A207820 -id:A207820 - cp's OEIS frontend

A210465 Let p_(3,1)(m) be the m-th prime == 1(mod 3). Then a(n) is the smallest p_(3,1)(m) such that the interval(p_(3,1)(m)n, p_(3,1)(m+1)n) contains exactly one prime == 1(mod 3).

7, 13, 193, 271, 157, 193, 1297, 1741, 1231, 1033, 3541, 1447, 727, 2341, 9337, 1747, 9007, 2287, 3307, 14401, 8887, 8161, 8461, 28753, 23623, 23893, 10861, 59233, 70111, 28927, 44257, 101113, 152947, 41941, 65167, 41263, 183301, 409573, 150517, 35803, 138883, 81547, 79693

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Jan 22 2013

nonn

The limit of a(n) as n goes to infinity is infinity.
Conjectures: (1) If q is the nearest prime>a(n), then q-a(n)=4 or 6 and both of these cases occur infinitely many times. (2) If q-a(n)=4 then q is the lesser of twin primes.
Thus, if the conjectures are true, then there exist infinitely many triples of primes of the form {p,p+4,p+6}.

Cf. A195325, A207820, A210467, A210475, A210476.

bPrime=Select[Table[Prime[n],{n,1000000}],Mod[#,3]==1&];(*A002476*)
binarySearch[lst_,find_]:=Module[{lo=1,up=Length[lst],v},(While[lo<=up,v=Floor[(lo+up)/2];If[lst[[v]]-find==0,Return[v]];If[lst[[v]]0&]]]+offset-1]];
z=1;(*example for "contains exactly ONE b-
primes"*)Table[bPrime[[NestWhile[#1+1&,1,!((nextBPrime[n bPrime[[#1]],z]n bPrime[[#1+1]]))&]]],{n,2,20}]

A210467 Let p_(3,2)(m) be the m-th prime == 2(mod 3). Then a(n) is the smallest p_(3,2)(m) such that the interval(p_(3,2)(m)n, p_(3,2)(m+1)n) contains exactly one prime == 2 (mod 3).

Original entry on oeis.org

2, 2, 101, 263, 1097, 251, 311, 461, 641, 941, 1601, 2351, 2543, 5003, 2837, 4787, 5711, 4283, 7901, 10331, 8831, 2687, 7877, 54287, 5711, 5501, 5303, 56087, 69827, 15641, 63611, 138581, 106427, 91571, 69827, 266177, 142421, 177533, 179687, 309311, 55691, 119291, 509543, 593987, 1393913

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Jan 22 2013

nonn

The limit of a(n) as n goes to infinity is infinity.
Conjectures: (1) If q is the nearest prime > a(n), then q-a(n) = 2 or 6 and both of these cases occur infinitely many times. (2) If q-a(n) = 2, then also q is lesser of a pair of cousin primes q and q+4, see A023200.
Thus, if the conjectures are true, then there exist infinitely many triples of primes of the form {p,p+2,p+6}.

Cf. A195325, A210465, A207820, A210475, A210476.

bPrime=Select[Table[Prime[n], {n, 1000000}], Mod[#, 3]==2&];
binarySearch[lst_, find_]:=Module[{lo=2, up=Length[lst], v}, (While[lo<=up, v=Floor[(lo+up)/2]; If[lst[[v]]-find==0, Return[v]]; If[lst[[v]]0&]]]+offset-1]];
z=1; (*example for "contains exactly ONE b-
primes"*)Table[bPrime[[NestWhile[#1+1&, 1, !((nextBPrime[n bPrime[[#1]], z]n bPrime[[#1+1]]))&]]], {n, 2, 20}]

A210475 Let p_(4,1)(m) be the m-th prime == 1 (mod 4). Then a(n) is the smallest p_(4,1)(m) such that the interval(p_(4,1)(m)n, p_(4,1)(m+1)n) contains exactly one prime == 1 (mod 4).

Original entry on oeis.org

13, 13, 29, 13, 193, 97, 97, 277, 457, 1193, 109, 229, 937, 397, 349, 1597, 2137, 937, 5569, 5737, 2833, 1549, 6733, 7477, 5077, 3457, 877, 4153, 12277, 11113, 8689, 14029, 11113, 5233, 24109, 14737, 26713, 1297, 77797, 12097, 51577, 57973, 33409, 30493, 49429, 112237, 10333, 143137

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Jan 23 2013

nonn

The limit of a(n) as n goes to infinity is infinity.

Conjecture: for n >= 12, every a(n) is the lesser of a pair of cousin primes p and p+4, (see A023200).

Cf. A195325, A207820, A210465, A210467.

myPrime=Select[Table[Prime[n],{n,3000000}],Mod[#,4]==1&];
binarySearch[lst_,find_]:=Module[{lo=1,up=Length[lst],v},(While[lo<=up,v=Floor[(lo+up)/2];If[lst[[v]]-find==0,Return[v]];If[lst[[v]]0&]]]+offset-1]];
z=1;(*contains exactly ONE myPrime in the interval*)
Table[myPrime[[NestWhile[#1+1&,1,!((nextMyPrime[n myPrime[[#1]],z+1]>n myPrime[[#1+1]]))&]]],{n,2,30}]

A210476 Let p_(4,3)(m) be the m-th prime == 3 (mod 4). Then a(n) is the smallest p_(4,3)(m) such that the interval(p_(4,3)(m)n, p_(4,3)(m+1)n) contains exactly one prime == 3(mod 4).

Original entry on oeis.org

7, 67, 43, 67, 67, 191, 883, 43, 643, 379, 739, 103, 463, 643, 487, 883, 1303, 3847, 1447, 13963, 1087, 8863, 1999, 8167, 7687, 8443, 2707, 2203, 11083, 3463, 7687, 31387, 8419, 15919, 12979, 10099, 26683, 22027, 46687, 79687, 15439, 65839, 46723, 44683, 14887, 58963, 13879, 26947, 77587

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Jan 23 2013

nonn

The limit of a(n) as n goes to infinity is infinity.

Conjecture: every a(n), except for a(7) = 191, is the lesser of a pair of cousin primes p and p+4, (see A023200).

Cf. A195325, A207820, A210465, A210467, A210475.

myPrime=Select[Table[Prime[n],{n,3000000}],Mod[#,4]==3&];
binarySearch[lst_,find_]:=Module[{lo=1,up=Length[lst],v},(While[lo<=up,v=Floor[(lo+up)/2];If[lst[[v]]-find==0,Return[v]];If[lst[[v]]0&]]]+offset-1]];
z=1;(*contains exactly ONE myPrime in the interval*)
Table[myPrime[[NestWhile[#1+1&,1,!((nextMyPrime[n myPrime[[#1]],z+1]>n myPrime[[#1+1]]))&]]],{n,2,30}]

cp's OEIS Frontend

A210465 Let p_(3,1)(m) be the m-th prime == 1(mod 3). Then a(n) is the smallest p_(3,1)(m) such that the interval(p_(3,1)(m)n, p_(3,1)(m+1)n) contains exactly one prime == 1(mod 3).

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A210467 Let p_(3,2)(m) be the m-th prime == 2(mod 3). Then a(n) is the smallest p_(3,2)(m) such that the interval(p_(3,2)(m)n, p_(3,2)(m+1)n) contains exactly one prime == 2 (mod 3).

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A210475 Let p_(4,1)(m) be the m-th prime == 1 (mod 4). Then a(n) is the smallest p_(4,1)(m) such that the interval(p_(4,1)(m)n, p_(4,1)(m+1)n) contains exactly one prime == 1 (mod 4).

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A210476 Let p_(4,3)(m) be the m-th prime == 3 (mod 4). Then a(n) is the smallest p_(4,3)(m) such that the interval(p_(4,3)(m)n, p_(4,3)(m+1)n) contains exactly one prime == 3(mod 4).

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cp's OEIS Frontend

A210465 Let p_(3,1)(m) be the m-th prime == 1(mod 3). Then a(n) is the smallest p_(3,1)(m) such that the interval(p_(3,1)(m)*n, p_(3,1)(m+1)*n) contains exactly one prime == 1(mod 3).

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A210467 Let p_(3,2)(m) be the m-th prime == 2(mod 3). Then a(n) is the smallest p_(3,2)(m) such that the interval(p_(3,2)(m)*n, p_(3,2)(m+1)*n) contains exactly one prime == 2 (mod 3).

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A210475 Let p_(4,1)(m) be the m-th prime == 1 (mod 4). Then a(n) is the smallest p_(4,1)(m) such that the interval(p_(4,1)(m)*n, p_(4,1)(m+1)*n) contains exactly one prime == 1 (mod 4).

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A210476 Let p_(4,3)(m) be the m-th prime == 3 (mod 4). Then a(n) is the smallest p_(4,3)(m) such that the interval(p_(4,3)(m)*n, p_(4,3)(m+1)*n) contains exactly one prime == 3(mod 4).

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A210465 Let p_(3,1)(m) be the m-th prime == 1(mod 3). Then a(n) is the smallest p_(3,1)(m) such that the interval(p_(3,1)(m)n, p_(3,1)(m+1)n) contains exactly one prime == 1(mod 3).

A210467 Let p_(3,2)(m) be the m-th prime == 2(mod 3). Then a(n) is the smallest p_(3,2)(m) such that the interval(p_(3,2)(m)n, p_(3,2)(m+1)n) contains exactly one prime == 2 (mod 3).

A210475 Let p_(4,1)(m) be the m-th prime == 1 (mod 4). Then a(n) is the smallest p_(4,1)(m) such that the interval(p_(4,1)(m)n, p_(4,1)(m+1)n) contains exactly one prime == 1 (mod 4).

A210476 Let p_(4,3)(m) be the m-th prime == 3 (mod 4). Then a(n) is the smallest p_(4,3)(m) such that the interval(p_(4,3)(m)n, p_(4,3)(m+1)n) contains exactly one prime == 3(mod 4).