A088070 -id:A088070 - cp's OEIS frontend

A088390 Start of the first run of a string of exactly n successive integers in A088070.

6, 3, 47, 10, 324, 23, 1338, 3261, 2779, 2314, 13067, 526096, 4308, 43369732, 205883268, 418113170, 1636119595, 1938732984, 35048134702

Offset: 1

Ray Chandler, Oct 06 2003

nonn

No more terms < 156*10^6. - David Wasserman, Aug 08 2005

a(20) > 2*10^11. [From Donovan Johnson, Jun 19 2010]

			a(4)=10 since 10 is the start of a run of 4 successive integers in A088070.

Cf. A088070.

One more term from David Wasserman, Aug 08 2005

a(15)-a(19) from Donovan Johnson, Jun 19 2010

A280382 Numbers k such that k-1 has the same number of prime factors counted with multiplicity as k+1.

Original entry on oeis.org

4, 5, 6, 12, 18, 19, 29, 30, 34, 42, 43, 50, 51, 55, 56, 60, 67, 69, 72, 77, 86, 89, 92, 94, 102, 108, 115, 120, 122, 138, 142, 144, 150, 151, 160, 171, 173, 180, 184, 186, 187, 189, 192, 197, 198, 202, 204, 214, 216, 218, 220, 228, 233, 236, 237, 240, 243, 245, 248, 249, 266, 267, 270, 271, 274, 282

Offset: 1

Rick L. Shepherd, Jan 01 2017

nonn

			Unlike for A088070, 5 is a term here because 4 = 2^2 and 6 = 2*3 each have two prime factors when counted with multiplicity. Similarly, 3 is not a term of this sequence (but is in A088070) because 2 and 4 have different numbers of prime factors as counted by A001222.

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A001222, A088070 (similar but prime factors counted without multiplicity), A280383 (prime factor count is same both ways), A280469 (subsequence of current with k-1 and k+1 squarefree also), A045920 (similar but for k and k+1).

Cf. A115167 (subsequence of odd terms).

Select[Range[2, 300], Equal @@ PrimeOmega[# + {-1, 1}] &] (* Amiram Eldar, May 20 2021 *)

IsInA280382(n) = n > 1 && bigomega(n-1) == bigomega(n+1)

from sympy import primeomega
def aupto(limit):
  prv, cur, nxt, alst = 1, 1, 2, []
  for n in range(3, limit+1):
    if prv == nxt: alst.append(n)
    prv, cur, nxt = cur, nxt, primeomega(n+2)
  return alst
print(aupto(282)) # Michael S. Branicky, May 20 2021

A280383 Numbers n such that n-1 has the same count of prime factors as n+1 when including multiplicity and also when not.

Original entry on oeis.org

4, 6, 12, 18, 19, 30, 34, 42, 51, 55, 56, 60, 72, 86, 92, 94, 102, 108, 138, 142, 144, 150, 160, 180, 184, 186, 192, 198, 202, 204, 214, 216, 218, 220, 228, 236, 240, 243, 248, 249, 266, 270, 282, 300, 302, 304, 307, 312, 320, 322, 328, 340, 341, 348, 349, 392, 394, 412, 414, 416, 420, 424, 432, 446, 452, 462, 470, 472, 476, 491, 516, 518, 522, 534, 536, 544, 552, 570, 580, 582, 590, 600, 604, 618, 634, 638, 642, 660, 664, 668, 670, 680, 686, 688, 696, 698, 701, 722

Offset: 1

Rick L. Shepherd, Jan 02 2017

nonn

First differs from its subsequence A074997 at a(97) = 701 because A074997(97) = 722.

			The number 19 is a term because 18 = 2*3^2 and 20 = 2^2*5 each have two distinct prime factors and each have three prime factors when counted with multiplicity.

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A074997, A088070, A280382.

Select[Range[800],PrimeNu[#]==PrimeNu[#+2]&&PrimeOmega[#]==PrimeOmega[#+2]&]+1 (* Harvey P. Dale, Jul 12 2023 *)

IsInA280383(n) = n > 1 && bigomega(n-1) == bigomega(n+1) && omega(n-1) == omega(n+1)

Sequence is A088070 INTERSECT A280382.

A115165 Odd numbers k such that k-1 and k+1 have the same number of distinct prime divisors.

Original entry on oeis.org

3, 11, 13, 19, 21, 23, 25, 27, 35, 37, 39, 45, 47, 49, 51, 53, 55, 57, 73, 75, 81, 87, 93, 95, 97, 99, 105, 107, 117, 123, 131, 135, 139, 143, 145, 147, 155, 159, 161, 163, 165, 169, 177, 181, 193, 195, 201, 207, 213, 215, 217, 221, 225, 229, 239, 243, 249, 259, 265

Offset: 1

Cino Hilliard, Mar 03 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A088070.

Cf. A001221.

[k: k in [3..265 by 2]| #PrimeDivisors(k-1) eq #PrimeDivisors(k+1)]; // Marius A. Burtea, Feb 19 2020

Select[Range[1, 301, 2], PrimeNu[#-1] == PrimeNu[#+1]&] (* Jean-François Alcover, Oct 18 2016 *)

g(n) = forstep(x=3,n,2,p1=omega(x-1);p2=omega(x+1);if(p1==p2,print(x",")))

Offset corrected by Michel Marcus, Oct 18 2016

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

A088390 Start of the first run of a string of exactly n successive integers in A088070.

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A280382 Numbers k such that k-1 has the same number of prime factors counted with multiplicity as k+1.

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A280383 Numbers n such that n-1 has the same count of prime factors as n+1 when including multiplicity and also when not.

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Mathematica

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Formula

A115165 Odd numbers k such that k-1 and k+1 have the same number of distinct prime divisors.

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Magma

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PARI

Extensions

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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GAP

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Mathematica

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