A045933 -id:A045933 - cp's OEIS frontend

A045932 Numbers n such that n through n+3 are divisible by the same number of distinct primes.

2, 33, 54, 55, 74, 85, 91, 92, 93, 115, 116, 133, 141, 142, 143, 144, 145, 158, 159, 175, 200, 205, 206, 212, 213, 214, 215, 216, 247, 295, 296, 301, 302, 323, 324, 325, 326, 332, 391, 392, 422, 445, 451, 535, 536, 542, 565, 632, 685, 686, 721, 722, 799, 800

Offset: 1

David W. Wilson

nonn

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

Cf. A001221, A006049, A006073, A045933-A045938.

Transpose[Select[Partition[Range[900],4,1],Length[Union[PrimeNu[#]]] == 1&]][[1]] (* Harvey P. Dale, Apr 12 2013 *)

A045934 Numbers n such that n through n+5 have the same number of distinct prime factors.

Original entry on oeis.org

91, 141, 142, 143, 212, 213, 214, 323, 324, 2302, 2303, 6850, 9061, 10280, 10281, 15740, 16130, 16164, 16682, 16683, 19052, 19053, 20212, 20213, 21195, 21196, 21790, 22055, 23064, 25779, 25780, 25991, 28608, 28674, 29971, 31442, 33084

Offset: 1

David W. Wilson

nonn

			The numbers from 91 to 96 all have 2 distinct prime factors: 91=7*13, 92=2^2*23, 93=3*31, 94=2*47, 95=5*19, and 96=2^5*3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2620 from Zak Seidov)
Roger B. Eggleton and James A. MacDougall, Consecutive integers with equally many principal divisors, Math. Mag. 81 (2008), 235-248.

Cf. A001221, A006049, A006073, A045932-A045933, A045935-A045938, A088983.

Select[Range[35000],Length[Union[Length/@FactorInteger[Range[#,#+5]]]]==1&]  (* Harvey P. Dale, Feb 27 2011 *)

A088983 Numbers n such that each of the 6 consecutive numbers n through n+5 has exactly two distinct prime factors.

Original entry on oeis.org

91, 141, 142, 143, 212, 213, 214, 323, 324, 2302, 2303

Offset: 1

Labos Elemer, Sep 30 2003

Initial segment of A045934 is identical to this sequence but in A045934 the 12th term is divisible by 3 prime factors. Is the present sequence complete?
No more terms < 3*10^8. - David Wasserman, Aug 29 2005
a(12) > 10^40, if it exists. - Giovanni Resta, May 10 2017
From David A. Corneth, May 14 2017: (Start)
We're looking for at least 6 consecutive positive integers that each have exactly two distinct prime divisors. I.e. 6 consecutive positive integers m with omega(m) = 2. Now of exactly 6 consecutive integers, exactly one of them is divisible by 6, i.e. m is of the form 2*3*k. However m has exactly 2 distinct prime divisors, so k can only have prime divisors 2 or 3. Now, suppose m ends in 6 or higher. Then one of the consecutive integers is divisible by 10 = 2*5. I.e. it's of the form 2*5*t. Then t can only have prime divisors 2 and 5. (End)
This sequence has no run of four consecutive integers, since Eggleton and MacDougall prove that there are no more than 9 consecutive integers with A001221(k) = 2. They conjecture that A007774 contains no runs of 9 consecutive integers, and has only two runs of size 8 (at 141 and 212) and two maximal runs of size 7 (at 323 and 2302); they add that the maximal run of size 6 at 91 might be the only such run, so A088983 might be complete. - Roger Eggleton via Jason Kimberley, Jul 12 2017

Roger B. Eggleton and James A. MacDougall, Consecutive integers with equally many principal divisors, Math. Mag. 81 (2008), 235-248.

Cf. A064709, A045932, A045933, A045934, A045935, A074851, A006073, A003586.

Select[Range[3000], AllTrue[# + Range[0, 5], Length@FactorInteger[#] == 2 &] &] (* Giovanni Resta, May 09 2017 *)

Definition simplified by Roger Eggleton via Jason Kimberley, Jul 12 2017

A088986 Numbers k such that each of k through k+4 are divisible by exactly two primes.

Original entry on oeis.org

54, 91, 92, 115, 141, 142, 143, 144, 158, 205, 212, 213, 214, 215, 295, 301, 323, 324, 325, 391, 535, 685, 721, 799, 1135, 1345, 1465, 1535, 1711, 1941, 1981, 2101, 2215, 2302, 2303, 2304, 2425, 2641, 3865, 4411, 5461, 6505, 6625, 6925, 7165, 7231, 7261

Offset: 1

Labos Elemer, Sep 30 2003

nonn

Identical with A045933 from first-to 38th terms, but deviates later because A045933 includes start of chains with more than 2 prime-factors.
Contrary to longer chains (6, 7, 8, ...) of omega = 2 this sequence seems to be either infinite or very long. See comments in A088983 [especially Eggleton via Kimberley, 2017].
Primes counted without multiplicity. - Harvey P. Dale, Oct 20 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..750 from Harvey P. Dale)

Cf. A001221, A045933, A088983.

Transpose[Select[Partition[Transpose[Select[Table[{n,PrimeNu[n]},{n,10000}],Last[#]==2&]][[1]],5,1],Last[#]-First[#]==4&]][[1]] (* Harvey P. Dale, Oct 20 2011 *)

lista(kmax) = {my(q = vector(5)); for(k = 6, kmax, q = concat(vecextract(q, "^1"), omega(k) == 2); if(vecmin(q) == 1, print1(k-4, ", ")));} \\ Amiram Eldar, Jul 11 2024

from sympy import primefactors
def ok(n):
    return all(len(primefactors(n + i))==2 for i in range(5))
print([n for n in range(1, 8001) if ok(n)]) # Indranil Ghosh, Jul 17 2017

A338455 Starts of runs of 5 consecutive numbers with the same total binary weight of their divisors (A093653).

Original entry on oeis.org

1307029927, 2116078861, 2665774183, 2809370965, 4108623302, 4493733751, 5333670902, 5497285284, 5679049670, 8209799382, 9665369455, 9708528486, 10353426151, 10606564910, 12777118615, 12795699493, 13660293367, 13847206214, 14351020663, 15735895813, 17912257013

Offset: 1

Amiram Eldar, Oct 28 2020

Numbers k such that A093653(k) = A093653(k+1) = A093653(k+2) = A093653(k+3) = A093653(k+4).

Can 6 consecutive numbers have the same total binary weight of their divisors? If they exist, then they are larger than 10^11.

			1307029927 is a term since A093653(1307029927) = A093653(1307029928) = A093653(1307029929) = A093653(1307029930) = A093653(1307029931) = 72.

Cf. A093653.
Subsequence of A338452, A338453 and A338454.
Similar sequences: A045933, A045941, A049051.

f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; m = 5; fs = f /@ Range[m]; Do[If[Equal @@  fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 10^7}]; s

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A045932 Numbers n such that n through n+3 are divisible by the same number of distinct primes.

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A045934 Numbers n such that n through n+5 have the same number of distinct prime factors.

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A088983 Numbers n such that each of the 6 consecutive numbers n through n+5 has exactly two distinct prime factors.

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A088986 Numbers k such that each of k through k+4 are divisible by exactly two primes.

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A338455 Starts of runs of 5 consecutive numbers with the same total binary weight of their divisors (A093653).

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