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 1-3 of 3 results.

A145784 Numbers with property that their number of prime factors counted with multiplicity is a multiple of 3.

Original entry on oeis.org

1, 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 64, 66, 68, 70, 75, 76, 78, 92, 96, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 144, 147, 148, 153, 154, 160, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 216, 222
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 19 2008

Keywords

Comments

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 02 2024

Links

Crossrefs

Cf. A001222, A010872, A373975 (characteristic function).
Subsequences: A000578, A014612, A046316, A121307, A373589, A373590, A373597, A373837.
Cf. also A028260, A214195, A297845.

Programs

  • Haskell
    a145784 n = a145784_list !! (n-1)
a145784_list = filter ((== 0) . a010872 . a001222) [1..]
-- Reinhard Zumkeller, May 26 2012
  • Mathematica
    Join[{1}, Select[Range[2,230], Mod[Total[Transpose[FactorInteger[#]][[2]]], 3] == 0 &]] (* T. D. Noe, May 21 2012 *)
  • PARI
    isok(k) = !(bigomega(k) % 3); \\ Amiram Eldar, May 16 2025

Formula

A010872(A001222(a(n))) = 0.

A373597 Non-multiples of 3 whose multiplicies of prime factors of types 3m-1 and 3m+1 are both multiples of 3.

Original entry on oeis.org

1, 8, 20, 44, 50, 64, 68, 92, 110, 116, 125, 160, 164, 170, 188, 212, 230, 236, 242, 275, 284, 290, 332, 343, 352, 356, 374, 400, 404, 410, 425, 428, 452, 470, 506, 512, 524, 530, 544, 548, 575, 578, 590, 596, 605, 637, 638, 668, 692, 710, 716, 725, 736, 764, 782, 788, 830, 880, 890, 902, 908, 928, 931, 932, 935
Offset: 1

Views

Author

Antti Karttunen, Jun 10 2024

Keywords

Comments

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. This is generated by semigroups A373589 and A373590.

Examples

			20 = 2*2*5 has 0 primes of type 3m+1 (A002476) and 3 primes of type 3m-1 (A003627) in its prime factorization, and as 0 and 3 are both multiples of 3, 20 is included as a term.
21952 = 2^6 * 7^3 is a term because there are 3 primes of type 3m+1 and 6 primes of type 3m-1, and as 6 and 3 are both multiples of 3, 21952 is included as a term.

Links

Crossrefs

Cf. A002476, A003627, A373596 (characteristic function).
Subsequences: A373589 and A373590.
Subsequence of A001651, and of A145784.
Subsequence of the sequences A369659, A369644, A327863, A289142, A373385, and some of their intersections: A373473, A373475, A373478, A373492, A373494.
Differs from A373492 for the first time at n=91, where a(91) = 1325, which skips the value A373492(91) = 1323 present in A373492.
Cf. also A046337 (roughly analogous sequence for k=2, instead of k=3).

Programs

A373589 Numbers whose number of prime factors (with multiplicity) is a multiple of 3, and all of them are of the type 3m-1 (in A003627).

Original entry on oeis.org

1, 8, 20, 44, 50, 64, 68, 92, 110, 116, 125, 160, 164, 170, 188, 212, 230, 236, 242, 275, 284, 290, 332, 352, 356, 374, 400, 404, 410, 425, 428, 452, 470, 506, 512, 524, 530, 544, 548, 575, 578, 590, 596, 605, 638, 668, 692, 710, 716, 725, 736, 764, 782, 788, 830, 880, 890, 902, 908, 928, 932, 935, 956, 986, 1000
Offset: 1

Views

Author

Antti Karttunen, Jun 10 2024

Keywords

Comments

A multiplicative semigroup: if m and n are in the sequence, then so is m*n.

Examples

			1 is a term since it has no prime factors, 0 is a multiple of 3, and "the empty set has every property". - _N. J. A. Sloane_, Dec 16 2024

Links

Crossrefs

Cf. A001222, A003627, A121307 (subsequence), A373588 (characteristic function).
Intersection of A004612 and A145784.
Subsequence of A373597, which in turn is a subsequence of many other sequences.
Cf. also A373590.

Programs

  • Mathematica
    Join[{1},Select[Range[1000],Mod[PrimeOmega[#],3]==0&&Union[Mod[FactorInteger[#][[;;,1]],3]]=={2}&]] (* Harvey P. Dale, Dec 16 2024 *)
  • PARI
    isA373589 = A373588;
Showing 1-3 of 3 results.

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

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