A243106 a(n) = Sum_{k=1..n} (-1)^isprime(k)*10^k.

10, -90, -1090, 8910, -91090, 908910, -9091090, 90908910, 1090908910, 11090908910, -88909091090, 911090908910, -9088909091090, 90911090908910, 1090911090908910, 11090911090908910, -88909088909091090, 911090911090908910, -9088909088909091090

Offset: 1

Author

R. J. Cano, Aug 19 2014

Keywords

• sign
• base

Comments

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

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]

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}.

Examples

			n=1 is not prime x^1 = (10)^1 = 10, therefore a(1)=10;
n=2 is prime and x^2 = (10)^2 = 100, taking it negative, a(2) = 10 - 100 = -90;
n=3 also is prime, x^3 = 1000, and we have a(3) = 10 - 100 - 1000 = -1090;
n=4 is not prime, so a(4) = 10 - 100 - 1000 + 10000 = 8910;
n=5 is prime, then a(5) = 10 - 100 - 1000 + 10000 - 100000 = -91090;
Examples of analysis for the concatenation patterns among the terms can be found at the "Additional Information" link.
		

Links

• R. J. Cano, Table of n, a(n) for n = 1..100
• R. J. Cano, Additional information.
• Eric Weisstein's World of Mathematics, Alternating Series
• Eric Weisstein's World of Mathematics, Tutte Polynomial

Crossrefs

Cf. A097454.

The same kind of base-independent behavior: A215940, A217626.

Partial sums of alternating series: A181482, A222739, A213203.

Programs

Mathematica

Table[Sum[ (-1)^Boole@ PrimeQ@ k*10^k, {k, n}], {n, 19}] (* Michael De Vlieger, Jan 03 2016 *)

PARI

ap(n, x)={my(s); forprime(p=1, n, s+=x^p); s}
a=(n, x=10)->(x^(n+1)-1)/(x-1)-2*ap(n, x)-1;

PARI

Delta=(i, j)->(i==j); /* Kronecker's Delta function */
t=n->matrix(n, n, i, j, Delta(i, j)*((-1)^isprime(i))); /* coeffs t[i, j] */
/* Tutte polynomial over n */
T(n, x, y)={my(t0=t(n)); sum(i=1, n, sum(j=1, n, t0[i, j]*(x^i)*(y^j)))};
a=(n, x=10)->T(n, x, 1);

PARI

A243106(n,b=10)=sum(k=1,n,(-1)^isprime(k)*b^k) \\ M. F. Hasler, Aug 20 2014

Formula

a(n,x) = Sum_{k=1..n} (-1)^isprime(k)*(x^k), for x=10 in decimal.

Extensions

Definition simplified by N. J. A. Sloane, Aug 19 2014

Definition further simplified and more terms from M. F. Hasler, Aug 20 2014