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

Author

Torlach Rush, Feb 10 2018

Keywords

• nonn

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

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..1478

Crossrefs

Intersection of A067888, A088070, and A055574.

Cf. A229254, A008683, A000010.

Programs

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

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 *)

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