|
%I #33 Sep 23 2021 01:26:44
%S 10,-90,-1090,8910,-91090,908910,-9091090,90908910,1090908910,
%T 11090908910,-88909091090,911090908910,-9088909091090,90911090908910,
%U 1090911090908910,11090911090908910,-88909088909091090,911090911090908910,-9088909088909091090
%N a(n) = Sum_{k=1..n} (-1)^isprime(k)*10^k.
%C Alternative definition: a(n,x)=T(x,1) for a dichromate or Tutte-Whitney polynomial in which the matrix t[i,j] is defined as t[i,j]=Delta(i,j)*((-1)^isprime(i)) and "Delta" is the Kronecker Delta function. - _Michel Marcus_, Aug 19 2014
%C If 10 is replaced by 1, then this becomes A097454. If it is replaced by 2, one gets A242002. Choosing powers of the base b=10, as done here, allows one to easily read off the equivalent for any other base b > 4, by simply replacing digits 8,9 with b-2,b-1 (when terms are written in base b). [Comment extended by _M. F. Hasler_, Aug 20 2014]
%C There are 2^n ways of taking the partial sum of the first n powers of b=10 if exponent zero is excluded and the signs can be assigned arbitrarily. Conjecture: When expressed in base b, the absolute value for any of these terms only contains digits belonging to {0,1,b-2,b-1}; here {0,1,8,9}.
%H R. J. Cano, <a href="/A243106/b243106.txt">Table of n, a(n) for n = 1..100</a>
%H R. J. Cano, <a href="https://oeis.org/w/images/d/d7/AdditionalInfo_primeSignSwitchingSeq00_.txt">Additional information.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlternatingSeries.html">Alternating Series</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TuttePolynomial.html">Tutte Polynomial</a>
%F a(n,x) = Sum_{k=1..n} (-1)^isprime(k)*(x^k), for x=10 in decimal.
%e n=1 is not prime x^1 = (10)^1 = 10, therefore a(1)=10;
%e n=2 is prime and x^2 = (10)^2 = 100, taking it negative, a(2) = 10 - 100 = -90;
%e n=3 also is prime, x^3 = 1000, and we have a(3) = 10 - 100 - 1000 = -1090;
%e n=4 is not prime, so a(4) = 10 - 100 - 1000 + 10000 = 8910;
%e n=5 is prime, then a(5) = 10 - 100 - 1000 + 10000 - 100000 = -91090;
%e Examples of analysis for the concatenation patterns among the terms can be found at the "Additional Information" link.
%t Table[Sum[ (-1)^Boole@ PrimeQ@ k*10^k, {k, n}], {n, 19}] (* _Michael De Vlieger_, Jan 03 2016 *)
%o (PARI) ap(n, x)={my(s); forprime(p=1, n, s+=x^p); s}
%o a=(n, x=10)->(x^(n+1)-1)/(x-1)-2*ap(n, x)-1;
%o (PARI) Delta=(i, j)->(i==j); /* Kronecker's Delta function */
%o t=n->matrix(n, n, i, j, Delta(i, j)*((-1)^isprime(i))); /* coeffs t[i, j] */
%o /* Tutte polynomial over n */
%o T(n, x, y)={my(t0=t(n)); sum(i=1, n, sum(j=1, n, t0[i, j]*(x^i)*(y^j)))};
%o a=(n, x=10)->T(n, x, 1);
%o (PARI) A243106(n,b=10)=sum(k=1,n,(-1)^isprime(k)*b^k) \\ _M. F. Hasler_, Aug 20 2014
%Y Cf. A097454.
%Y The same kind of base-independent behavior: A215940, A217626.
%Y Partial sums of alternating series: A181482, A222739, A213203.
%K sign,base
%O 1,1
%A _R. J. Cano_, Aug 19 2014
%E Definition simplified by _N. J. A. Sloane_, Aug 19 2014
%E Definition further simplified and more terms from _M. F. Hasler_, Aug 20 2014
|
