A216497 -id:A216497 - cp's OEIS frontend

A094383 Primes p such that d>0 exists and p-d, p-2d and p-3d are also primes.

23, 29, 41, 43, 53, 59, 79, 83, 97, 101, 103, 107, 113, 127, 131, 139, 149, 151, 157, 163, 167, 173, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 313, 317, 331, 347, 349, 353, 359, 367, 373, 383

Offset: 1

Reinhard Zumkeller, Apr 28 2004

nonn

Conjecture: only 25 primes are not in the sequence, namely 2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 47, 61, 67, 71, 73, 89, 109, 137, 179, 211, 277, 337, 379, 499, 557. - Alex Ratushnyak, Sep 08 2012

			59=prime(17) -> 59-6=53=prime(16) -> 53-6=47=prime(15) ->
47-6=41=prime(13), therefore 59 is a term; also 59 -> 59-18=41=prime(13) ->
41-18=23=prime(9) -> 23-18=5=prime(3).

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

Cf. A094382, A216495, A216496, A216497, A216498, A216468.

prms = 3; fQ[p_] := Module[{d = 1}, While[prms*d < p && Union[PrimeQ[p - Range[prms]*d]] != {True}, d++]; prms*d < p]; Select[Prime[Range[2, PrimePi[383]]], fQ] (* T. D. Noe, Sep 08 2012 *)

is(n)=my(t); forprime(p=2,n-6,if((n-p)%3==0 && isprime((t=(n-p)/3)+p) && isprime(2*t+p) && isprime(n), return(1))); 0 \\ Charles R Greathouse IV, Sep 10 2014

A216495 Primes p with property that there exists a number d>0 such that numbers p-d, p-2*d are primes.

Original entry on oeis.org

7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Alex Ratushnyak, Sep 08 2012

nonn

Conjecture: only 5 primes are not in the sequence: 2, 3, 5, 13, 37.

Vincenzo Librandi, Table of n, a(n) for n = 1..9617

Cf. A094383, A216497, A216498, A216468.

prms = 2; fQ[p_] := Module[{d = 1}, While[prms*d < p && Union[PrimeQ[p - Range[prms]*d]] != {True}, d++]; prms*d < p]; Select[Prime[Range[2, PrimePi[283]]], fQ] (* T. D. Noe, Sep 08 2012 *)

is(n)=my(t); forprime(p=2,n-4,if(isprime((t=(n-p)\2)+p) && isprime(2*t+p) && isprime(n), return(1))); 0 \\ Charles R Greathouse IV, Sep 10 2014

A216468 Primes p with property that there exists a number d>0 such that numbers p-k*d, k=1...6, are six primes.

Original entry on oeis.org

907, 1307, 1439, 1459, 1669, 1879, 2089, 2141, 2351, 2713, 4139, 4759, 4969, 5179, 5417, 6047, 6101, 6353, 6779, 6793, 7919, 8369, 8663, 9049, 9349, 9491, 9533, 9623, 9769, 10099, 10691, 10883, 11083, 11213, 11369, 11399, 11621, 11789, 11887, 11923, 12097, 12119

Offset: 1

Alex Ratushnyak, Sep 07 2012

nonn

Conjecture: only 312722 primes are not in the sequence: 2, 3, ..., 198702899.

			907 is in the sequence because with d = 150: 7, 157, 307, 457, 607, 757 are all primes.

Cf. A215895, A216495, A094383, A216497, A216498.

fQ[p_] := Module[{d = 1}, While[6*d < p && Union[PrimeQ[p - Range[6]*d]] != {True}, d++]; 6*d < p]; Select[Prime[Range[4, PrimePi[12119]]], fQ] (* T. D. Noe, Sep 07 2012 *)

is(n)=my(t); forprime(p=2,n-20,if((n-p)%6==0 && isprime((t=(n-p)/6)+p) && isprime(2*t+p) && isprime(3*t+p) && isprime(4*t+p) && isprime(5*t+p) && isprime(n), return(1))); 0 \\ Charles R Greathouse IV, Sep 10 2014

A216498 Primes p with property that there exists a number d>0 such that numbers p-k*d, k=1...5, are five primes.

Original entry on oeis.org

157, 257, 311, 353, 463, 509, 691, 757, 823, 839, 881, 907, 941, 953, 1063, 1097, 1223, 1229, 1249, 1297, 1301, 1307, 1439, 1459, 1531, 1583, 1669, 1723, 1777, 1879, 1907, 1913, 1931, 2027, 2087, 2089, 2141, 2143, 2179, 2207, 2293, 2351, 2371, 2377, 2399, 2411

Offset: 1

Alex Ratushnyak, Sep 08 2012

nonn

Conjecture: only 9198 primes are not in the sequence: 2, 3, ..., 2521081.

			157 is in the sequence because with d=30: 127, 97, 67, 37, 7 are all primes.

Cf. A216495, A094383, A216497, A216468.

prms = 5; fQ[p_] := Module[{d = 1}, While[prms*d < p && Union[PrimeQ[p - Range[prms]*d]] != {True}, d++]; prms*d < p]; Select[Prime[Range[2, PrimePi[2411]]], fQ] (* T. D. Noe, Sep 08 2012 *)

is(n)=my(t); forprime(p=2,n-16,if((n-p)%5==0 && isprime((t=(n-p)/5)+p) && isprime(2*t+p) && isprime(3*t+p) && isprime(4*t+p) && isprime(n), return(1))); 0 \\ Charles R Greathouse IV, Sep 10 2014

A216496 Primes that are not in A094383.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 47, 61, 67, 71, 73, 89, 109, 137, 179, 211, 277, 337, 379, 499, 557

Offset: 1

Alex Ratushnyak, Sep 08 2012

nonn

Conjecture: a(25)=557 is the last term.

Cf. A216495, A094383, A216497, A216498, A216468.

prms = 3; fQ[p_] := Module[{d = 1}, While[prms*d < p && Union[PrimeQ[p - Range[prms]*d]] != {True}, d++]; prms*d < p]; Select[Prime[Range[PrimePi[10000]]], ! fQ[#] &] (* T. D. Noe, Sep 08 2012 *)

A216590 Primes p with property that there exists a number d>0 such that numbers p-k*d, k=1...7, are seven primes.

Original entry on oeis.org

1669, 1879, 2089, 2351, 4969, 5179, 6047, 10883, 11923, 12097, 12143, 12329, 12539, 12763, 13049, 13183, 15413, 15923, 16187, 16547, 16741, 17189, 17581, 18481, 19993, 20201, 21433, 21727, 22303, 22483, 23021, 23053, 23831, 24023, 24749, 25579, 25633, 26111, 26561

Offset: 1

Alex Ratushnyak, Sep 09 2012

nonn

Conjecture: only 5254157 primes are not in the sequence: 2, 3, ..., 5082095279.

Conjecture: for any k>0 there exists p0 such that for any prime p>p0 there exists a k-term arithmetic progression of primes with p at the end.

			1669 is in the sequence because with d=210: 1459, 1249, 1039, 829, 619, 409, 199 are all primes.

Cf. A216495, A094383, A216497, A216498, A216468.

is(n)=my(t); forprime(p=2,n-26,if((n-p)%7==0 && isprime((t=(n-p)/7)+p) && isprime(2*t+p) && isprime(3*t+p) && isprime(4*t+p) && isprime(5*t+p) && isprime(6*t+p) && isprime(n), return(1))); 0 \\ Charles R Greathouse IV, Sep 10 2014

A216539 Largest prime p such that there is no n-term arithmetic progression of primes ending with p.

Original entry on oeis.org

2, 37, 557, 100291, 2521081, 198702899, 5082095279

Offset: 2

Alex Ratushnyak, Sep 12 2012

For n>=3, the value given is only a conjecture.

			It is conjectured (A216495) that 37 is the largest prime p with no 3-term arithmetic progression ending with p. This would imply a(3)=37. (7,19,31, for example, is a progression ending with 31.)

A216495 conjectures a(3).
A094383 conjectures a(4).
A216497 conjectures a(5).
A216498 conjectures a(6).
A216468 conjectures a(7).
A216590 conjectures a(8).
Cf. A005115.

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A094383 Primes p such that d>0 exists and p-d, p-2*d and p-3*d are also primes.

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A216495 Primes p with property that there exists a number d>0 such that numbers p-d, p-2*d are primes.

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A216468 Primes p with property that there exists a number d>0 such that numbers p-k*d, k=1...6, are six primes.

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A216498 Primes p with property that there exists a number d>0 such that numbers p-k*d, k=1...5, are five primes.

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A216496 Primes that are not in A094383.

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A216590 Primes p with property that there exists a number d>0 such that numbers p-k*d, k=1...7, are seven primes.

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A216539 Largest prime p such that there is no n-term arithmetic progression of primes ending with p.

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A094383 Primes p such that d>0 exists and p-d, p-2d and p-3d are also primes.