A245615 Sum_{lpf(j)=11, 11<j<=11*prime(n)}(-1)^(A000120(j)+1), where lpf=least prime factor.

1, 2, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 9, 8, 7, 8, 9, 10, 10, 11, 12, 11, 13, 12, 13, 12, 13, 13, 14, 13, 15, 16, 17, 18, 16, 16, 17, 16, 15, 16, 17, 17, 17, 18, 17, 18, 19, 20, 21, 23, 23, 22, 21, 22, 23, 24, 22, 23, 22, 23, 23, 22, 22, 21

Offset

5,2

Comments

Since lpf(j)=11, the smallest prime divisor of j is always 11, while every other prime divisor of j, if one exists, is >=11. Since j>11, then the least suitable j>=121. Offset 5 is natural since prime(5)=11.

This is a special Newman's sum.

Conjecture: All terms are nonnegative.

A similar conjecture arises by changing 11 in the definition of the sequence by some other primes: 19,41,67,107,173,...

Links

Robert Israel, Table of n, a(n) for n = 5..4905

D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.

V. Shevelev, Generalized Newman phenomena and digit conjectures on primes, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12.

Example

For n=5, j=121. So we have a(5)=(-1)^(A000120(121)+1)=(-1)^(5+1)=1;
for n=6, there is a second summand with j=11*13.
So a(6)=1+(-1)^(A000120(143)+1)=1+(-1)^(5+1)=2, etc.

Maple

N:= 100: # to get terms up to a(N)
Ks:= select(t -> min(numtheory:-factorset(t))>=11, {$11..ithprime(N)}):
f:= proc(n) local j, Kn, p;
    p:= ithprime(n);
    Kn:= map(`*`, select(`<=`, Ks, p), 11);
    add(2*(convert(convert(j, base, 2), `+`) mod 2)-1, j=Kn);
end proc:
seq(f(n), n=5..N); # Robert Israel, Aug 04 2014

Prog

(PARI) a(n) = sum(j=12, 11*prime(n), if (factor(j)[1, 1]==11, (-1)^(hammingweight(j)+1), 0)); \\ Michel Marcus, Aug 04 2014

Crossrefs

Cf. A000120.

Sequence in context: A362598 A179765 A004074 * A053646 A080776 A360659

Adjacent sequences: A245612 A245613 A245614 * A245616 A245617 A245618

Keyword

nonn

Author

Vladimir Shevelev, Jul 27 2014

Extensions

Corrected and extended by Robert Israel, Aug 04 2014

Status

approved