A320708 -id:A320708 - cp's OEIS frontend

A031938 Lower prime of a difference of 20 between consecutive primes.

887, 1637, 3089, 3413, 3947, 5717, 5903, 5987, 6803, 7649, 8243, 8543, 8783, 8867, 9257, 10223, 10433, 10667, 11027, 11093, 11177, 11447, 11597, 11657, 11867, 11987, 13127, 13421, 13649, 14087, 14177, 15473, 16943, 17519, 17807, 18149, 18461, 18617, 18839

Offset: 1

Jeff Burch

nonn

			887 is a term as the next prime is 907.

Remi Eismann, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

[p: p in PrimesUpTo(20000) | NextPrime(p)-p eq 20]; // Bruno Berselli, Apr 09 2013

(*M6*) a=887; S={}; Do[b=NextPrime[a]; If[b-a==20,AppendTo[S,a]]; a=b,{10^4}]; S (* Zak Seidov, Aug 14 2009 *)
Transpose[Select[Partition[Prime[Range[2000]], 2, 1], Last[#] - First[#] == 20 &]][[1]] (* Bruno Berselli, Apr 09 2013 *)

a(n) = prime(A320708(n)). - R. J. Mathar, Apr 30 2024

Entries and b-file checked by Zak Seidov, Aug 14 2009

A107730 Numbers n such that prime(n+1) has the same last digit as prime(n).

Original entry on oeis.org

34, 42, 53, 61, 68, 80, 82, 101, 106, 115, 125, 127, 138, 141, 145, 154, 157, 172, 175, 177, 191, 193, 204, 222, 233, 258, 259, 266, 269, 279, 289, 306, 308, 310, 316, 324, 369, 383, 397, 399, 403, 418, 422, 431, 442, 443, 474, 480, 491, 497, 500, 502, 518

Offset: 1

Jonathan Vos Post, Jun 12 2007

			a(1) = 34 because prime(34) = 139, prime(35) = 149, both end with the digit 9.
a(2) = 42 because prime(42) = 181, prime(43) = 191, both end with the digit 1.
a(4) = 61 because prime(61) = 283, prime(62) = 293, both end with the digit 3.
a(5) = 68 because prime(68) = 337, prime(69) = 347, both end with the digit 7.

Muniru A Asiru, Table of n, a(n) for n = 1..10000
Fred B. Holt, On the last digits of consecutive primes, arXiv:1604.02443 [math.NT], 2016.
Robert J. Lemke Oliver, Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
Index entries for primes, gaps between

Cf. A000040, A129750.

Union of rows r == 0 (mod 5) of A174349. Indices of multiples of 10 (A008592) in A001223.

P:=List(Filtered([1..4000],IsPrime),n->Reversed(ListOfDigits(n)));;
a:=Filtered([1..Length(P)-1],i->P[i+1][1]=P[i][1]); # Muniru A Asiru, Oct 31 2018

isA107730 := proc(n) local ldign, ldign2 ; ldign := convert(ithprime(n),base,10) ; ldign2 := convert(ithprime(n+1),base,10) ; if op(1,ldign) = op(1,ldign2) then true ; else false ; fi ; end: for n from 1 to 600 do if isA107730(n) then printf("%d, ",n) ; fi ; od ; # R. J. Mathar, Jun 15 2007

Select[Range[200],IntegerDigits[Prime[ # ]][[ -1]]==IntegerDigits[Prime[ #+1]][[ -1]]&] (* Stefan Steinerberger, Jun 14 2007 *)
Flatten[Position[Partition[Prime[Range[600]],2,1],?(Mod[#[[1]],10] == Mod[#[[2]],10]&),{1},Heads->False]] (* _Harvey P. Dale, Aug 20 2015 *)

isok(n) = (prime(n) % 10) == prime(n+1) % 10; \\ Michel Marcus, Feb 16 2017

is_A107730(n)=!((nextprime(1+n=prime(n))-n)%10) \\ This (...) is twice as fast as prime(n+1)-prime(n), and prime(n) becomes very slow for n > 41538, even with primelimit = 10^7. - M. F. Hasler, Oct 24 2018

Numbers n such that A000040(n)==A000040(n+1) mod 10, or A000040(n+1) - A000040(n) = 10*k for some integer k, or n such that A129750(n) = 0. [Corrected and edited by M. F. Hasler, Oct 24 2018]
A107730 = A001223^(-1)(A008592) = { i > 0 | A001223(i) == 0 (mod 10)} = U_{k>0} {A174349(5k,j); j >= 1}. - M. F. Hasler, Oct 24 2018
Union of A320703, A320708, A320713, A320718, ... A116493,..., A116496 ... etc. - R. J. Mathar, Apr 30 2024

More terms from Stefan Steinerberger and R. J. Mathar, Jun 14 2007

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A031938 Lower prime of a difference of 20 between consecutive primes.

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