OFFSET
1,1
COMMENTS
mu(k-1) = mu(k+1), where mu(k) = A008683(k), since k-1 and k+1 have the same number of distinct prime factors.
tau(k-1) = tau(k+1) = abs(phi(k-1) - phi(k+1)) iff abs(phi(k-1) - phi(k+1)) = 4, where phi(j) is A000010. When tau(j) = 4 omega(j) = 2 and phi(j), the product of two even numbers is divisible by 4.
For known elements:
- sigma(k +- 1) and tau(k +- 1) the greatest common divisor is 4.
- sigma(k +- 1) is divisible by tau(k +- 1).
- the digital root of sigma(k +- 1) is either 3 or 9.
- the prime signature of k +- 1 is the same (see question below).
The first prime terms are 919, 110495719, 2587274227, 3908452759, 4020447619, and 9314901619. - Giovanni Resta, Feb 12 2018
Are the prime signatures of k +- 1 always the same? - Andrey Zabolotskiy, Feb 14 2018
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..1478
EXAMPLE
34 is in the sequence because tau(33)=tau(35)=4, omega(33)=omega(35)=2, and sigma(33)=sigma(35)=48.
919 is in the sequence because tau(918)=tau(920)=16, omega(918)=omega(920)=3, and sigma(918)=sigma(920)=2160.
MAPLE
with(numtheory):
select(k->sigma(k-1)=sigma(k+1) and mobius(k-1)=mobius(k+1) and tau(k-1)=tau(k+1), [$2..2000000]); # Muniru A Asiru, Feb 17 2018
MATHEMATICA
1 + Position[Partition[Array[{DivisorSigma[0, #], DivisorSigma[1, #], PrimeOmega[#]} &, 10^6], 3, 1], _?(#[[1]] == +#[[-1]] &), {1}, Heads -> False][[All, 1]] (* Michael De Vlieger, Feb 17 2018 *)
PROG
(GAP) Filtered([2..2000000], k->Sigma(k-1)=Sigma(k+1) and Number(FactorsInt(k-1))=Number(FactorsInt(k+1)) and Tau(k-1)=Tau(k+1)); # Muniru A Asiru, Feb 17 2018
(PARI) list(lim)=my(v=List(), k2=7, s2=sigma(k2), k1=8, s1=sigma(k1), s); forfactored(k=9, 1+lim\1, s=sigma(k); if(s==s2 && numdiv(k)==numdiv(k2) && omega(k)==omega(k2), listput(v, k1[1])); k2=k1; k1=k; s2=s1; s1=s); Vec(v) \\ Charles R Greathouse IV, Feb 20 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Torlach Rush, Feb 10 2018
STATUS
approved