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A090897 Next n digits of Pi.

%I A090897 #60 Jan 27 2023 19:47:58
%S A090897 3,14,159,2653,58979,323846,2643383,27950288,419716939,9375105820,
%T A090897 97494459230,781640628620,8998628034825,34211706798214,
%U A090897 808651328230664,7093844609550582,23172535940812848,111745028410270193,8521105559644622948,95493038196442881097
%N A090897 Next n digits of Pi.
%C A090897 More precisely: the integer resulting from reading the "next n digits of Pi" in base 10, so leading zeros cannot be directly seen, but easily be "reconstructed" from the fact that the term will have less than n digits although it is made from n digits of Pi. - _M. F. Hasler_, Jan 06 2023
%C A090897 It seems that all terms have at least one prime factor that does not appear in the combined list of prime factors of the preceding terms of the sequence. - _Mario Cortés_, Aug 20 2020 [Checked up to n=65. - _Michel Marcus_, Aug 21 2020]
%H A090897 Michel Marcus, <a href="/A090897/b090897.txt">Table of n, a(n) for n = 1..100</a>
%F A090897 a(n) = floor( Pi * 10^(n*(n+1)/2-1) ) mod (10^n). - _Carl R. White_, Aug 13 2010
%e A090897 a(3) = 159 because after the first (a(1) = 3) and the next two digits of Pi (a(2) = 14) the next three are 159.
%e A090897 From _Aaron T Cowan_, Jan 03 2023: (Start)
%e A090897 Other examples are as follows and fall into a triangular digit pattern, though there is no guarantee that they will remain triangular in all cases
%e A090897   a(1) = 3;
%e A090897   a(2) = 14;
%e A090897   a(3) = 159;
%e A090897   a(4) = 2653;
%e A090897   a(5) = 58979;
%e A090897    (End)
%e A090897 Indeed, precisely whenever A086639(n) = 0, then the corresponding term of this sequence will lack one or more leading zeros and therefore the above list will deviate from the triangular shape. - _M. F. Hasler_, Jan 06 2023
%p A090897 Partitioner := proc(cons, len) local i, R, spl; R := []; i:=0;
%p A090897 spl := L -> [seq([seq(L[i], i=1 + n*(n+1)/2..(n+1)*(n+2)/2)], n=0..len)]:
%p A090897 ListTools:-Reverse(convert(floor(cons*10^((len+1)*(len+2)/2)), base, 10)):
%p A090897 map(`@`(parse, cat, op), spl(%)) end:
%p A090897 aList := -> Partitioner(Pi, 20); aList(20); # _Peter Luschny_, Aug 22 2020
%t A090897 With[{pi=RealDigits[Pi,10,500][[1]]},FromDigits/@Table[Take[pi,{n (n-1)/2+1, (n(n+1))/2}],{n,25}]] (* _Harvey P. Dale_, Dec 24 2011 *)
%o A090897 (PARI) lista(nn) = {my(nd = 5+nn*(nn+1)/2); default(realprecision, nd); my(vd = digits(floor(Pi*10^nd))); my(pos = 1); my(vr = vector(nn)); for (n=1, nn, vr[n] = fromdigits(vector(n, k, vd[k+ pos-1])); pos += n;); vr;} \\ _Michel Marcus_, Aug 21 2020
%Y A090897 Cf. A000796 (Pi), A016062, A081368, A086639.
%K A090897 easy,nonn,base
%O A090897 1,1
%A A090897 _Michael Joseph Halm_, Feb 26 2004