A185934 -id:A185934 - cp's OEIS frontend

A039701 a(n) = n-th prime modulo 3.

2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1

Offset: 1

Clark Kimberling

If n > 2 and prime(n) is a Mersenne prime then a(n) = 1. Proof: prime(n) = 2^p - 1 for some odd prime p, so prime(n) = 2*4^((p-1)/2) - 1 == 2 - 1 = 1 (mod 3). - Santi Spadaro, May 03 2002; corrected and simplified by Dean Hickerson, Apr 20 2003
Except for n = 2, a(n) is the smallest number k > 0 such that 3 divides prime(n)^k - 1. - T. D. Noe, Apr 17 2003
a(n) <> 0 for n <> 2; a(A049084(A003627(n))) = 2; a(A049084(A002476(n))) = 1; A134323(n) = (1 - 0^a(n)) * (-1)^(a(n)+1). - Reinhard Zumkeller, Oct 21 2007
Probability of finding 1 (or 2) in this sequence is 1/2. This follows from the Prime Number Theorem in arithmetic progressions. Examples: There are 4995 1's in terms (10^9 +1) to (10^9+10^4); there are 10^9/2-1926 1's in the first 10^9 terms. - Jerzy R Borysowicz, Mar 06 2022

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A091178 (indices of 1's), A091177 (indices of 2's).
Cf. A120326 (partial sums).
Cf. A185934, A217659.
Cf. A010872.
Other moduli: A039702-A039706, A038194, A007652, A039709-A039715.

a039701 = (`mod` 3) . a000040
a039701_list = map (`mod` 3) a000040_list
-- Reinhard Zumkeller, Nov 16 2012

[p mod(3): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 3, n=1..105); # Nathaniel Johnston, Jun 29 2011

Mathematica
```
Table[Mod[Prime[n], 3], {n, 100}]
```

primes(100)%3 \\ Charles R Greathouse IV, May 06 2014

Sum_k={1..n} a(k) ~ (3/2)*n. - Amiram Eldar, Dec 11 2024

A030459 Prime p concatenated with next prime is also prime.

Original entry on oeis.org

2, 31, 83, 151, 157, 167, 199, 233, 251, 257, 263, 271, 331, 353, 373, 433, 467, 509, 523, 541, 601, 653, 661, 677, 727, 941, 971, 1013, 1033, 1181, 1187, 1201, 1223, 1259, 1367, 1453, 1459, 1657, 1669, 1709, 1741, 1861, 1973, 2069, 2161

Offset: 1

Patrick De Geest

Terms 157, 257, 263, 541, 1187, 1459, 2179 also belong to A030460. - Carmine Suriano, Jan 27 2011

All terms, except for the first one, must be either in A185934 or in A185935, i.e., have the same residue (mod 6) as the subsequent prime. - M. F. Hasler, Feb 06 2011

M. F. Hasler and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 4110 terms from Hasler)

See A030461 for the concatenated primes.

Cf. A000720, A030460, A045533, A151799.

Select[Prime[Range[500]],PrimeQ[FromDigits[Join[IntegerDigits[#], IntegerDigits[ NextPrime[#]]]]]&] (* Harvey P. Dale, Jun 20 2011 *)

o=2;forprime(p=3,1e4, isprime(eval(Str(o,o=p))) & print1(precprime(p-1)","))  \\ M. F. Hasler, Feb 06 2011

a(n) = A151799(A030460(n)).

A030461(n) = concat(a(n), A030460(n)) = A045533(A000720(a(n))).

A030461 Primes that are concatenations of two consecutive primes.

Original entry on oeis.org

23, 3137, 8389, 151157, 157163, 167173, 199211, 233239, 251257, 257263, 263269, 271277, 331337, 353359, 373379, 433439, 467479, 509521, 523541, 541547, 601607, 653659, 661673, 677683, 727733, 941947, 971977, 10131019

Offset: 1

Patrick De Geest

Any term in the sequence (apart from the first) must be a concatenation of consecutive primes differing by a multiple of 6. - Francis J. McDonnell, Jun 26 2005

			a(2) is 3137 because 31 and 37 are consecutive primes and after concatenation 3137 is also prime. - _Enoch Haga_, Sep 30 2007

Georg Fischer, Table of n, a(n) for n = 1..5720 [First 1000 terms from Zak Seidov]

Cf. A030459.
Cf. A185934, A185935.
Subsequence of A045533.

a030461 n = a030461_list !! (n-1)
a030461_list = filter ((== 1) . a010051') a045533_list
-- Reinhard Zumkeller, Apr 20 2012

[Seqint( Intseq(NthPrime(n+1)) cat Intseq(NthPrime(n)) ): n in [1..200 ]| IsPrime(Seqint( Intseq(NthPrime(n+1)) cat Intseq(NthPrime(n)) )) ]; // Marius A. Burtea, Mar 21 2019

conc:=proc(a,b) local bb: bb:=convert(b,base,10): 10^nops(bb)*a+b end: p:=proc(n) local w: w:=conc(ithprime(n),ithprime(n+1)): if isprime(w)=true then w else fi end: seq(p(n),n=1..250); # Emeric Deutsch

Select[Table[p=Prime[n]; FromDigits[Join[Flatten[IntegerDigits[{p,NextPrime[p]}]]]],{n,170}],PrimeQ] (* Jayanta Basu, May 16 2013 *)

{digits(n) = if(n==0,[0],u=[];while(n>0,d=divrem(n,10);n=d[1];u=concat(d[2],u));u)} {m=1185;p=2;while(pKlaus Brockhaus

o=2;forprime(p=3,1e4, isprime(eval(Str(o,o=p))) & print1(precprime(p-1),p",")) \\ M. F. Hasler, Feb 06 2011

A030461(n) = concat(A030459(n),A030460(n)) = A045533( A000720( A030459(n))). - M. F. Hasler, Feb 06 2011

Edited by N. J. A. Sloane, Apr 19 2009 at the suggestion of Zak Seidov

A217659 Larger of two consecutive primes which both equal 1 (mod 3).

Original entry on oeis.org

37, 67, 79, 157, 163, 211, 223, 277, 337, 373, 379, 439, 541, 547, 577, 607, 613, 631, 673, 733, 739, 757, 997, 1009, 1039, 1069, 1087, 1123, 1129, 1213, 1237, 1249, 1297, 1327, 1399, 1459, 1471, 1543, 1549, 1627, 1663, 1693, 1747, 1753, 1759, 1777, 1783

Offset: 1

Reinhard Zumkeller, Nov 16 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A002476, A151800, A185934, A219244.

a217659 = a151800 . fromInteger . a185934

With[{p = Prime[Range[350]]}, ind = Position[Mod[p, 3], 1] // Flatten; p[[1 + ind[[Position[Differences[ind], 1] // Flatten]]]]] (* Amiram Eldar, Apr 12 2025 *)

a(n) = A151800(A185934(n)) = 6*A219244(n) + A185934(n).

A185936 First of a run of 3 or more consecutive primes which all equal 1 (mod 3).

Original entry on oeis.org

151, 199, 367, 523, 601, 727, 991, 1063, 1117, 1231, 1453, 1531, 1741, 1747, 1753, 1759, 2161, 2281, 2671, 3049, 3061, 3169, 3301, 3307, 3499, 3631, 3727, 4093, 4159, 4423, 4549, 4591, 4597, 4651, 4987, 5101, 5107, 5197, 5419, 5557, 5743, 5821, 6067, 6361, 6367, 6397, 6607, 6661, 7351, 7369, 7393, 7951, 8179, 8311

Offset: 1

M. F. Hasler, Feb 06 2011

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

A subsequence of A185934.

Select[Partition[Prime[Range[1500]], 3, 1], Mod[#, 3] == {1, 1, 1} &][[All, 1]] (* Paolo Xausa, Mar 07 2025 *)

s=Mod([1,1,1],3); o=vector(3); i=0; forprime( p=1,1e4, o[i++%3+1]=p; o-s || print1( o[(i+1)%3+1]","))

A219244 Differences of two consecutive primes which both equal 1 modulo 3, divided by 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Nov 16 2012

nonn

a(n) = (A217659(n) - A185934(n)) / 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A185934, A217659.

a219244 n = a219244_list !! (n-1)
a219244_list = map (`div`  6) $ zipWith (-) a217659_list a185934_list

(Last[#]-First[#])/6&/@Select[Partition[Prime[Range[800]],2,1], Mod[First[#],3] == Mod[Last[#],3]==1&] (* Harvey P. Dale, Feb 26 2013 *)

A185942 First of a run of 4 or more consecutive primes which all equal 1 (mod 3).

Original entry on oeis.org

1741, 1747, 1753, 3049, 3301, 4591, 5101, 6361, 7351, 7369, 8311, 8707, 8713, 8887, 9067, 9091, 9103, 9631, 10639, 11287, 12577, 12823, 12829, 13267, 15187, 15583, 15817, 15889, 15901, 16363, 16369, 16561, 16729, 16981, 17041, 17419, 17431, 17839, 18199, 18211, 19213, 19219, 19471, 19477, 19483, 19489, 20071

Offset: 1

M. F. Hasler, Feb 06 2011

nonn

Subsequence of terms of A185936 such that A185936(k+1) = nextprime(A185936(k)).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A185934 - A185941.

mod3Q[l_]:=Union[Mod[#,3]&/@l]=={1}
Transpose[Select[Partition[Prime[Range[2500]],4,1],mod3Q]][[1]]  (* Harvey P. Dale, Feb 16 2011 *)

s=Mod([1,1,1,1],3);o=vector(#s);i=0;forprime(p=1,3e4,o[i++%#o+1]=p;o-s|print1(o[(i+1)%#o+1]","))

cp's OEIS Frontend

A039701 a(n) = n-th prime modulo 3.

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A030459 Prime p concatenated with next prime is also prime.

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A030461 Primes that are concatenations of two consecutive primes.

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A217659 Larger of two consecutive primes which both equal 1 (mod 3).

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A185936 First of a run of 3 or more consecutive primes which all equal 1 (mod 3).

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A219244 Differences of two consecutive primes which both equal 1 modulo 3, divided by 6.

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A185942 First of a run of 4 or more consecutive primes which all equal 1 (mod 3).

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