A229254 -id:A229254 - cp's OEIS frontend

A276714 Numbers n such that n and n+3 have the same number and sum of divisors (A000005 and A000203).

42677635, 276742235, 6439057062, 7512673242, 43592652562, 48847956255, 48880963215, 55018687182, 60184185702, 91484515395, 100774916235, 101926379835, 111886551315, 122388340095, 133012188855, 137978601142, 247631352255, 263171068875, 293467635615, 305946896255

Offset: 1

Jaroslav Krizek, Sep 16 2016

nonn

Intersection of A015861 and A276713.

Also numbers n such that A229335(n) = A229335(n+3).

			42677635 is in sequence because tau(42677635) = tau(42677638) = 32 and sigma(42677635) = sigma(42677638) = 68769792.

Martin Ehrenstein, Table of n, a(n) for n = 1..30

Cf. A015861, A229335, A276713, A276715.

Cf. Similar sequences with numbers n such that n and n+k have the same number and sum of divisors for k=1: A054004, for k=2: A229254.

[n: n in [A015861(k)] | NumberOfDivisors(n) eq  NumberOfDivisors(n+3) and SumOfDivisors(n) eq  SumOfDivisors(n+3)]

More terms from Martin Ehrenstein, Jul 12 2024

A276715 a(n) = the smallest number k such that k and k + n have the same number and sum of divisors (A000005 and A000203).

Original entry on oeis.org

1, 14, 33, 42677635, 51, 46, 155, 62, 69, 46, 174, 154, 285, 182, 141, 62, 138, 142, 235, 158, 123, 94, 213, 322, 295, 94, 177, 118, 159, 406, 376, 266, 177, 891528365, 321, 310, 355, 248, 249, 166, 213, 418, 376, 602, 426, 142, 570, 310, 445, 248, 249, 158

Offset: 0

Jaroslav Krizek, Sep 16 2016

nonn

If a(33) exists, it must be greater than 2*10^8.
a(n) for n >= 34: 321, 310, 355, 248, 249, 166, 213, 418, 376, 602, 426, 142, 570, 310, 445, 248, 249, 158, 267, 406, 632, 166, 267, ...
The records occur at indices 0, 1, 2, 3, 33, 207, 471, ... with values 1, 14, 33, 42677635, 891528365, 2944756815, 3659575815, ... - Amiram Eldar, Feb 17 2019

			a(2) = 33 because 33 is the smallest number such that tau(33) = tau(35) = 4 and simultaneously sigma(33) = sigma(35) = 48.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A065559 (smallest k such that tau(k) = tau(k+n)), A007365 (smallest k such that sigma(k) = sigma(k+n)).

Cf. Sequences with numbers n such that n and n+k have the same number and sum of divisors for k=1: A054004, for k=2: A229254, k=3: A276714.

A276715:=func; [A276715(n):n in[0..32]]

a[k_] := Module[{n=1}, While[DivisorSigma[0,n] != DivisorSigma[0,n+k] || DivisorSigma[1,n] != DivisorSigma[1,n+k], n++]; n]; Array[a, 50, 0] (* Amiram Eldar, Feb 17 2019 *)

from itertools import count
from sympy import divisor_sigma
def A276715(n): return next(k for k in count(1) if all(divisor_sigma(k,i)==divisor_sigma(n+k,i) for i in (0,1))) # Chai Wah Wu, Jul 25 2022

a(33) onwards from Amiram Eldar, Feb 17 2019

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.

Original entry on oeis.org

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

Torlach Rush, Feb 10 2018

nonn

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

			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.

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

Intersection of A067888, A088070, and A055574.

Cf. A229254, A008683, A000010.

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

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

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 *)

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

cp's OEIS Frontend

A276714 Numbers n such that n and n+3 have the same number and sum of divisors (A000005 and A000203).

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A276715 a(n) = the smallest number k such that k and k + n have the same number and sum of divisors (A000005 and A000203).

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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.

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