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.

A029747 Numbers of the form 2^k times 1, 3 or 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 4096, 5120, 6144, 8192, 10240, 12288, 16384, 20480, 24576, 32768, 40960, 49152, 65536, 81920, 98304, 131072, 163840, 196608
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Fixed points of the Doudna sequence: A005940(a(n)) = A005941(a(n)) = a(n). - Reinhard Zumkeller, Aug 23 2006
Subsequence of A103969. - R. J. Mathar, Mar 06 2010
Question: Is there a simple proof that A005940(c) = c would never allow an odd composite c as a solution? See also my comments in A163511 and in A335431 concerning similar problems, also A364551 and A364576. - Antti Karttunen, Jul 28 & Aug 11 2023

Examples

			128 = 2^7 * 1 is in the sequence as well as 160 = 2^5 * 5. - _David A. Corneth_, Sep 18 2020

Links

Crossrefs

Cf. A122132, A000079, A007283, A020714, A130793.
Cf. A005940, A005941, A163511, A335431, A364499, A364551, A364576.
Subsequence of the following sequences: A103969, A253789, A364541, A364542, A364544, A364546, A364548, A364550, A364560, A364565.
Even terms form a subsequence of A320674.

Programs

  • Mathematica
    m = 200000; Select[Union @ Flatten @ Outer[Times, {1, 3, 5}, 2^Range[0, Floor[Log2[m]]]], # < m &] (* Amiram Eldar, Oct 15 2020 *)
  • PARI
    is(n) = n>>valuation(n, 2) <= 5 \\ David A. Corneth, Sep 18 2020
  • Python
    def A029747(n):
    if n<3: return n
    a, b = divmod(n,3)
    return 1<Chai Wah Wu, Apr 02 2025

Formula

a(n) = if n < 6 then n else 2*a(n-3). - Reinhard Zumkeller, Aug 23 2006
G.f.: (1+x+x^2)^2/(1-2*x^3). - R. J. Mathar, Mar 06 2010
Sum_{n>=1} 1/a(n) = 46/15. - Amiram Eldar, Oct 15 2020

Extensions

Edited by David A. Corneth and Peter Munn, Sep 18 2020

A364558 a(n) = A364557(n) - A000010(n), where A364557 is the Möbius transform of A005941, and A000010 (Euler phi) is the Möbius transform of A000027.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 6, 0, 20, 2, -4, 0, 48, -2, 110, 0, -4, 6, 234, 0, -12, 20, -10, 4, 484, -4, 994, 0, -4, 48, -16, -4, 2012, 110, 8, 0, 4056, -4, 8150, 12, -16, 234, 16338, 0, -26, -12, 32, 40, 32716, -10, -24, 8, 92, 484, 65478, -8, 131012, 994, -20, 0, -16, -4, 262078, 96, 212, -16, 524218, -8, 1048504
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2023

Keywords

Links

Crossrefs

Cf. A000010, A005941, A364557, A364559 (inverse Möbius transform), A364565 (positions of 0's), A364566 (of terms < 0).

Programs

  • PARI
    A364558(n) = (A364557(n)-eulerphi(n));
  • Python
    from math import prod
from sympy import factorint, primepi
def A364558(n): return (1<1 else 0 # Chai Wah Wu, Jul 29 2023

A364566 Numbers k such that A364557(k) < A000010(k), where A364557 is the Möbius transform of A005941, and A000010 is Euler totient function phi.

Original entry on oeis.org

9, 15, 18, 21, 25, 27, 30, 33, 35, 36, 42, 45, 49, 50, 54, 55, 60, 63, 65, 66, 70, 72, 75, 77, 81, 84, 90, 91, 98, 99, 100, 105, 108, 110, 117, 119, 120, 121, 125, 126, 130, 132, 135, 140, 143, 144, 147, 150, 154, 162, 165, 168, 169, 175, 180, 182, 187, 189, 195, 196, 198, 200, 209, 210, 216, 220, 221, 225, 231
Offset: 1

Views

Author

Antti Karttunen, Jul 29 2023

Keywords

Crossrefs

Positions of negative terms in A364558.
Cf. A000010, A364557, A364565.

Programs

  • PARI
    A364557(n) = if(1==n, 1, 2^(primepi(vecmax(factor(n)[, 1]))+(bigomega(n)-omega(n))-1));
isA364566(n) = (A364557(n)

A364962 Odd numbers k such that A005941(k) is either k itself or its descendant in Doudna-tree, A005940.

Original entry on oeis.org

1, 3, 5, 11, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 127
Offset: 1

Views

Author

Antti Karttunen, Aug 14 2023

Keywords

Comments

Questions: Is 85 the only composite in this sequence? (See also A364565). Are there any more terms after 127, or is the sequence finite?

Examples

			85 = 5*17 is a term, because A005941(85) = 133 = 7*19 = A003961(85), thus 133 is a left hand side child of 85 in the tree depicted in A005940, and therefore 85 is included in this sequence. (See also the last example in A364959).

Crossrefs

Cf. A005940, A005941, A029747, A156552, A252464.
Cf. also A364959, A364961.

Programs

  • PARI
    A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA364962(n) = if(!(n%2),0,my(k=A005941(n)); while(k>n, k = A252463(k)); (k==n));
Showing 1-4 of 4 results.

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

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