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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 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 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 by 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}.

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)/(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

Cf. A097454.

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

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

Sequence in context: A063945 A218127 A322647 * A046706 A116348 A043088

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

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Last modified September 21 21:56 EDT 2019. Contains 327282 sequences. (Running on oeis4.)