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
Examples
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.
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.
- Index entries for primes, gaps between
Crossrefs
Programs
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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
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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
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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 *)
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PARI
isok(n) = (prime(n) % 10) == prime(n+1) % 10; \\ Michel Marcus, Feb 16 2017
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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
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]
Extensions
More terms from Stefan Steinerberger and R. J. Mathar, Jun 14 2007