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

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

1,1

Links

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.

Formula

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A000040, A129750.

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

Sequence in context: A045044 A108303 A062695 * A320703 A095419 A232581

Adjacent sequences: A107727 A107728 A107729 * A107731 A107732 A107733

Keyword

base,easy,nonn

Author

Jonathan Vos Post, Jun 12 2007

Extensions

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

Status

approved