

A243106


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


2



10, 90, 1090, 8910, 91090, 908910, 9091090, 90908910, 1090908910, 11090908910, 88909091090, 911090908910, 9088909091090, 90911090908910, 1090911090908910, 11090911090908910, 88909088909091090, 911090911090908910, 9088909088909091090
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Alternative definition: a(n,x)=T(x,1) for a dichromate or TutteWhitney 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 b2,b1 (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,b2,b1}; here {0,1,8,9}.


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


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)/(x1)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

Cf. A097454.
The same kind of baseindependent behavior: A215940, A217626.
Partial sums of alternating series: A181482, A222739, A213203.
Sequence in context: A063945 A218127 A322647 * A046706 A337867 A116348
Adjacent sequences: A243103 A243104 A243105 * A243107 A243108 A243109


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



