A294173 Numbers k whose nearest neighbors have the same number of divisors, the same number of distinct prime factors, and the same sum of divisors.

34, 55, 919, 1241, 4149, 4188, 7170, 12566, 15086, 24882, 25020, 26610, 51836, 53964, 59988, 77058, 143370, 150420, 167561, 170562, 205728, 215070, 220818, 418308, 564858, 731321, 907255, 910316, 986154, 1239870, 1569336, 1622914, 1841861, 1887240, 1979307, 2229012, 2262108

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

Intersection of A067888, A088070, and A055574.

Cf. A229254, A008683, A000010.

Sequence in context: A259676 A227304 A055574 * A176687 A224896 A103558

Adjacent sequences: A294170 A294171 A294172 * A294174 A294175 A294176

Keyword

nonn

Author

Torlach Rush, Feb 10 2018

Status

approved