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 1-5 of 5 results.

A235992 Numbers with an even arithmetic derivative, cf. A003415.

Original entry on oeis.org

0, 1, 4, 8, 9, 12, 15, 16, 20, 21, 24, 25, 28, 32, 33, 35, 36, 39, 40, 44, 48, 49, 51, 52, 55, 56, 57, 60, 64, 65, 68, 69, 72, 76, 77, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 100, 104, 108, 111, 112, 115, 116, 119, 120, 121, 123, 124, 128, 129, 132, 133
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 11 2014

Keywords

Comments

A165560(a(n)) = 0; A003415(a(n)) mod 2 = 0.
For n > 1: A007814(a(n)) <> 1, A006519(a(n)) <> 2.
Union of multiples of 4 and odd numbers with an even number of prime factors with multiplicity. - Charlie Neder, Feb 25 2019
After two initial terms (0 and 1), numbers n such that A086134(n) = 2. - Antti Karttunen, Sep 30 2019
A multiplicative semigroup; if m and n are in the sequence then so is m*n. (See also comments in A359780.) - Antti Karttunen, Jan 17 2023

Links

Crossrefs

Cf. A235991 (complement).
Union of A327862 and A327864.
Union of A359829 (primitive elements) and A359831 (nonprimitive elements).
Cf. A003415, A086134, A327863, A327865, A327933, A327935, A358680 (characteristic function).
Positions of multiples of 4 in A358669 (and in A358765).
Cf. also A028260, A036349, A046337, A332820 (other multiplicative semigroups), and comments in A359780.

Programs

  • Haskell
    a235992 n = a235992_list !! (n-1)
a235992_list = filter (even . a003415) [0..]
  • Mathematica
    Select[Range[0, 133], EvenQ@ If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]] &] (* Michael De Vlieger, Sep 30 2019 *)
  • Python
    from itertools import count, islice
from sympy import factorint
def A235992_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n: not n&3 or (n&1 and not sum(factorint(n).values())&1), count(max(startvalue,0)))
A235992_list = list(islice(A235992_gen(),40)) # Chai Wah Wu, Nov 04 2022

A067019 Odd numbers with an odd number of prime factors (counted with multiplicity).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 45, 47, 53, 59, 61, 63, 67, 71, 73, 75, 79, 83, 89, 97, 99, 101, 103, 105, 107, 109, 113, 117, 125, 127, 131, 137, 139, 147, 149, 151, 153, 157, 163, 165, 167, 171, 173, 175, 179, 181, 191, 193, 195, 197, 199
Offset: 1

Views

Author

Shyam Sunder Gupta, Feb 16 2002

Keywords

Comments

Subsequence of odd terms of A026424. - Michel Marcus, Jul 04 2015
The sequence a(1)=0, for n>1 a(n) is smallest number such that for all s,t,mAnders Hellström, Jul 08 2015

Examples

			a(9) = 27, which is odd with an odd number of prime factors, i.e., 3.

Links

Crossrefs

Intersection of A005408 and A026424.
Setwise difference A005408 \ A046337.
Cf. A353558 (characteristic function).
Positions of the terms of the form 4u+2 (A016825) in A358669 (and in A358765).

Programs

  • Mathematica
    Select[Range[1,301,2],OddQ[PrimeOmega[#]]&] (* Harvey P. Dale, Feb 15 2025 *)
  • PARI
    isok(k) = { k%2 == 1 && bigomega(k)%2 == 1 } \\ Harry J. Smith, Apr 25 2010

A358669 Pointwise product of the arithmetic derivative and the primorial base exp-function.

Original entry on oeis.org

0, 0, 3, 6, 36, 18, 25, 10, 180, 180, 315, 90, 400, 50, 675, 1200, 7200, 450, 2625, 250, 9000, 7500, 14625, 2250, 27500, 12500, 28125, 101250, 180000, 11250, 217, 14, 1680, 588, 1197, 1512, 2100, 70, 2205, 3360, 21420, 630, 7175, 350, 25200, 40950, 39375, 3150, 98000, 24500, 118125, 105000, 441000
Offset: 0

Views

Author

Antti Karttunen, Dec 05 2022

Keywords

Links

Crossrefs

Cf. A003415, A059841, A121262, A152822, A276086, A327858, A353558, A358680, A358765 (= a(n) mod 60), A359423, A359603 [Dirichlet inverse of 1+a(n)].
Cf. A016825 (positions of odd terms), A042965 (of even terms), A235992 (of multiples of 4), A067019 (of terms of the form 4k+2), A358748 (of the form 4k+1), A358749 (of the form 4k+3).

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A358669(n) = (A003415(n)*A276086(n));

Formula

a(n) = A003415(n) * A276086(n).
From Antti Karttunen, Jan 09 2023: (Start)
a(n) = A327858(n) * A359423(n).
For all n >= 0, A059841(a(n)) = A152822(n).
For all n >= 1, 1-A152822(a(n)) = A353558(n).
For all n >= 0, A121262(a(n)) = A358680(n).
(End)

A359604 a(n) = A359603(n) mod 60.

Original entry on oeis.org

1, 56, 53, 39, 41, 30, 49, 51, 48, 16, 29, 49, 9, 12, 25, 36, 29, 38, 49, 21, 33, 22, 29, 53, 40, 2, 0, 53, 29, 30, 45, 33, 25, 10, 45, 20, 49, 42, 53, 33, 29, 0, 9, 33, 48, 52, 29, 58, 40, 58, 13, 13, 29, 26, 17, 33, 33, 22, 29, 31, 21, 8, 44, 57, 41, 14, 49, 9, 13, 52, 29, 16, 9, 42, 40, 53, 21, 20
Offset: 1

Views

Author

Antti Karttunen, Jan 11 2023

Keywords

Links

Crossrefs

Cf. A358669, A359603, A359590 (parity of terms).
Cf. also A358765.

Programs

A359424 The least common multiple of the arithmetic derivative and the primorial base exp-function, reduced modulo 60.

Original entry on oeis.org

0, 0, 3, 6, 36, 18, 5, 10, 0, 30, 15, 30, 40, 50, 45, 0, 0, 30, 45, 10, 0, 30, 45, 30, 20, 50, 15, 30, 0, 30, 37, 14, 0, 42, 57, 12, 0, 10, 45, 0, 0, 30, 35, 50, 0, 30, 15, 30, 20, 10, 15, 0, 0, 30, 15, 40, 0, 30, 45, 30, 8, 38, 57, 18, 24, 42, 5, 10, 0, 30, 15, 30, 0, 50, 15, 30, 0, 30, 55, 10, 0, 0, 15
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2023

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327858, A358669, A359423.
Cf. A016825 (positions of odd terms), A042965 (of even terms), A327864 (of multiples of 4).
Cf. also A358765.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A359423(n) = lcm(A003415(n), A276086(n));
A359424(n) = (A359423(n)%60);
Showing 1-5 of 5 results.

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