A066417 -id:A066417 - cp's OEIS frontend

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

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.

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

A093394 a(n) is the GCD of n and the product of the anti-divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 15, 1, 1, 6, 1, 4, 21, 2, 1, 8, 5, 2, 27, 4, 1, 30, 1, 1, 33, 2, 35, 12, 1, 2, 39, 8, 1, 42, 1, 4, 45, 2, 1, 16, 7, 10, 51, 4, 1, 54, 55, 8, 57, 2, 1, 60, 1, 2, 63, 1, 65, 66, 1, 4, 69, 70, 1, 24, 1, 2, 75, 4

Offset: 3

Lior Manor, Mar 28 2004

nonn

See A066272 for definition of anti-divisor.

			The anti-divisors of 18 are 4, 5, 7, 12. Hence a(18) = gcd(4*5*7*12, 18) = 6.

Cf. A066417, A091507, A093395, A093396.

a(n) = gcd(n, A091507(n)).

A093395 Numerators of n divided by the product of the anti-divisors of n.

Original entry on oeis.org

3, 4, 5, 3, 7, 8, 3, 5, 11, 3, 13, 7, 1, 16, 17, 3, 19, 5, 1, 11, 23, 3, 5, 13, 1, 7, 29, 1, 31, 32, 1, 17, 1, 3, 37, 19, 1, 5, 41, 1, 43, 11, 1, 23, 47, 3, 7, 5, 1, 13, 53, 1, 1, 7, 1, 29, 59, 1, 61, 31, 1, 64, 1, 1, 67, 17, 1, 1, 71, 3, 73, 37, 1, 19

Offset: 3

Lior Manor, Mar 28 2004

See A066272 for definition of anti-divisor.

			The anti-divisors of 18 are 4, 5, 7, 12. Hence a(18) = 18/GCD(4*5*7*12, 18) = 3.

Cf. A066417, A091507, A093394, A093396 (denominators).

a(n) = n/GCD(n, A091507(n)) = n/A093394(n)

Name changed by Franklin T. Adams-Watters, Aug 21 2013

A191581 Numbers whose sum of their anti-divisors divides the sum of their divisors.

Original entry on oeis.org

3, 6, 11, 22, 30, 33, 65, 82, 117, 218, 354, 483, 508, 537, 3276, 6430, 21541, 117818, 130356, 753612, 1007328, 2113416, 2379540, 3589646, 7231219, 7346148, 8515767, 13050345, 20199648, 34424166, 44575896, 47245905, 50414595, 104335023, 217728002, 1217532421

Offset: 1

Paolo P. Lava, Jun 07 2011

A161917 is a subsequence of this sequence.

			6-> sum divisors=sigma(6)=12; sum anti-divisors=4; 12/4=3.
30-> sum divisors=sigma(30)=72; sum anti-divisors=4+12+20=36; 72/36=2.

Cf. A000203, A066272, A066417, A161917, A191580.

with(numtheory): P:=proc(i) local a, b, j, k, s, n;
for n from 3 to i do b:=divisors(n); s:=add(b[k],k=1..nops(b));
k:=0; j:=n; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
if type(s/a,integer) then print(n); fi; od; end: P(10^6);

f[n_] := Total@ Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[Range[3, 10^3], Mod[DivisorSigma[1, #], f@ #] == 0 &] (* _Michael De Vlieger, Oct 08 2015 *)

{n: A066417(n) | A000203(n)}. - R. J. Mathar, Oct 01 2011

a(21)-a(36) from Donovan Johnson, Jun 24 2012

A192266 Decimal expansion of Sum_{k >= 1} 1/k^sigma_(k) where sigma_(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

2, 1, 2, 7, 8, 2, 7, 8, 0, 2, 4, 2, 5, 0, 7, 1, 7, 8, 3, 0, 4, 4, 1, 3, 1, 7, 4, 6, 9, 6, 6, 0, 9, 9, 2, 6, 2, 4, 5, 0, 7, 7, 3, 5, 3, 0, 8, 3, 4, 1, 9, 8, 9, 7, 3, 0, 9, 4, 3, 0, 6, 8, 3, 7, 1, 7, 1, 8, 7, 1, 8, 2, 8, 4, 3, 0, 3, 2, 7, 1, 4, 2, 5, 6, 4, 8

Offset: 1

Paolo P. Lava, Jun 27 2011

Continued fraction (2,7,1,4,1,1,1,6,4,1,11,1,2...).

			1/1^sigma*(1)+ 1/2^sigma*(2) + 1/3^sigma*(3) + 1/4^sigma*(4) + 1/5^sigma*(5) + 1/6^sigma*(6) + ... = 1/1^0 + 1/2^0 + 1/3^2 + 1/4^3 + 1/5^5 + 1/6^4 + ... = 2.12782780242507..

Cf. A066417, A192265.

with(numtheory): P:=proc(i) local a,j,k,n,s; d:=2; for n from 3 to i do k:=0; j:=n;
while j mod 2 <> 1 do k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
d:=d+1/n^a; od; print(evalf(d, 300)); end: P(100);

f[n_] := Total@ Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; First@ RealDigits@ N[Sum[1/k^f@ k, {k, 120}], 86] (* _Michael De Vlieger, Oct 08 2015 *)

Corrected and edited by R. J. Mathar, Jun 27 2011

A192269 Super anti-abundant numbers.

Original entry on oeis.org

1, 3, 4, 5, 7, 13, 17, 32, 38, 45, 67, 77, 143, 203, 247, 473, 682, 787, 1463, 2678, 2992, 3465, 8662, 10868, 16065, 25987, 26163, 29452, 112613, 157658, 202702, 233415, 363825, 795217, 1148647, 1914412, 2139637, 5743237, 5743238, 8393963, 11869357, 64353712

Offset: 1

Paolo P. Lava, Jun 28 2011

nonn

Like A004394 but using anti-divisors. A super anti-abundant number is a number n such that sigma*(n)/n > sigma*(k)/k for all kA066417(n)/n.

			1 -> sigma*(1)/1 = 0/1 = 0;
3 -> sigma*(3)/3 = 2/3 = 0.6666...;
4 -> sigma*(4)/4 = 3/4 = 0.75;
5 -> sigma*(5)/5 = 5/5 = 1;
7 -> sigma*(7)/7 = 10/7 = 1.4285...; etc.

Jud McCranie, Table of n, a(n) for n = 1..66

Cf. A004394, A066417, A192268.

with(numtheory); P:= proc(n) local a,k,i,j,s; s:=0; print(1);
for i from 3 to n do
k:=0; j:=i; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; a:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2;
if a/i>s then s:=a/i; print(i); fi; od; end: P(50000);

a(26)-a(42) from Donovan Johnson, Sep 07 2011

A192818 Numbers which are both deficient (A005100) and anti-deficient (A192267).

Original entry on oeis.org

1, 2, 3, 4, 9, 16, 19, 26, 29, 34, 44, 51, 61, 64, 69, 79, 89, 106, 131, 134, 139, 141, 146, 159, 166, 169, 191, 194, 201, 209, 211, 219, 226, 236, 239, 244, 251, 254, 261, 271, 274, 289, 296, 299, 309, 316, 321, 334, 339, 341, 344, 349, 359, 376, 381, 386

Offset: 1

Jonathan Vos Post, Jul 10 2011

			24 is anti-deficient because its anti-divisors are 7, 16 and their sum is 23 < 24.  26 is deficient because its proper divisors are 1, 2, 13 which sum to 16 and 16 < 26.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1500 from Nathaniel Johnston)

Cf. A005100, A066417, A192267.

q[n_] := Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]] < n && DivisorSigma[1, n] < 2*n; Select[Range[300], q] (* _Amiram Eldar, Jan 13 2022 after Michael De Vlieger at A066417 *)

A005100 INTERSECTION A192267.

More terms and inserted a(1)=1 from Nathaniel Johnston, Sep 26 2011

cp's OEIS Frontend

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

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

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

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

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A093394 a(n) is the GCD of n and the product of the anti-divisors of n.

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A093395 Numerators of n divided by the product of the anti-divisors of n.

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A191581 Numbers whose sum of their anti-divisors divides the sum of their divisors.

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A192266 Decimal expansion of Sum_{k >= 1} 1/k^sigma_*(k) where sigma_*(n) is the sum of the anti-divisors of n.

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A192269 Super anti-abundant numbers.

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A192266 Decimal expansion of Sum_{k >= 1} 1/k^sigma_(k) where sigma_(n) is the sum of the anti-divisors of n.