A280382 -id:A280382 - cp's OEIS frontend

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

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.

A115167 Odd numbers k such that k-1 and k+1 have the same number of prime divisors with multiplicity.

Original entry on oeis.org

5, 19, 29, 43, 51, 55, 67, 69, 77, 89, 115, 151, 171, 173, 187, 189, 197, 233, 237, 243, 245, 249, 267, 271, 283, 285, 291, 295, 307, 317, 329, 341, 343, 349, 355, 403, 405, 411, 427, 429, 435, 437, 461, 489, 491, 507, 569, 571, 593, 597, 603, 605, 653, 665

Offset: 1

Cino Hilliard, Mar 03 2006

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

Subsequence of A280382.

s = {}; o1 = 0; Do[o2 = PrimeOmega[n]; If[o1 == o2, AppendTo[s, n-1]]; o1 = o2, {n, 2, 666, 2}]; s (* Amiram Eldar, Sep 23 2019 *)
Select[Mean/@SequencePosition[PrimeOmega[Range[700]],{x_,,x}],OddQ] (* Harvey P. Dale, Jan 11 2024 *)

g(n) = forstep(x=3, n, 2, p1=bigomega(x-1); p2=bigomega(x+1); if(p1==p2, print1(x",")))

from sympy import primeomega
def aupto(limit):
  prv, nxt, alst = 1, 2, []
  for n in range(3, limit+1, 2):
    if prv == nxt: alst.append(n)
    prv, nxt = nxt, primeomega(n+3)
  return alst
print(aupto(665)) # Michael S. Branicky, May 19 2021

A280469 Numbers n such that n-1 and n+1 are squarefree and have the same number of prime factors.

Original entry on oeis.org

4, 6, 12, 18, 30, 34, 42, 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, 248, 266, 270, 282, 300, 302, 304, 312, 320, 322, 328, 340, 348, 392, 394, 412, 414, 416, 420, 432, 446

Offset: 1

Rick L. Shepherd, Jan 03 2017

nonn

For a given term n of this sequence, n-1 and n+1 are both squarefree k-almost primes for the same k. The sequence is thus the union of the averages (arithmetic means) of twin prime pairs (A014574), the averages of twin squarefree semiprime pairs, the averages of twin squarefree 3-almost prime pairs, ... (where "twin ... pairs" means the members of each pair differ by two). A subsequence of A280382 and of A280383.

			The number 34 is a term because 33 = 3*11 and 35 = 5*7, a twin semiprime pair. Unlike A280382 and A280383, 19 is not a term here because 18 = 2*3^2 and 20  = 2^2*5, neither of which is squarefree.

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

Cf. A014574, A280382, A280383.

Select[Range@ 500, And[Times @@ First@ # == 1, SameQ @@ Last@ #] &@ Transpose@ Map[{Boole@ SquareFreeQ@ #, PrimeNu@ #} &, # + {-1, 1}] &] (* Michael De Vlieger, Jan 30 2017 *)

IsInA280469(n) = n > 1 && issquarefree(n-1) && issquarefree(n+1) && omega(n-1) == omega(n+1)

A328359 Numbers k such that Omega(k - 2) = Omega(k) = Omega(k + 2) where Omega = A001222.

Original entry on oeis.org

5, 68, 93, 121, 143, 172, 185, 188, 203, 215, 217, 219, 244, 284, 289, 301, 303, 321, 342, 393, 404, 413, 415, 428, 436, 471, 490, 517, 535, 570, 581, 604, 669, 687, 697, 788, 791, 815, 858, 870, 892, 1014, 1057, 1079, 1135, 1137, 1139, 1147, 1167, 1205, 1206, 1208, 1210, 1255, 1268, 1276

Offset: 1

Juri-Stepan Gerasimov, Oct 14 2019

nonn

			5 is a term because A001222(3) = A001222(5) = A001222(7) = 1;
68 is a term because A001222(66) = A001222(68) = A001222(70) = 3;
93 is a term because A001222(91) = A001222(93) = A001222(95) = 2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001222, A280382, A278311.

[k: k in [4..1300]| forall{m:m in [-2,2]| &+[p[2]: p in Factorization(k+m)] eq &+[p[2]: p in Factorization(k)] }]; // Marius A. Burtea, Oct 15 2019

Select[Range[10^4],PrimeOmega[#-2]==PrimeOmega[#]==PrimeOmega[#+2]&] (* Metin Sariyar, Oct 14 2019 *)
Flatten[Position[Partition[PrimeOmega[Range[2000]],5,1],?(#[[1]]== #[[3]] == #[[5]]&),1,Heads->False]]+2 (* _Harvey P. Dale, Nov 02 2021 *)

A086806 Sarrus numbers k such that k-1 and k+1 have the same number of prime divisors (counted with multiplicity).

Original entry on oeis.org

341, 13747, 19951, 35333, 60787, 137149, 150851, 387731, 458989, 617093, 769757, 1104349, 1251949, 1277179, 1397419, 1463749, 1507963, 1826203, 2134277, 2205967, 2617451, 2976487, 3345773, 4361389, 6474691, 6955541, 8095447

Offset: 1

Jason Earls, Aug 05 2003

nonn

			341 is a pseudoprime to base 2 while 340 = 2^2*5*17 and 342 = 2*3^2*19 each have four primes dividing them.

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

Intersection of A001567 and A280382.

Cf. A001222.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[9224390], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PrimeFactorExponentsAdded[ # - 1] == PrimeFactorExponentsAdded[ # + 1] & ]

More terms from Robert G. Wilson v, Aug 13 2003

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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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A115167 Odd numbers k such that k-1 and k+1 have the same number of prime divisors with multiplicity.

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A280469 Numbers n such that n-1 and n+1 are squarefree and have the same number of prime factors.

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A328359 Numbers k such that Omega(k - 2) = Omega(k) = Omega(k + 2) where Omega = A001222.

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A086806 Sarrus numbers k such that k-1 and k+1 have the same number of prime divisors (counted with multiplicity).

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