A294278 -id:A294278 - cp's OEIS frontend

A006049 Numbers k such that k and k+1 have the same number of distinct prime divisors.

2, 3, 4, 7, 8, 14, 16, 20, 21, 31, 33, 34, 35, 38, 39, 44, 45, 50, 51, 54, 55, 56, 57, 62, 68, 74, 75, 76, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 111, 115, 116, 117, 118, 122, 123, 127, 133, 134, 135, 141, 142, 143, 144, 145, 146, 147, 152, 158, 159, 160, 161, 171, 175

Offset: 1

N. J. A. Sloane

Sequence is infinite, as proved by Schlage-Puchta, who comments: "Buttkewitz found a non-computational proof, and the Goldston-Pintz-Yildirim-sieve yields more precise information". - Charles R Greathouse IV, Jan 09 2013

The asymptotic density of this sequence is 0 (Erdős, 1936). - Amiram Eldar, Sep 17 2024

Calvin C. Clawson, Mathematical mysteries, Plenum Press, 1996, p. 250.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from T. D. Noe)
Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.
Jan-Christoph Schlage-Puchta, The equation ω(n)=ω(n+1), Mathematika, Vol. 50, No. 1-2 (2003), pp. 99-101; arXiv preprint, arXiv:1105.1621 [math.NT], 2011.

Cf. A001221, A052215, A107800, A006073, A294277, A294278.

Subsequence of A062974.

import Data.List (elemIndices)
a006049 n = a006049_list !! (n-1)
a006049_list = map (+ 1) $ elemIndices 0 $
   zipWith (-) (tail a001221_list) a001221_list
-- Reinhard Zumkeller, Jan 22 2013

f[n_] := Length@FactorInteger[n];t = f /@ Range[175];Flatten@Position[Rest[t] - Most[t], 0] (* Ray Chandler, Mar 27 2007 *)
Select[Range[200],PrimeNu[#]==PrimeNu[#+1]&] (* Harvey P. Dale, May 09 2012 *)
Flatten[Position[Partition[PrimeNu[Range[200]],2,1],?(#[[1]]==#[[2]]&),{1},Heads->False]] (* _Harvey P. Dale, May 22 2015 *)
SequencePosition[PrimeNu[Range[200]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 02 2019 *)

is(n)=omega(n)==omega(n+1) \\ Charles R Greathouse IV, Jan 09 2013

A001221(a(n)) = A001221(a(n)+1). - Reinhard Zumkeller, Jan 22 2013

Extended by Ray Chandler, Mar 27 2007

A067003 Number of numbers <= n with same number of distinct prime factors as n.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 5, 6, 7, 2, 8, 3, 9, 4, 5, 10, 11, 6, 12, 7, 8, 9, 13, 10, 14, 11, 15, 12, 16, 1, 17, 18, 13, 14, 15, 16, 19, 17, 18, 19, 20, 2, 21, 20, 21, 22, 22, 23, 23, 24, 25, 26, 24, 27, 28, 29, 30, 31, 25, 3, 26, 32, 33, 27, 34, 4, 28, 35, 36, 5, 29, 37, 30, 38, 39, 40, 41

Offset: 1

Henry Bottomley, Dec 21 2001

			a(11)=8 since 2,3,4,5,7,8,9,11 each have one distinct prime factor. a(12)=3 since 6,10,12 each have two distinct prime factors.
From _Gus Wiseman_, Dec 28 2018: (Start)
Column n lists the a(n) positive integers less than or equal to n with the same number of distinct prime factors as n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
        2  3  4     5  7  8  6   9   10  11  12  14  13  16  15  17  18
           2  3     4  5  7      8   6   9   10  12  11  13  14  16  15
              2     3  4  5      7       8   6   10  9   11  12  13  14
                    2  3  4      5       7       6   8   9   10  11  12
                       2  3      4       5           7   8   6   9   10
                          2      3       4           5   7       8   6
                                 2       3           4   5       7
                                         2           3   4       5
                                                     2   3       4
                                                         2       3
                                                                 2
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Positions of 1's are A002110.
Cf. A001221, A008479, A058933, A067004. Inverse of A000961, A007774, A033992, A033993, A051270 etc.
Cf. A000010, A006049, A061142, A294277, A294278, A302242, A322837, A322841.

Table[Length[Select[Range[n],PrimeNu[#]==PrimeNu[n]&]],{n,100}] (* Gus Wiseman, Dec 28 2018 *)

a(n) = my(nb = #factor(n)~); sum(k=1, n, #factor(k)~ == nb); \\ Michel Marcus, Jul 13 2019

a(A002110(n)) = 1.

A294277 Numbers k such that omega(k) < omega(k+1) (where omega(m) = A001221(m), the number of distinct primes dividing m).

Original entry on oeis.org

1, 5, 9, 11, 13, 17, 19, 23, 25, 27, 29, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 89, 97, 101, 103, 104, 107, 109, 113, 119, 121, 125, 128, 129, 131, 137, 139, 149, 151, 153, 155, 157, 163, 164, 167, 169, 173, 179, 181, 185

Offset: 1

Rémy Sigrist, Oct 26 2017

This sequence, alongside A006049 and A294278, form a partition of the positive integers.

The asymptotic density of this sequence is 1/2 (Erdős, 1936). - Amiram Eldar, Sep 17 2024

			omega(1) = 0 < omega(2) = 1, hence 1 belongs to this sequence.
omega(4) = 1 = omega(5) = 1, hence 4 does not belong to this sequence.
omega(6) = 2 > omega(7) = 1, hence 6 does not belong to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Cf. A001221, A006049, A294278.

Position[Partition[PrimeNu[Range[200]],2,1],?(#[[1]]<#[[2]]&),1,Heads-> False]//Flatten (* _Harvey P. Dale, May 06 2018 *)

for (n=1, 185, if (omega(n) < omega(n+1), print1 (n ", ")))

A322837 Number of positive integers less than n with fewer distinct prime factors than n.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 1, 8, 1, 9, 1, 10, 10, 1, 1, 12, 1, 13, 13, 13, 1, 14, 1, 15, 1, 16, 1, 29, 1, 1, 19, 19, 19, 19, 1, 20, 20, 20, 1, 40, 1, 22, 22, 22, 1, 23, 1, 24, 24, 24, 1, 25, 25, 25, 25, 25, 1, 57, 1, 27, 27, 1, 28, 62, 1, 29, 29, 65, 1, 30, 1, 31

Offset: 1

Gus Wiseman, Dec 28 2018

			Column n lists the a(n) positive integers less than n with fewer distinct prime factors than n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
     1  1  1  1  5  1  1  1  9   1   11  1   13  13  1   1   17  1   19
                 4           8       9       11  11          16      17
                 3           7       8       9   9           13      16
                 2           5       7       8   8           11      13
                 1           4       5       7   7           9       11
                             3       4       5   5           8       9
                             2       3       4   4           7       8
                             1       2       3   3           5       7
                                     1       2   2           4       5
                                             1   1           3       4
                                                             2       3
                                                             1       2
                                                                     1

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program

Positions of 1's are A246655.

Cf. A000010, A001221, A006049, A067003, A294277, A294278, A302242, A322838, A322841.

Table[Length[Select[Range[n],PrimeNu[#]

PARI
```
\\ See Corneth link
```

A322839 Numbers n with more prime factors (counted with multiplicity) than n+1.

Original entry on oeis.org

4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 45, 46, 48, 50, 52, 54, 56, 58, 60, 64, 66, 68, 70, 72, 76, 78, 80, 81, 82, 84, 88, 90, 92, 96, 100, 102, 104, 105, 106, 108, 110, 112, 114, 117, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

First differs from A074827 in having 104.

			104 has four prime factors (2, 2, 2, 13), while 105 has only three (3, 5, 7), so 104 belongs to the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000961, A001221, A001222, A045920, A294277, A294278, A302242, A322838, A322840.

R:= NULL: count:= 0: w:= 0:
for n from 2 while count < 100 do
  v:= numtheory:-bigomega(n);
  if v < w then R:= R,n-1; count:= count+1; fi;
  w:= v;
od:
R; # Robert Israel, Jan 16 2023

Select[Range[100],PrimeOmega[#]>PrimeOmega[#+1]&]
Position[Partition[PrimeOmega[Range[200]],2,1],?(#[[1]]>#[[2]]&),1,Heads -> False]//Flatten (* _Harvey P. Dale, Jan 02 2021 *)

A322840 Positive integers n with fewer prime factors (counted with multiplicity) than n + 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 26, 29, 31, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 134, 137, 139, 143, 146, 149

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

			49 = 7*7 has two prime factors, while 50 = 2*5*5 has three, so 49 belongs to the sequence.

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

Cf. A001222, A006049, A045920, A294277, A294278, A302242, A322837, A322838, A322839.

Select[Range[100],PrimeOmega[#]?(#[[1]]< #[[2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 23 2021 *)

isok(n) = bigomega(n) < bigomega(n+1); \\ Michel Marcus, Dec 29 2018

A293460 a(n) = Sum_{k=1..n} sign(omega(n+1) - omega(n)) (where omega(m) = A001221(m), the number of distinct primes dividing m).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 1, 2, 3, 2, 3, 3, 4, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 4, 4

Offset: 0

Rémy Sigrist, Oct 26 2017

sign

The sign function is defined by:
- sign(0) = 0,
- sign(n) = +1 for any n > 0,
- sign(n) = -1 for any n < 0.
a(n) corresponds to the number of integers up to n in A294277 minus the number of integers up to n in A294278.
The first negative value occurs at a(178) = -1.
Will this sequence change sign indefinitely?

			The following table shows the first terms of the sequence, alongside sign(omega(n+1)-omega(n)), omega(n+1) and omega(n):
n       a(n)    sign    w(n+1)  w(n)
-       ----    ----    ------  ----
0       0
1       1       1       1       0
2       1       0       1       1
3       1       0       1       1
4       1       0       1       1
5       2       1       2       1
6       1       -1      1       2
7       1       0       1       1
8       1       0       1       1
9       2       1       2       1
10      1       -1      1       2
11      2       1       2       1
12      1       -1      1       2
13      2       1       2       1
14      2       0       2       2
15      1       -1      1       2
16      1       0       1       1
17      2       1       2       1
18      1       -1      1       2
19      2       1       2       1
20      2       0       2       2

Georg Fischer, Table of n, a(n) for n = 0..1000
Rémy Sigrist, Line graph of the first 10000 terms
Rémy Sigrist, Line graph of the first 100000000 terms
Rémy Sigrist, Line graph of the first 1000000000 terms
Rémy Sigrist, Line graph of the first 10000000000 terms

Cf. A001221, A006049, A294277, A294278.

Accumulate[Join[{0},Sign[Differences[PrimeNu[Range[90]]]]]] (* Harvey P. Dale, Dec 25 2024 *)

s = 0; for (n=1, 87, print1 (s ", "); s += sign(omega(n+1)-omega(n)))

a(0) = 0, and for any n > 0:
- a(A294277(n)) = a(A294277(n)-1) + 1,
- a(A006049(n)) = a(A006049(n)-1),
- a(A294278(n)) = a(A294278(n)-1) - 1.
Also: a(n) = #{ k / A294277(k) <= n } - #{ k / A294278(k) <= n }.

cp's OEIS Frontend

A006049 Numbers k such that k and k+1 have the same number of distinct prime divisors.

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A067003 Number of numbers <= n with same number of distinct prime factors as n.

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A294277 Numbers k such that omega(k) < omega(k+1) (where omega(m) = A001221(m), the number of distinct primes dividing m).

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A322837 Number of positive integers less than n with fewer distinct prime factors than n.

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A322839 Numbers n with more prime factors (counted with multiplicity) than n+1.

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A322840 Positive integers n with fewer prime factors (counted with multiplicity) than n + 1.

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A293460 a(n) = Sum_{k=1..n} sign(omega(n+1) - omega(n)) (where omega(m) = A001221(m), the number of distinct primes dividing m).

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