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.

A260489 a(n) = 3n - A260488(n).

Original entry on oeis.org

0, 2, 4, 5, 5, 7, 5, 7, 8, 8, 5, 7, 8, 8, 10, 8, 10, 8, 5, 7, 8, 8, 10, 8, 10, 11, 11, 8, 10, 11, 11, 8, 10, 8, 5, 7, 8, 8, 10, 8, 10, 11, 11, 8, 10, 11, 11, 13, 11, 13, 11, 8, 10, 11, 11, 13, 11, 13, 11, 8, 10, 11, 11, 8, 10, 8, 5, 7, 8, 8, 10
Offset: 0

Views

Author

Franklin T. Adams-Watters, Jul 27 2015

Keywords

Comments

The sequence is nonnegative for at least the first 10000 terms, so the nonn keyword was used. That it is always nonnegative is a conjecture. That it is unbounded is also a conjecture.
The sequence is nonnegative for essentially the reason given in A225838 for its terms exceeding the terms of its complement. Here, the customers are labeled with numbers, m, that are powers of 2 and they each deposit 2 coins at each epoch (6k+1)*m and withdraw them separately at epoch (6k+3)*m and (6k+5)*m. The terms here are the number of coins in the vault after the n-th deposit. Simple accountancy says the sum of nonnegative bank balances is nonnegative. - Peter Munn, Nov 15 2023

Crossrefs

Cf. A008585, A225838, A260488.

A225837 Numbers of form 2^i*3^j*(6k+1), i, j, k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 18, 19, 21, 24, 25, 26, 27, 28, 31, 32, 36, 37, 38, 39, 42, 43, 48, 49, 50, 52, 54, 55, 56, 57, 61, 62, 63, 64, 67, 72, 73, 74, 75, 76, 78, 79, 81, 84, 85, 86, 91, 93, 96, 97, 98, 100, 103, 104, 108, 109, 110, 111, 112
Offset: 1

Views

Author

Ralf Stephan, May 16 2013

Keywords

Comments

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022
From Peter Munn, Nov 16 2023: (Start)
Contains all nonzero squares.
Dividing by 5 the terms that are multiples of 5 gives its complement, A225838.
(A352272, 2*A352272, 3*A352272, 6*A352272) is a partition of the terms.
The terms form a subgroup of the positive integers under the operation A059897(.,.) and are the positive integers in an index 2 multiplicative subgroup of rationals that is generated by 2, 3 and integers congruent to 1 modulo 6. See A225857 and A352272 for further information about such subgroups.
(End)

Links

Crossrefs

Complement of A225838.
Subsequences: A003136\{0}, A083854\{0}, A260488\{0}, A352272.
Symmetric difference of A026225 and A036554; of A036668 and A189716.
Cf. A016921, A059897, A225857.

Programs

  • Magma
    [n: n in [1..200] | IsOne(d mod 6) where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // Bruno Berselli, May 16 2013
  • Mathematica
    mx = 122; t = {}; Do[n = 2^i*3^j (6 k + 1); If[n <= mx, AppendTo[t, n]], {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, mx/6}]; Union[t] (* T. D. Noe, May 16 2013 *)
  • PARI
    for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%6==1),print1(n,",")))
  • Python
    from sympy import integer_log
def A225837(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x//3**i>>j)+5)//6 for i in range(integer_log(x,3)[0]+1) for j in range((x//3**i).bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Feb 02 2025

A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

Original entry on oeis.org

9, 18, 25, 36, 49, 50, 72, 81, 98, 100, 121, 144, 162, 169, 196, 200, 225, 242, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1058, 1089, 1152, 1156, 1225, 1250, 1296, 1352, 1369
Offset: 1

Views

Author

Benoit Cloitre, Apr 18 2002

Keywords

Comments

Previous name: sum(d|n,6d/(2+mu(d))) is odd, where mu(.) is the Moebius function, A008683.
From Peter Munn, Jul 06 2020: (Start)
Numbers that have an odd number of odd nonsquarefree divisors.
[Proof of equivalence to the name, where m denotes a positive integer:
(1) These properties are equivalent: (a) m has an even number of odd squarefree divisors; (b) m has a nontrivial odd part.
(2) These properties are equivalent: (a) m has an odd number of odd divisors; (b) the odd part of m is square.
(3) m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are both true or both false.
(4) The trivial odd part, 1, is a square, so (1)(b) and (2)(b) cannot both be false, which (from (1), (2)) means (1)(a) and (2)(a) cannot both be false.
(5) From (3), (4), m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are true.
(6) m satisfies the condition in the name if and only if (1)(b) and (2)(b) are true, which (from (1), (2)) is equivalent to (1)(a) and (2)(a) being true, and hence from (5), to m satisfying the condition at the start of this comment.]
(End)
Numbers whose sum of non-unitary divisors (A048146) is odd. - Amiram Eldar, Sep 16 2024

Examples

			To determine the odd part of 18, remove all factors of 2, leaving 9. 9 is a nontrivial square, so 18 is in the sequence. - _Peter Munn_, Jul 06 2020

Links

Crossrefs

A000265, A008683 are used in definitions of this sequence.
Lists of numbers whose odd part satisfies other conditions: A028982 (square), A028983 (nonsquare), A029747 (less than 6), A029750 (less than 8), A036349 (even number of prime factors), A038550 (prime), A070776 U {1} (power of a prime), A072502 (square of a prime), A091067 (has form 4k+3), A091072 (has form 4k+1), A093641 (noncomposite), A105441 (composite), A116451 (greater than 4), A116882 (less than or equal to even part), A116883 (greater than or equal to even part), A122132 (squarefree), A229829 (7-rough), A236206 (11-rough), A260488\{0} (has form 6k+1), A325359 (proper prime power), A335657 (odd number of prime factors), A336101 (prime power).
Cf. A048146, A091476.

Programs

  • Mathematica
    Select[Range[1000], (odd = #/2^IntegerExponent[#, 2]) > 1 && IntegerQ @ Sqrt[odd] &] (* Amiram Eldar, Sep 29 2020 *)
  • PARI
    upto(n) = { my(res = List()); forstep(i = 3, sqrtint(n), 2, for(j = 0, logint(n\i^2, 2), listput(res, i^2<David A. Corneth, Sep 28 2020

Formula

Sum_{n>=1} 1/a(n) = 2 * Sum_{k>=1} 1/(2*k+1)^2 = Pi^2/4 - 2 = A091476 - 2 = 0.467401... - Amiram Eldar, Feb 18 2021

Extensions

New name from Peter Munn, Jul 06 2020
Showing 1-3 of 3 results.

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