A077800 -id:A077800 - cp's OEIS frontend

A045533 Concatenate the n-th and (n+1)st prime.

23, 35, 57, 711, 1113, 1317, 1719, 1923, 2329, 2931, 3137, 3741, 4143, 4347, 4753, 5359, 5961, 6167, 6771, 7173, 7379, 7983, 8389, 8997, 97101, 101103, 103107, 107109, 109113, 113127, 127131, 131137

Offset: 1

N. J. A. Sloane, Dec 11 1999

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077800, A095958 (subsequence), A030461 (primes).

a045533 n = a045533_list !! (n-1)
a045533_list = f $ map show a000040_list :: [Integer] where
   f (t:ts@(t':_)) = read (t ++ t') : f ts
-- Reinhard Zumkeller, Apr 20 2012

[Seqint(Intseq(NthPrime(n+1)) cat Intseq(NthPrime(n))): n in [1..50 ] ]; // Marius A. Burtea, Mar 21 2019

#[[1]]*10^IntegerLength[#[[2]]]+#[[2]]&/@Partition[Prime[Range[40]],2,1] (* Harvey P. Dale, Jun 06 2015 *)

a(n) = eval(concat(Str(prime(n)), Str(prime(n+1)))); \\ Michel Marcus, May 11 2014

a(n) = prime(n)*(10^floor(log_10(prime(n+1)))+1) + prime(n+1). - Conner L. Delahanty, May 10 2014

Offset changed to 1, in agreement with (almost?) all references to this sequence, by M. F. Hasler

A256753 Numbers n such that n is both the average of some twin prime pair p, q (q = p+2) (i.e., n = p+1 = q-1) and is also the average of the prime before p and the prime after q.

Original entry on oeis.org

12, 18, 30, 42, 60, 102, 108, 228, 270, 312, 420, 462, 570, 600, 858, 882, 1050, 1092, 1230, 1290, 1302, 1428, 1488, 1620, 1872, 1998, 2028, 2340, 2550, 2688, 2730, 3390, 3462, 3540, 3582, 4020, 4230, 4242, 4272, 4338, 4518, 4650, 4788

Offset: 1

Karl V. Keller, Jr., Apr 09 2015

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs).

			For n=12: 7, 11, 13, 17 are four consecutive primes with 13 = 11 + 2 and (7+17)/2 = 12.
For n=18: 13, 17, 19, 23 are four consecutive primes with 19 = 17 + 2 and (13+23)/2 = 18.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..500000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A077800 (twin primes), A014574.

Select[Prime[Range[10^3]],PrimeQ[#+2]&&2*#+2==NextPrime[#,-1]+NextPrime[#,2]&]+1 (* Ivan N. Ianakiev, Apr 23 2015 *)
Select[Partition[Prime[Range[700]],4,1],#[[3]]-#[[2]]==2&&(#[[1]]+#[[4]])/2 == (#[[2]]+#[[3]])/2&][[All,2]]+1 (* Harvey P. Dale, May 06 2022 *)

lista(nn) = {forprime(p=3, nn, if (isprime(p+2), if (precprime(p-1)+nextprime(p+3) == 2*(p+1), print1(p+1, ", "));););} \\ Michel Marcus, Apr 12 2015

from sympy import isprime,prevprime,nextprime
for i in range(5,12001,2):
  if isprime(i) and isprime(i+2):
    if prevprime(i)+nextprime(i,2) == 2*(i+1): print(i+1,end=', ')

A138389 Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 20, 21, 23, 24, 25, 29, 31, 33, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Vladimir Shevelev, May 08 2008

nonn

Note that every i not exceeding n/2 for which (n,i)=1 divides binomial(n-i-1,i-1). For n>33, a(n) is either prime or square of a prime or a product of twin primes. For a proof, see link of V. Shevelev.

Numbers n such that A178105(n) = 0. - Michel Marcus, Feb 07 2016

Peter J. C. Moses, Table of n, a(n) for n = 1..10000
V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Intl. J. of Number Theory, 3, no.1 (2007), 119-139.

Cf. A000040, A001248, A077800, A037074, A178105.

Select[Range@ 200, Function[n, NoneTrue[Select[Range@ Floor[n/2], ! CoprimeQ[#, n] &], Divisible[Binomial[n - # - 1, # - 1], #] &]]] (* Michael De Vlieger, Feb 07 2016, Version 10 *)

isok(n) = {my(md = -1); for (d=2, n\2, if (((binomial(n-d-1,d-1) % d) == 0) && (gcd(n, d) > 1), if (md == -1, md = d, md = min(d, md)));); (md == -1);} \\ Michel Marcus, Feb 07 2016

A175612 Pairs of cousin semiprimes (m, m+4).

Original entry on oeis.org

6, 10, 10, 14, 21, 25, 22, 26, 34, 38, 35, 39, 51, 55, 58, 62, 65, 69, 82, 86, 87, 91, 91, 95, 111, 115, 115, 119, 118, 122, 119, 123, 129, 133, 141, 145, 142, 146, 155, 159, 183, 187, 201, 205, 202, 206, 205, 209, 209, 213, 213, 217, 214, 218, 215, 219, 217, 221

Offset: 1

Juri-Stepan Gerasimov, Jul 24 2010

nonn

Cf. A077800.

SequencePosition[PrimeOmega[Range[300]], {2, , , , 2}] // Flatten (* _Jean-François Alcover, Feb 10 2018 *)

Edited and terms checked by D. S. McNeil, Nov 26 2010

A095958 Twin prime pairs concatenated in decimal representation.

Original entry on oeis.org

35, 57, 1113, 1719, 2931, 4143, 5961, 7173, 101103, 107109, 137139, 149151, 179181, 191193, 197199, 227229, 239241, 269271, 281283, 311313, 347349, 419421, 431433, 461463, 521523, 569571, 599601, 617619, 641643, 659661

Offset: 1

Reinhard Zumkeller, Jul 14 2004

a(n) mod 3 = 0 for n>1, proof: A007953(a(n)) = 2*A007953(A001359(n)+1) and A007953(A001359(n)) mod 3 = 2 for n>1, therefore A007953(a(n)) mod 3 = 0.

			29 = A001359(5), 29 + 2 = 31 = A006512(5): a(5) = 2931.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A045533, A001359, A006512.

Cf. A077800, subsequence of A045533.

a095958 n = a095958_list !! (n-1)
a095958_list = f $ map show a077800_list :: [Integer] where
   f (t:t':ts) = read (t ++ t') : f ts
-- Reinhard Zumkeller, Apr 20 2012

[Seqint( Intseq(NthPrime(n+1)) cat Intseq(NthPrime(n)) ): n in [1..150 ]| NthPrime(n+1)-NthPrime(n) eq 2 ]; // Marius A. Burtea, Mar 21 2019

concat[{a_,b_}]:=FromDigits[Flatten[IntegerDigits/@{a,b}]]; concat/@ Select[Partition[ Prime[ Range[150]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Apr 20 2012 *)

A182481 a(n) is the least k such that 6kn-1 and 6kn+1 are twin primes, and a(n)=0, if such k does not exist.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 4, 2, 1, 3, 1, 4, 5, 2, 2, 1, 1, 2, 2, 7, 5, 1, 3, 1, 2, 5, 16, 2, 1, 7, 1, 1, 5, 2, 2, 9, 1, 8, 1, 5, 9, 4, 5, 1, 3, 1, 4, 3, 2, 7, 1, 20, 5, 2, 8, 14, 1, 3, 21, 43, 4, 6, 3, 5, 8, 4, 9, 2, 1, 3, 1, 14, 15, 9, 30, 1, 4, 22, 7, 20, 21, 9

Offset: 1

Vladimir Shevelev, May 01 2012

nonn

Conjecture: a(n)>0; equivalently, for every n, the arithmetic progression {6*k*n-1} contains infinitely many lessers of twin primes (A001359).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A014574, A001097, A077800, A002822.

Table[k = 0; While[! (PrimeQ[6*k*n - 1] && PrimeQ[6*k*n + 1]), k++]; k, {n, 100}] (* T. D. Noe, May 02 2012 *)

a(n)=my(k);n*=6;until(isprime(n*k++-1)&&isprime(n*k+1),);k \\ Charles R Greathouse IV, May 01 2012

A232880 Twin primes with digital root 2 or 4.

Original entry on oeis.org

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

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 *)

A232882 Twin primes with digital root 8 or 1.

Original entry on oeis.org

17, 19, 71, 73, 107, 109, 179, 181, 197, 199, 269, 271, 431, 433, 521, 523, 809, 811, 827, 829, 881, 883, 1061, 1063, 1151, 1153, 1277, 1279, 1619, 1621, 1871, 1873, 1997, 1999, 2087, 2089, 2141, 2143, 2267, 2269, 2339, 2341, 2591, 2593, 2789, 2791, 2969

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 A232881 for {5,7}.

Twin primes of the form 9n +- 1. [Bruno Berselli, Aug 26 2014]

			17 and 19 are in the sequence because they form a twin prime pair in which 17 has a digital root of 8 and 19 has one of 1. Likewise 71 and 73 form a twin prime pair where 71 has 8 for a digital root and 73 has 1.

Cf. A001097, A077800, A232880, A232881.

dRoot[n_] := 1 + Mod[n - 1, 9]; tw = Select[Prime[Range[1000]], PrimeQ[# + 2] &]; Select[Union[tw, tw + 2], MemberQ[{1, 8}, dRoot[#]] &] (* T. D. Noe, Dec 10 2013 *)

A074040 Product of first n twin prime pair products.

Original entry on oeis.org

15, 525, 75075, 24249225, 21800053275, 38433493923825, 138322144631846175, 716923675626858725025, 7458156997546211316435075, 86984485062381462583582279725, 1656445549042930191979157352803175

Offset: 1

Reinhard Zumkeller, Aug 13 2002

nonn

			The first two twin prime pairs are (3,5) and (5,7), their products: 15 and 35, therefore a(2)=15*35=525.

Cf. A074041, A001359, A006512 & A014574.

Cf. A037074, A077800, A079164.

a = {4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150} (* A014574 *); Table[ Product[a[[k]]^2 - 1, {k, 1, n}], {n, 1, 12}]
Rest[FoldList[Times,1,Times@@@Select[Partition[Prime[Range[50]],2,1],#[[2]]-#[[1]]==2&]]] (* Harvey P. Dale, Jan 19 2015 *)
step[{list_, q_}] := Module[{p=NextPrime[q]}, {Join[list, If[PrimeQ[p+2], {{p,p+2}}, {}]], p}]
pairList[n_] := First[NestWhile[step, {{{3, 5}}, 3}, Length[First[step[#]]]<=n&]]
a037074[n_] := Map[Apply[Times, #]&, pairList[n]]
a074040[n_] := Rest[FoldList[Times, 1, a037074[n]]]
a074040[11] (* Hartmut F. W. Hoft, Apr 27 2021 *)

a(1) = A037074(1) and a(n) = a(n-1)*A037074(n) for n>1.

a(n) = A079164(2*n).

Edited by Robert G. Wilson v, Aug 17 2002

Corrections in Comment and Example, and added references. Hartmut F. W. Hoft, Apr 27 2021

cp's OEIS Frontend

A045533 Concatenate the n-th and (n+1)st prime.

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A256753 Numbers n such that n is both the average of some twin prime pair p, q (q = p+2) (i.e., n = p+1 = q-1) and is also the average of the prime before p and the prime after q.

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A138389 Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1).

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A175612 Pairs of cousin semiprimes (m, m+4).

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A095958 Twin prime pairs concatenated in decimal representation.

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A182481 a(n) is the least k such that 6*k*n-1 and 6*k*n+1 are twin primes, and a(n)=0, if such k does not exist.

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A232880 Twin primes with digital root 2 or 4.

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A232881 Twin primes with digital root 5 or 7.

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A182481 a(n) is the least k such that 6kn-1 and 6kn+1 are twin primes, and a(n)=0, if such k does not exist.