A066466 -id:A066466 - cp's OEIS frontend

A092679 Numbers k such that 3*2^k has only one anti-divisor.

0, 1, 5, 17

Offset: 1

Lior Manor, Mar 03 2004

Next term should be greater than 3*10^6 (cf. A181490).

See A066272 for definition of anti-divisor.

A092679 = [i for i,n in enumerate(map(lambda x:3*2**x,range(20))) if len([d for d in range(2,n,2) if n%d and not 2*n%d]+[d for d in range(3,n,2) if n%d and 2*n%d in [d-1,1]])==1] # Chai Wah Wu, Aug 09 2014

A092680(n) = 3*2^a(n).

a(n) = A181490(n) - 1. - Max Alekseyev, Feb 14 2025

A092680 Numbers of the form 3*2^k with a single anti-divisor.

Original entry on oeis.org

3, 6, 96, 393216

Offset: 1

Lior Manor, Mar 03 2004

See A066272 for definition of anti-divisor.

If it exists, a(5) > 3*2^(1000). See A092679. - J.W.L. (Jan) Eerland, Nov 13 2022

Cf. A066272, A066466, A092679, A181490, A181491, A181492.

from itertools import count, islice
from sympy.ntheory.factor_ import antidivisor_count
def A092680_gen(): return filter(lambda n: antidivisor_count(n)==1,(3*2**k for k in count(0)))
A092680_list = list(islice(A092680_gen(),4)) # Chai Wah Wu, Jan 04 2022

a(n) = 3*2^A092679(n).

a(n) = 3*2^(A181490(n)-1) = (A181491(n)+1)/2 = (A181492(n)-1)/2. - Max Alekseyev, Feb 14 2025

A203616 Numbers k such that the reversal of sigma(k) equals the sum of the reversals of the anti-divisors of k, where sigma(k) is the sum of the anti-divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 20, 63, 96, 97, 317, 596, 1473, 3934, 26777, 27684, 50867, 51767, 62417, 322001, 393216, 1308775, 1420260, 1851474, 2651867, 2659067, 3040656, 3227267, 3289277, 3376007, 4626917, 4639067, 5378507, 6054521, 6227027, 6239839, 6439067, 6581929

Offset: 1

Paolo P. Lava, Jan 20 2012

A066466 is a subsequence of this sequence.

			n=317. Anti-divisors: 2, 3, 5, 127, 211.
Sum of the reversals of the anti-divisors: 2+3+5+721+112=843.
Sigma*(317)=348 and its reversal is 843.
n=1473. Anti-divisors: 2, 5, 6, 7, 19, 31, 95, 155, 421, 589, 982.
Sum of the reversals of the anti-divisors:
2+5+6+7+91+13+59+551+124+985+289=2132.
Sigma*(1473)=2312 and its reversal is 2132.

Dumitru Damian, Table of n, a(n) for n = 1..235 (terms up to 10^9)

Cf. A004086, A066272, A066466, A069942, A195144, A203615.

isA203616:=proc(j) local a,b,c;   a:=0; b:=0;
   for c from 2 to j-1 do
     if abs((j mod c)-c/2)<1 then a:=a+A004086(c); b:=b+c; fi;
   od;
evalb(A004086(b)=a) end: # simplified by M. F. Hasler, Jan 29 2012
for n to 10^7 do if isA203616(n) then lprint(n) fi od: # simplified by M. F. Hasler, Jan 29 2012

from itertools import count, islice
from sympy.ntheory.factor_ import antidivisors
def a203616():
    isa = lambda n: str(sum((a:=antidivisors(n))))[::-1]==str(sum(map(int, (str()[::-1] for  in a))))
    yield from (n for n in count(1) if isa(n))
a203616_list = [*islice(a203616(), 20)] # Dumitru Damian, Feb 12 2024

a(22)-a(40) from Dumitru Damian, Feb 12 2024

A131489 Partial products of A092680.

Original entry on oeis.org

3, 18, 1728, 679477248

Offset: 1

Jonathan Vos Post, Jul 28 2007

nonn

Max Alekseyev points out that every term of A066466, except 4, must be of the form 3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. There are no such new k+1 (i.e., except known 1,2,6,18) below 1000. In other words, 3*2^n - 1, 3*2^n + 1 are twin primes for n=1,2,6,18. According to these tables in the Keller links there are no other such n up to 18*10^6. Therefore the next term of A066466 (if it exists) is greater than 3*2^(18*10^6) ~= 10^5418540. Hence the next element of the anti-primorials (if it exists) is greater than 679477248 * 10^5418540 > 10^5418548. [Updated by Max Alekseyev, May 23 2023]

			a(1) = 3.
a(2) = 3 * 6 = 18.
a(3) = 3 * 6 * 96 = 1728.
a(4) = 3 * 6 * 96 * 393216 = 679477248.

Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
Wilfrid Keller, List of primes k.2^n - 1 for k < 300

Cf. A092680.

a(n) = Product_{k=1..n} A092680(k).

A263940 Numbers such that the product of the sum of their anti-divisors and the sum of the reciprocals of their anti-divisors is an integer.

Original entry on oeis.org

3, 4, 6, 37, 96, 937, 2760, 393216

Offset: 1

Paolo P. Lava, Oct 30 2015

nonn

A066466 is a subset of this sequence.

The sums are 1, 1, 1, 57, 1, 1457, 385, 1, ...

			Anti-divisors of 937 are 2, 3, 5, 15, 25, 75, 125, 375 and 625. Their sum is 1250 while the sum of their reciprocals is 1/2 + 1/3 + 1/5 + 1/15 + 1/25 + 1/75 + 1/125 + 1/375 + 1/625 = 1457/1250. Finally 1250 * 1457/1250 = 1457.

Cf. A066272, A066466, A263928.

with(numtheory); P:=proc(q) local a,b,k,n;
for n from 3 to q do a:=0; b:=0;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; b:=b+1/k;
fi; od; if type(a*b,integer) then print(n); fi; od; end: P(10^4);

f[n_] := Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[Range[3, 3000], IntegerQ@ Times[Total@ f@ #, Total[1/f@ #]] &] (* _Michael De Vlieger, Nov 11 2015 *)

cp's OEIS Frontend

A092679 Numbers k such that 3*2^k has only one anti-divisor.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

Formula

A092680 Numbers of the form 3*2^k with a single anti-divisor.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

Formula

A203616 Numbers k such that the reversal of sigma*(k) equals the sum of the reversals of the anti-divisors of k, where sigma*(k) is the sum of the anti-divisors of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

Extensions

A131489 Partial products of A092680.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A263940 Numbers such that the product of the sum of their anti-divisors and the sum of the reciprocals of their anti-divisors is an integer.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A203616 Numbers k such that the reversal of sigma(k) equals the sum of the reversals of the anti-divisors of k, where sigma(k) is the sum of the anti-divisors of k.