cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A045938 Numbers n such that n through n+9 are divisible by the same number of distinct primes.

Original entry on oeis.org

48919, 184171, 218972, 218973, 320085, 320671, 343443, 353944, 397322, 403117, 435721, 492037, 526095, 526096, 526097, 526098, 526099, 534078, 534079, 534080, 583340, 607116, 636332, 693841, 701595, 761492, 822260, 919998, 942528
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A001221, A006049, A006073, A045932-A045937.

Extensions

Offset corrected by Amiram Eldar, Oct 26 2019

A294278 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

6, 10, 12, 15, 18, 22, 24, 26, 28, 30, 36, 40, 42, 46, 48, 52, 58, 60, 63, 66, 70, 72, 78, 80, 82, 84, 88, 90, 96, 100, 102, 105, 106, 108, 110, 112, 114, 120, 124, 126, 130, 132, 136, 138, 140, 148, 150, 154, 156, 162, 165, 166, 168, 170, 172, 174, 178, 180
Offset: 1

Views

Author

Rémy Sigrist, Oct 26 2017

Keywords

Comments

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

Examples

			omega(1) = 0 < omega(2) = 1, hence 1 does not belong 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 belongs to this sequence.

Links

Crossrefs

Cf. A001221, A006049, A294277.

Programs

A344314 Number k such that k and k+1 have the same number of nonunitary divisors (A048105).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 27, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 124, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157
Offset: 1

Views

Author

Amiram Eldar, May 14 2021

Keywords

Examples

			1 is a term since A048105(1) = A048105(2) = 0.
27 is a term since A048105(27) = A048105(28) = 2.

Links

Crossrefs

Cf. A048105, A064115, A335328, A344315.
Similar sequences: A005237, A006049, A343819, A344312, A344313.

Programs

  • Mathematica
    nd[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; Select[Range[200], nd[#] == nd[# + 1] &]

A045933 Numbers n such that n through n+4 are divisible by the same number of distinct 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, 2664, 2714, 3865, 3912, 4411, 5450, 5461, 6354, 6505
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A001221, A006049, A006073, A045932, A045934-A045938.

Programs

  • Mathematica
    SequencePosition[PrimeNu[Range[7000]],{x_,x_,x_,x_,x_}][[All,1]] (* Harvey P. Dale, Jun 13 2022 *)

A101932 Numbers n with omega(n) equal to omega(n-1) and omega (n+1).

Original entry on oeis.org

3, 4, 8, 21, 34, 35, 39, 45, 51, 55, 56, 57, 75, 76, 86, 87, 92, 93, 94, 95, 99, 116, 117, 118, 123, 134, 135, 142, 143, 144, 145, 146, 147, 159, 160, 161, 176, 177, 184, 188, 201, 202, 206, 207, 208, 213, 214, 215, 216, 217, 218, 225
Offset: 1

Views

Author

Neil Fernandez, Dec 21 2004

Keywords

Examples

			45 is in the sequence because it has 2 prime factors (3 and 5) as do 44 (2 and 11) and 46 (2 and 23).

Links

Crossrefs

Cf. A001221.
Subsequence of A006049.

Programs

  • Mathematica
    For[i=1, i<1000, If[And[Length[FactorInteger[i-1]]==Length[FactorInteger[i]], Length[FactorInteger[i+1]]==Length[FactorInteger[i]]], Print[i]];i++ ]
Select[Range[2, 225], PrimeNu[#] == PrimeNu[# - 1] == PrimeNu[# + 1] &] (* Jayanta Basu, Aug 11 2013 *)
SequencePosition[PrimeNu[Range[300]],{x_,x_,x_}][[All,1]]+1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 18 2018 *)
  • PARI
    isok(n) = (omega(n) == omega(n-1)) && (omega(n)==omega(n+1)) \\ Michel Marcus, May 05 2017

A344312 Number k such that k and k+1 have the same number of exponential divisors (A049419).

Original entry on oeis.org

1, 2, 5, 6, 8, 10, 13, 14, 21, 22, 24, 27, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 120, 122, 124, 125, 129, 130, 133, 135, 137, 138, 141
Offset: 1

Views

Author

Amiram Eldar, May 14 2021

Keywords

Examples

			1 is a term since A049419(1) = A049419(2) = 1.
8 is a term since A049419(8) = A049419(9) = 2.

Links

Crossrefs

Cf. A049419.
Similar sequences: A005237, A006049, A343819, A344313, A344314.

Programs

  • Mathematica
    f[p_, e_] := DivisorSigma[0, e]; ed[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[200], ed[#] == ed[# + 1] &]

A344313 Number k such that k and k+1 have the same number of bi-unitary divisors (A286324).

Original entry on oeis.org

2, 3, 4, 14, 15, 20, 21, 26, 27, 33, 34, 35, 38, 44, 45, 50, 51, 57, 62, 68, 74, 75, 76, 81, 85, 86, 91, 92, 93, 94, 98, 99, 104, 115, 116, 117, 118, 122, 123, 124, 133, 135, 141, 142, 145, 146, 147, 158, 171, 177, 187, 189, 201, 202, 205, 206, 212, 213, 214
Offset: 1

Views

Author

Amiram Eldar, May 14 2021

Keywords

Examples

			2 is a term since A286324(2) = A286324(3) = 2.
14 is a term since A286324(14) = A286324(15) = 4.

Links

Crossrefs

Cf. A286324, A293183, A304463.
Similar sequences: A005237, A006049, A343819, A344312, A344314.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], e + 1, e]; bd[1] = 1; bd[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[200], bd[#] == bd[# + 1] &]

A273879 Numbers k such that k and k+1 have 6 distinct prime factors.

Original entry on oeis.org

11243154, 13516580, 16473170, 16701684, 17348330, 19286805, 20333495, 21271964, 21849905, 22054515, 22527141, 22754589, 22875489, 24031370, 25348070, 25774329, 28098245, 28618394, 28625960, 30259229, 31846269, 32642805
Offset: 1

Views

Author

Charles R Greathouse IV, Jun 02 2016

Keywords

Comments

Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite (Theorem 2).

Examples

			13516580 = 2^2 * 5 * 7 * 11 * 67 * 131 and 13516581 = 3 * 13 * 17 * 19 * 29 * 37 so 13516580 is in this sequence.

Links

Crossrefs

Numbers k such that k and k+1 have j distinct prime factors: A006549 (j=1, apart from the first term), A074851 (j=2), A140077 (j=3), A140078 (j=4), A140079 (j=5).
Cf. A006049, A093548.

Programs

  • Mathematica
    SequencePosition[PrimeNu[Range[3265*10^4]],{6,6}][[All,1]] (* Harvey P. Dale, Nov 20 2021 *)
  • PARI
    is(n)=omega(n)==6 && omega(n+1)==6

Formula

a(1) = A138206(2). - R. J. Mathar, Jul 15 2023
{k: k in A074969 and k+1 in A074969.} - R. J. Mathar, Jul 19 2023

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

Views

Author

Gus Wiseman, Dec 28 2018

Keywords

Examples

			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

Links

Crossrefs

Positions of 1's are A246655.
Cf. A000010, A001221, A006049, A067003, A294277, A294278, A302242, A322838, A322841.

Programs

  • Mathematica
    Table[Length[Select[Range[n],PrimeNu[#]
  • PARI
    \\ See Corneth link

A322841 Number of positive integers less than n with more distinct prime factors than n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 3, 0, 0, 5, 5, 0, 6, 0, 0, 0, 9, 0, 10, 0, 11, 0, 12, 0, 13, 13, 1, 1, 1, 1, 17, 1, 1, 1, 20, 0, 21, 2, 2, 2, 24, 2, 25, 2, 2, 2, 28, 2, 2, 2, 2, 2, 33, 0, 34, 3, 3, 36, 3, 0, 38, 4, 4, 0, 41, 5, 42, 5, 5, 5, 5, 0, 47, 6, 48
Offset: 1

Views

Author

Gus Wiseman, Dec 28 2018

Keywords

Examples

			Column n lists the a(n) positive integers less than n with more 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
  ---------------------------------------------------------------------
                    6  6  6      10      12          15  15      18
                                  6      10          14  14      15
                                          6          12  12      14
                                                     10  10      12
                                                      6   6      10
                                                                  6

Links

Crossrefs

Positions of zeros are A116998.
Cf. A000010, A000961, A001221, A006049, A058933, A067003, A294277, A302242, A322837, A322838.

Programs

  • Maple
    b:= proc(n) option remember; nops(numtheory[factorset](n)) end:
a:= proc(n) option remember;
      (t-> add(`if`(b(i)>t, 1, 0), i=1..n-1))(b(n))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 28 2018
  • Mathematica
    Table[Length[Select[Range[n],PrimeNu[#]>PrimeNu[n]&]],{n,100}]
  • PARI
    a(n) = my(omegan=omega(n)); sum(k=1, n-1, omega(k) > omegan); \\ Michel Marcus, Dec 29 2018
  • PARI
    first(n) = {my(t = 1, pp = 1, res = vector(n)); forprime(p = 2, oo, pp*=p; if(pp > n, v = vector(t); break); t++); for(i = 1, n, o = omega(i); res[i] = v[o+1]; for(j = 1, o, v[j]++)); res} \\ David A. Corneth, Dec 29 2018
Previous Showing 11-20 of 38 results. Next

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