, 3167, 3257, 3467, 3527, 3557, 3767, 3917, 4127, 4157, 4217, 4337, 4517, 4547, 4637, 4787, 4967, 5417, 5477, 5657, 5867, 6197

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [this sequence and A282324] is A132242.
The union of [{3, 5}, A282321, this sequence and A060229] is the lesser of twin primes sequence A001359.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.
A181605 without the 7. The proof works along the same lines as the proof in A282322. - R. J. Mathar, Feb 14 2017
Number of terms < 10^k: 0, 0, 1, 9, 64, 414, 2734, 19674, 146953, ... - Muniru A Asiru, Jan 09 2018

			From _Muniru A Asiru_, Jan 25 2018: (Start)
17 is a member because the pair (17, 19) is a twin prime, 17 < 19 and 17 mod 30 = 17.
137 is a member because the pair (137, 139) is a twin prime, 137 < 139 and 137 mod 30 = 17.
197 is a member because the pair (197, 199) is a twin prime, 197 < 199 and 197 mod 30 = 17.
(End)

Muniru A Asiru, Table of n, a(n) for n = 1..3000

Subset of A001097, A001359, A039949, A132242 and A132247.

Cf. A006512, A232880, A232881, A232882, A282321, A282322, A282324, A282326.

P:=Filtered([1..400000], IsPrime);;
P1:=List(Filtered(Filtered(List([1..Length(P)-1],n->[P[n],P[n+1]]),i->i[2]-i[1]=2),j->j[1] mod 30=17),k->k[1]);; # Muniru A Asiru, Jul 08 2017

[p: p in PrimesUpTo(7000) | IsPrime(p+2) and p mod 30 eq 17 ]; // Vincenzo Librandi, Feb 13 2017

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)+2) and ithprime(i) mod 30 = 17 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;

Select[17 + 30 Range[0, 220], PrimeQ[#] && PrimeQ[# + 2] &] (* Robert G. Wilson v, Jan 09 2018 *)
Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2&&Mod[#[[1]],30]==17&][[;;,1]] (* or *) Select[Range[17,7000,30],AllTrue[#+{0,2},PrimeQ]&] (* Harvey P. Dale, Mar 02 2024 *)

list(lim)=my(v=List(), p=2); forprime(q=3, lim+2, if(q-p==2 && q%30==19, listput(v, p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A282324 Greater of twin primes congruent to 19 (mod 30).

Original entry on oeis.org

19, 109, 139, 199, 229, 349, 619, 829, 859, 1279, 1429, 1489, 1609, 1669, 1699, 1789, 1879, 1999, 2029, 2089, 2239, 2269, 2659, 2689, 3169, 3259, 3469, 3529, 3559, 3769, 3919, 4129, 4159, 4219, 4339, 4519, 4549, 4639, 4789, 4969, 5419, 5479, 5659, 5869, 6199

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [A282323 and this sequence] is A132242.
The union of [{5, 7}, A282322, this sequence and A282326] is the greater of twin primes sequence A006512.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.
Number of terms less than 10^k, k=2,3,4,...: 1, 9, 64, 414, 2734, 19674, 146953, ... - Muniru A Asiru, Feb 09 2018

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Subset of A001097, A006512, A132234, A132242 and A132247.

Cf. A001359, A132235, A132236, A232880, A232881, A232882, A282321, A282322, A282323, A060229, A282326.

Filtered(List([1..220], k -> 30*k-11), n -> IsPrime(n) and IsPrime(n-2));  # Muniru A Asiru, Feb 02 2018

[p: p in PrimesUpTo(7000) | IsPrime(p-2) and p mod 30 eq 19 ]; // Vincenzo Librandi, Feb 13 2017

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)-2) and ithprime(i) mod 30 = 19 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;
# More efficient
select(n -> isprime(n-2) and isprime(n), [seq(30*k+19, k=0..220)]); # Muniru A Asiru, Jan 30 2018

Select[Prime[Range[1000]], PrimeQ[# - 2] && Mod[#, 30] == 19 &] (* Vincenzo Librandi, Feb 13 2017 *)

list(lim)=my(v=List(), p=2); forprime(q=3, lim, if(q-p==2 && q%30==19, listput(v, q)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A282326 Greater of twin primes congruent to 1 (mod 30).

Original entry on oeis.org

31, 61, 151, 181, 241, 271, 421, 571, 601, 661, 811, 1021, 1051, 1231, 1291, 1321, 1621, 1951, 2131, 2311, 2341, 2551, 2731, 2791, 2971, 3001, 3121, 3301, 3331, 3361, 3391, 3541, 3931, 4021, 4051, 4231, 4261, 4651, 4801, 5011, 5101, 5281, 5521, 5641, 5851

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [A060229 and this sequence] is A132243.
The union of [{5, 7}, A282322, A282324 and this sequence] is the greater of twin primes sequence A006512.
The union of [{3, 5, 7}, A282321 to A060229 and this sequence] is the twin primes sequence A001097.
Number of terms less than 10^k, k=2,3,4,...: 2, 11, 72, 407, 2697, 19507, 146516, ... - Muniru A Asiru, Mar 05 2018

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Subset of A006512, A181603, A132230, A132243 and A132247.

Cf. A001359, A232880, A232881, A232882, A282321, A282322, A282323, A282324, A060229.

Filtered(List([0..300], k -> 30*k+1), n -> IsPrime(n-2) and IsPrime(n));  # Muniru A Asiru, Mar 05 2018

select(n -> isprime(n-2) and isprime(n), [seq(30*k+1, k=0..300)]); # Muniru A Asiru, Mar 05 2018

1 + Select[30 Range@ 200, AllTrue[# + {-1, 1}, PrimeQ] &] (* Michael De Vlieger, Mar 26 2018 *)

list(lim)=my(v=List(),p=2); forprime(q=3,lim, if(q-p==2 && q%30==1, listput(v,q)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A282322 Greater of twin primes congruent to 13 (mod 30).

Original entry on oeis.org

13, 43, 73, 103, 193, 283, 313, 433, 463, 523, 643, 823, 883, 1033, 1063, 1093, 1153, 1303, 1453, 1483, 1723, 1873, 1933, 2083, 2113, 2143, 2383, 2593, 2713, 2803, 3253, 3373, 3463, 3583, 3673, 3823, 3853, 4003, 4093, 4243, 4273, 4423, 4483, 4723, 4933, 5023, 5233, 5443, 5503, 5653, 5743

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [A282321 and this sequence] is A132241.
The union of [{5, 7}, this sequence, A282324 and A282326] is the greater of twin primes sequence A006512.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.
A181604 without the 3. [Proof: working mod 10 we see that each value here is in A181604. For the other direction: Except 3 all twin primes in A181604 are upper twin primes; they cannot be lower twin primes because the upper ones would be multiples of 5. The twin primes in A181604 could be == 3 (mod 30) or == 13 (mod 30) or == 23 (mod 30). The first case is excluded because they would be multiples of 3; the third case is excluded because the lower twin primes would be == 21 (mod 30) and also multiples of 3. So only the case == 13 (mod 30) remains.] - R. J. Mathar, Feb 14 2017
Number of terms < 10^k for k >= 1: 0, 3, 13, 67, 401, 2736, 19797, 146841, 1141217, 9137078, ..., . - Robert G. Wilson v, Jan 07 2018

Muniru A Asiru, Table of n, a(n) for n = 1..20000

Subset of A001097, A006512, A132233, A132241 and A132247.

Cf. A001359, A232880, A232881, A232882, A282321, A282323, A282324, A282326.

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)-2) and ithprime(i) mod 30 = 13 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;

Select[13 + 30 Range[0, 200], PrimeQ[# - 2] && PrimeQ[#] &] (* Robert G. Wilson v, Jan 07 2018 *)

list(lim)=my(v=List(), p=2); forprime(q=3, lim, if(q-p==2 && q%30==13, listput(v, q)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

cp's OEIS Frontend

A232880 Twin primes with digital root 2 or 4.

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A282321 Lesser of twin primes congruent to 11 (mod 30).

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A232881 Twin primes with digital root 5 or 7.

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A282323 Lesser of twin primes congruent to 17 (mod 30).

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A282324 Greater of twin primes congruent to 19 (mod 30).

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A282326 Greater of twin primes congruent to 1 (mod 30).

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A282322 Greater of twin primes congruent to 13 (mod 30).

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