A066272 -id:A066272 - cp's OEIS frontend

A242966 Composite numbers whose anti-divisors are all primes.

4, 8, 16, 64, 1024, 4096, 65536, 262144, 4194304, 1073741824, 1152921504606846976, 1267650600228229401496703205376, 85070591730234615865843651857942052864, 93536104789177786765035829293842113257979682750464

Offset: 1

Paolo P. Lava, May 28 2014

nonn

It appears they are all powers of 2.
Subset of A242965.
a(n) must be 2^k. - Hiroaki Yamanouchi, Mar 17 2015
The exponents are 2, 3, 4, 6, 10, 12, 16, 18, 22, 30, 60, 100, 126, 166, 198, ... - Michel Marcus, Mar 18 2015

			The anti-divisors of 1024 are all primes: 3, 23, 89, 683.
The same for 65536: 3, 43691.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..15

Cf. A066272, A242965.

P := proc(q) local k,ok,n; for n from 3 to q do if not isprime(n)
then ok:=1; for k from 2 to n-1 do if abs((n mod k)-k/2)<1
then if not isprime(k) then ok:=0; break; fi; fi; od;
if ok=1 then print(n); fi; fi; od; end: P(10^100);

antiDivisors[n_] := Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[2^Range[2, 20], AllTrue[antiDivisors@ #, PrimeQ] &] (* _Michael De Vlieger, Mar 18 2015 *)

from sympy import isprime, divisors
A242966 = [n for n in range(3,10**5) if not isprime(n) and list(filter(lambda x: not isprime(x), [2*d for d in divisors(n) if n > 2*d and n % (2*d)] + [d for d in divisors(2*n-1) if n > d >=2 and n % d] + [d for d in divisors(2*n+1) if n > d >=2 and n % d])) == []]
# Chai Wah Wu, Aug 13 2014

a(11)-a(14) from Hiroaki Yamanouchi, Mar 17 2015

A066416 Number of numbers m such that the sum of the anti-divisors of m is n+1.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 2, 1, 0, 0, 1, 1, 3, 0, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 3, 0, 0, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 1, 3, 1, 3, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2

Offset: 1

Jon Perry, Dec 28 2001

nonn

See A066272 for definition of anti-divisor.

			8 has anti-divisors 1, 3 and 5, whose sum is 9 and 9 has anti-divisors 1, 2 and 6, whose sum is 9 and there are no others. Therefore a(8)=2.

Jon Perry, Anti-divisor
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A066417, A066418, A058838, A066241.

A066464 Least number k such that k has n anti-divisors.

Original entry on oeis.org

1, 3, 5, 7, 13, 17, 32, 38, 85, 67, 162, 137, 338, 203, 760, 247, 1225, 472, 578, 682, 1012, 787, 9112, 1463, 2048, 2047, 2888, 2363, 5513, 3465, 5512, 6682, 8978, 5197, 17672, 5198, 71442, 9653, 29768, 8662, 40898, 13513, 81608, 15593, 131072, 35437

Offset: 0

Robert G. Wilson v, Jan 02 2002

nonn

See A066272 for definition of anti-divisor.

Donovan Johnson, Table of n, a(n) for n = 0..400

Cf. A066272.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n & ]; a = Table[0, {50} ]; Do[ b = Length[ antid[n]]; If[ b < Length[a] && a[[b + 1]] == 0, a[[b + 1]] = n], {n, 1, 2^17} ]; a

A066482 The smallest anti-divisor of n.

Original entry on oeis.org

2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 8, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 8, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 64, 2, 3, 2, 3

Offset: 3

Robert G. Wilson v, Jan 02 2002

nonn

Almost identical to A007978, least non-divisor of n, but there are some subtle differences.

See A066272 for definition of anti-divisor.

Seiichi Manyama, Table of n, a(n) for n = 3..10000

Cf. A066481.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 &], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 &], 2n/Select[ Divisors[2*n], OddQ[ # ] && # != 1 &]]], # < n & ]; Table[ First[ antid[n]], {n, 3, 100} ]

A066519 Gaps between successive numbers with an anti-divisor class sum of zero.

Original entry on oeis.org

1, 1, 3, 3, 6, 2, 4, 7, 2, 6, 7, 3, 1, 4, 3, 8, 7, 2, 1, 3, 5, 10, 2, 1, 3, 3, 2, 1, 5, 1, 1, 1, 4, 4, 2, 2, 2, 9, 2, 6, 9, 1, 1, 4, 4, 1, 3, 6, 1, 3, 22, 1, 9, 1, 1, 2, 2, 4, 7, 3, 5, 4, 1, 2, 20, 1, 2, 6, 1, 4, 4, 9, 5, 1, 4, 5, 2, 7, 8, 2, 2, 9, 2, 2, 1, 5, 3, 1, 4, 1, 12, 16, 13, 5, 1, 9, 2, 1, 3, 3

Offset: 1

Jon Perry, Jan 06 2002

nonn

See A066272 for definition of anti-divisor.

			f(1)=f(2)=f(3)=0, f(4)=1, f(5)=-1, f(6)=0, so the first 3 gaps are 1, 1, 3.

Jon Perry, Class sums

Cf. A066518.

a[ n_ ] := Sum[ Which[ Mod[ n, d ]==(d-1)/2, -1, Mod[ n, d ]==(d+1)/2, 1, True, 0 ], {d, 2, n-1} ]; z=Select[ Range[ 1, 500 ], a[ # ]==0& ]; Drop[ z, 1 ]-Drop[ z, -1 ]

Edited by Dean Hickerson, Jan 17 2002

A066542 Nonnegative integers all of whose anti-divisors are either 2 or odd.

Original entry on oeis.org

3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251

Offset: 1

John W. Layman, Jan 07 2002

nonn

See A066272 for definition of anti-divisor.
The following conjectures have been proved by Bob Selcoe. - Michael Somos, Feb 28 2014
Additional conjectures suggested by computational experiments:
1) Numbers all of whose anti-divisors (AD's) are odd => {2^k} (A000079).
2) Numbers with AD 2, all other AD's odd => primes (A000040).
3) Numbers none of whose AD's are multiples of 3 => 3*2^k (A007283).
4) Numbers all of whose AD's are even => 3*A002822 = A040040 (except for a(0)=1), both related to twin prime pairs.
Calculations suggest the following conjecture. This sequence consists of all odd primes and nonnegative powers of 2 and no other terms. This has been verified for to n=100000. Robert G. Wilson v extended the conjecture out to 2^20.
From Bob Selcoe, Feb 24 2014: (Start)
The sequence consists of all odd primes and powers of two (>=2^2) and no other terms.
Proof: Denote the even anti-divisors of n as ADe(n). ADe(n) is defined as the set of numbers x satisfying the equation n(mod x)=x/2. Substitute x = 2n/y, since it can be shown that ADe(n) => 2n divided by the odd divisors of n when n>1 (This is because 2j anti-divides only numbers of the form 3j+2j*k; j>=1, k>=0.  For example: j=7; 14 anti-divides only 21,35,49,63.... So in other words, even numbers anti-divide only odd multiples (>=3) of themselves, divided by 2). Therefore, ADe(n) is n(mod [2n/y])=n/y, and y must be an odd divisor of n and 2n, y>1.  Since y is the only odd divisor of n when y>1 iff n is prime, then ADe(n) => 2 when n is prime.  Since 2n has no odd divisors when n=2^k, then ADe(n) is null when n=2^k.  Therefore, the only numbers whose anti-divisors are either 2 or odd must be primes and powers of 2.
Similarly, for odd anti-divisors (ADo(n)): Given 2j+1 (odd numbers) anti-divide only numbers of the forms [(3j+1)+(2j+1)*k] and [(3j+2)+(2j+1)*k]; j>=1, k>=0. (For example: j=6; 13 anti-divides only 19,20, 32,33, 45,46...). Since odd n divided by its odd divisors ARE its odd divisors, then ADo(n) => the divisors of 2n-1 and 2n+1 (except 1, 2n-1 and 2n+1).
By extension:
1) Numbers all of whose anti-divisors (AD's) are odd => {2^k} (A000079).
2) Numbers with ADe(n)=2, all other AD's odd => primes (A000040).
3) Numbers none of whose AD's are multiples of j => j*2^k.
4) When 2n-1 and 2n+1 are twin primes, (A040040, except for a(0)=1) then n has only even AD's.
(End)
If 1 and 2 are included, this sequence contains all positive integers not contained in A111774. - Bob Selcoe, Sep 09 2014 [corrected by Wolfdieter Lang, Nov 06 2020]

			From _Bob Selcoe_, Feb 24 2014: (Start)
ADe(420): Odd divisors of 420 are: 3,5,7,15,21,35, 105. ADe(420) => 840/{3,5,7,15,21,35,105} = 8,24,40,56,120,168 and 280.
ADo(420) => the divisors of 839 and 841, which are (a) for 839: null (839 is prime); and (b) for 841: 29 (841 is 29^2).
All AD's (AD(420)) => 8,24,29,40,56,120,168 and 280 (End)

Cf. A000040, A000079, A002822, A007283, A040040.

Cf. A014545, A006862, A111774.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 &], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 &], 2n / Select[ Divisors[2*n], OddQ[ # ] && # != 1 &]]], # < n & ]; f[n_] := Select[ antid[n], EvenQ[ # ] && # > 2 & ]; Select[ Range[3, 300], f[ # ] == {} & ]

A073694 Numbers k such that the number of divisors of k equals the number of anti-divisors of k.

Original entry on oeis.org

5, 32, 50, 162, 512, 1984, 2450, 3784, 5408, 7564, 9248, 15488, 19208, 22684, 26680, 30752, 53792, 79600, 85698, 102604, 113764, 131584, 189112, 199712, 279752, 336200, 435244, 514098, 546012, 581042, 658952, 712818, 727218, 752764, 767560

Offset: 1

Jason Earls, Aug 30 2002

nonn

See A066272 for definition of anti-divisor.

			32 is here since it has 6 divisors: {1, 2, 4, 8, 16, 32} and 6 anti-divisors: {3, 5, 7, 9, 13, 21}.

Vincenzo Librandi and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 227 terms from Vincenzo Librandi)

Cf. A000005, A066272.

atd[n_] := Count[Flatten[Quotient[#, Rest[Select[Divisors[#], OddQ]]] & /@ (2 n + Range[-1, 1])], Except[1]]; Select[Range[770000], DivisorSigma[0, #] == atd[#] &] (* Jayanta Basu, Jul 06 2013 *)

{for(n=1,770000,v1=[]; v2=[]; v3=[]; ds=divisors(2*n-1); for(k=2,matsize(ds)[2]-1, if(ds[k]%2>0,v1=concat(v1,ds[k]))); ds=divisors(2*n); for(k=2,matsize(ds)[2]-1,if(ds[k]%2>0, v2=concat(v2,ds[k]))); ds=divisors(2*n+1); for(k=2,matsize(ds)[2]-1,if(ds[k]%2>0,v3=concat(v3,ds[k]))); v=vecsort(concat(v1,concat(v2,v3))); if(matsize(v)[2]==numdiv(n),print1(n,",")))}

Edited and extended by Klaus Brockhaus, Sep 01 2002

A073931 Numbers n such that the sum of the anti-divisors of n = 2n.

Original entry on oeis.org

77, 1568, 2768, 4775040

Offset: 1

Jason Earls, Sep 03 2002

See A066272 for definition of anti-divisor.

a(5) > 10^10 - Hiroaki Yamanouchi, Mar 18 2015

Cf. A066417.

antiDivisorSum[n_] := Total[Select[Range[2, n - 1], Abs[Mod[n, #] - #/2] < 1 &]]
Select[Range[1, 1600], antiDivisorSum[#] == 2*# &] (* Julien Kluge, Sep 19 2016 *)

from sympy import divisors
A073931 = [n for n in range(3,10**5) if sum([2*d for d in divisors(n) if n > 2*d and n % (2*d)] + [d for d in divisors(2*n-1) if n > d >=2 and n % d] + [d for d in divisors(2*n+1) if n > d >=2 and n % d]) == 2*n]
# Chai Wah Wu, Aug 13 2014

A074751 Numbers k such that the sum of the anti-divisors of k = sum of proper divisors (or aliquot parts) of k.

Original entry on oeis.org

1, 4, 44, 260, 1350, 6284, 6954, 13364, 273366, 333546, 466614, 4659934050

Offset: 1

Jason Earls, Sep 06 2002

Integers k such that A066417(k) = A001065(k)

a(13) > 10^10. - Hiroaki Yamanouchi, Mar 18 2015

Cf. A001065, A066417, A066272.

spd(n) = if( n==0, 0, sigma(n) - n); \\ A001065
sad(n) = my(k); if(n>1, k=valuation(n, 2); sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2, 0); \\ A066417
isok(k) = sad(k) == spd(k); \\ Michel Marcus, Mar 30 2025

a(12) from Hiroaki Yamanouchi, Mar 18 2015

A074898 Impossible values for sum of anti-divisors of n.

Original entry on oeis.org

1, 6, 7, 9, 11, 15, 17, 20, 21, 25, 26, 27, 29, 31, 33, 35, 37, 38, 43, 44, 45, 47, 49, 51, 53, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 75, 77, 79, 81, 82, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 103, 105, 109, 111, 113, 115, 117, 119, 120, 121, 123, 125, 127, 128, 129, 131, 133, 134, 135, 137, 139, 141, 143, 145, 146, 149, 151, 153, 155, 157, 158, 159, 161, 163, 165, 167, 168, 169, 170, 171

Offset: 1

Jason Earls, Sep 14 2002

nonn

See A066272 for definition of anti-divisor.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A005114, A066417.

More terms from Paolo P. Lava, Jul 06 2011

cp's OEIS Frontend

A242966 Composite numbers whose anti-divisors are all primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A066416 Number of numbers m such that the sum of the anti-divisors of m is n+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A066464 Least number k such that k has n anti-divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A066482 The smallest anti-divisor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A066519 Gaps between successive numbers with an anti-divisor class sum of zero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A066542 Nonnegative integers all of whose anti-divisors are either 2 or odd.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A073694 Numbers k such that the number of divisors of k equals the number of anti-divisors of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A073931 Numbers n such that the sum of the anti-divisors of n = 2n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica