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.

Showing 1-4 of 4 results.

A071814 Numbers k such that the number of 1's in the binary representation of k equals bigomega(k), the number of prime divisors of k (counted with multiplicity).

Original entry on oeis.org

2, 6, 9, 10, 28, 33, 34, 42, 44, 50, 52, 54, 60, 65, 70, 76, 90, 98, 129, 135, 138, 148, 150, 156, 164, 184, 198, 204, 210, 225, 228, 232, 261, 266, 268, 273, 290, 292, 294, 297, 306, 308, 322, 330, 340, 344, 385, 388, 390, 405, 424, 440, 468, 486, 496, 504
Offset: 1

Views

Author

Jason Earls, Jun 07 2002

Keywords

Comments

A115156 is a subsequence: A001222(A115156(n)) = A000120(A115156(n)) = n. - Reinhard Zumkeller, Jan 14 2006

Examples

			232 is a term because 232 = 11101000_2 and 232 = 2^3*29.

Links

Crossrefs

Cf. A000120, A001222, A071594, A115156.

Programs

  • Mathematica
    fQ[n_] := Count[IntegerDigits[n, 2], 1] == Plus @@ Last /@ FactorInteger@n; Select[ Range@517, fQ[ # ] &] (* Robert G. Wilson v, Jan 18 2006 *)
Select[Range[600],Count[IntegerDigits[#,2],1]==PrimeOmega[#]&] (* Harvey P. Dale, Mar 07 2019 *)

A071595 Odd numbers k such that the number of 1's in binary representation of k equals omega(k), the number of distinct primes in the factorization of k.

Original entry on oeis.org

33, 65, 129, 273, 385, 513, 1025, 1155, 1281, 2049, 2065, 2145, 4097, 4161, 4641, 8193, 8195, 8211, 8225, 8265, 8385, 8449, 9345, 10241, 16905, 17409, 21505, 24585, 32775, 32785, 32835, 32865, 33033, 33285, 33345, 33825, 34881, 36865, 36993
Offset: 1

Views

Author

Benoit Cloitre, Jun 01 2002

Keywords

Examples

			129 = 10000001 in base 2 and 129 = 3*43 hence 129 is in the sequence.

Links

Crossrefs

Cf. A071594, A071596.

Programs

  • Mathematica
    Select[Range[1, 37000, 2], DigitCount[#, 2, 1] == PrimeNu[#] &] (* Amiram Eldar, Jan 11 2020*)
  • PARI
    for(n=1,80000,if(sum(i=1,length(binary(n)), component(binary(n),i))==-(-1)^n*omega(n),print1(n,",")))

A071596 Even numbers k such that the number of 1's in binary representation of k equals omega(k), the number of distinct primes in the factorization of k.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 32, 34, 36, 40, 42, 48, 64, 68, 70, 72, 80, 84, 96, 128, 136, 138, 140, 144, 160, 168, 192, 210, 256, 266, 272, 276, 280, 288, 290, 320, 322, 330, 336, 384, 390, 420, 512, 514, 518, 522, 530, 532, 544, 552, 560, 576, 580, 640
Offset: 1

Views

Author

Benoit Cloitre, Jun 01 2002

Keywords

Examples

			532 = 1000010100 in base 2 and 532 = 2^2*7*19 hence 532 is in the sequence.

Links

Crossrefs

Cf. A071594, A071595.

Programs

  • Mathematica
    Select[Range[2, 640, 2], DigitCount[#, 2, 1] == PrimeNu[#] &] (* Amiram Eldar, Jan 11 2020 *)
  • PARI
    for(n=1,80000,if(sum(i=1,length(binary(n)), component(binary(n),i))==(-1)^n*omega(n),print1(n,",")))

A140707 A positive integer n is included if n written in binary contains the same number of 0's as the number of distinct primes that divide n.

Original entry on oeis.org

1, 2, 5, 10, 11, 12, 13, 21, 22, 23, 26, 27, 28, 29, 39, 42, 45, 46, 47, 51, 54, 57, 58, 59, 61, 78, 87, 90, 91, 93, 94, 102, 105, 114, 115, 117, 118, 120, 122, 124, 125, 159, 174, 175, 182, 183, 186, 187, 189, 191, 207, 210, 215, 219, 220, 221, 223, 230, 234, 235
Offset: 1

Views

Author

Leroy Quet, Jul 11 2008

Keywords

Examples

			90 written in binary is 1011010. There are three 0's in this binary representation. 90 has the prime factorization: 2^1 *3^2 *5^1. There are 3 distinct primes dividing 90. Since the number of 0's in the binary representation equals the number of distinct primes dividing 90, then 90 is in the sequence.

Links

Crossrefs

Cf. A071594, A001221, A023416.

Programs

  • Maple
    A080791 := proc(n) local dgs ; dgs := convert(n,base,2) ; nops(dgs)-add(i,i=dgs) ; end: A001221 := proc(n) nops(numtheory[factorset](n)) ; end: isA140707 := proc(n) RETURN( A080791(n) = A001221(n)) ; end: for n from 1 to 300 do if isA140707(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Aug 08 2008
  • Mathematica
    Select[Range[300],DigitCount[#,2,0]==PrimeNu[#]&] (* Harvey P. Dale, Dec 08 2017 *)

Formula

{n: A080791(n) = A001221(n)}. - R. J. Mathar, Aug 08 2008

Extensions

Extended beyond 42 by R. J. Mathar, Aug 08 2008
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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