OFFSET
1,1
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}.
This was proved by an autonomous AI agent, see the Tsoukalas paper and the Lean file. The proof uses strong induction on N to show any integer expressible with base-b digit coefficients in {-1, 0, 1} has all base-b digits in {0, 1, b-2, b-1}, via repeated division and modular arithmetic extracting each successive digit (Summary by Opus 4.7). - Ralf Stephan, May 26 2026
LINKS
R. J. Cano, Table of n, a(n) for n = 1..100
R. J. Cano, Additional information.
Google Deepmind, AlphaProof Nexus: A243106 Lean file
George Tsoukalas et al., Advancing Mathematics Research with AI-Driven Formal Proof Search, arXiv:2605.22763 [cs.AI], 2026.
Eric Weisstein's World of Mathematics, Alternating Series
Eric Weisstein's World of Mathematics, Tutte Polynomial
FORMULA
a(n,x) = Sum_{k=1..n} (-1)^isprime(k)*(x^k), for x=10 in decimal.
EXAMPLE
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.
MATHEMATICA
Table[Sum[ (-1)^Boole@ PrimeQ@ k*10^k, {k, n}], {n, 19}] (* Michael De Vlieger, Jan 03 2016 *)
PROG
(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
CROSSREFS
KEYWORD
sign,base
AUTHOR
R. J. Cano, Aug 19 2014
EXTENSIONS
Definition simplified by N. J. A. Sloane, Aug 19 2014
Definition further simplified and more terms from M. F. Hasler, Aug 20 2014
STATUS
approved