A066417 -id:A066417 - cp's OEIS frontend

A372753 Integers m, with k digits, such that m = Sum_{i=1..k} A066417(m without its i-th digit).

1528, 2970, 19486, 26062, 27670, 37898, 144186, 290814, 331884, 442598, 1010400, 2758596, 4046432, 4270842, 5518368, 5805182, 6826434, 7231218

Offset: 1

Paolo P. Lava, Jul 28 2024

Cf. A066417, A374308, A374309, 374368.

with(numtheory): P:=proc(q) local a,b,d,i,j,k,n,t;
for n from 1 to q do t:=0; a:=convert(n,base,10);
for b from 1 to nops(a) do d:=0;
for i from 1 to nops(a) do if i<>b then d:=d*10+a[-i]; fi; od; k:=0; j:=d;
if d>0 then while j mod 2<>1 do k:=k+1; j:=j/2; od;
t:=sigma(2*d+1)+sigma(2*d-1)+sigma(d/2^k)*2^(k+1)-6*d-2+t; fi; od;
if n=t then print(n); fi; od; end: P(10^8);

A066980 Numbers k such that antid(k) > antid(k+1) and santid(k) > santid(k+1), where antid(k) = A066272 and santid(k) = A066417.

Original entry on oeis.org

5, 7, 13, 15, 18, 23, 25, 28, 32, 33, 35, 38, 43, 45, 47, 50, 53, 55, 60, 63, 67, 68, 73, 77, 78, 83, 85, 88, 95, 98, 105, 108, 110, 113, 115, 117, 123, 127, 128, 130, 133, 137, 138, 140, 143, 145, 150, 153, 155, 158, 162, 163, 165, 168, 172, 173, 175, 176, 178, 182

Offset: 1

Jason Earls, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A066272, A066417.

A230373 Numbers n such that sigma(sigma(n)) = sigma(sigma(n)), where sigma*(n) is the sum of anti-divisors of n (A066417).

Original entry on oeis.org

3, 265, 450, 1989, 18278, 31639, 55474, 71306, 96639, 197518, 267026, 1620723, 1888235, 3605481, 4448715, 10837215, 12128451, 22598820, 84681074, 96503379, 130118331, 152234714, 162138375, 189149834, 211239421, 343379954, 353833749, 404994939, 599244123, 804486314

Offset: 1

Paolo P. Lava, Oct 23 2013

			Divisors of 450 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450 and sigma(450) = 1209; anti-divisors of 1209 are 2, 6, 26, 41, 59, 62, 78, 186, 806 and sigma*(1209) = 1266.
Anti-divisors of 450 are 4, 12, 17, 20, 29, 31, 36, 53, 60, 100, 180, 300 and sigma*(450) = 842; divisors of 842 are 1, 2, 421, 842 and sigma(842) = 1266.
Therefore 450 is part of the sequence because sigma(sigma*(450)) = sigma*(sigma(450)) = 1266.

Cf. A000203, A066417.

with(numtheory);P:=proc(q) local a,b,c,k,j,n;
for n from 3 to q do c:=sigma(n);
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;
k:=0; j:=c; while j mod 2<>1 do k:=k+1; j:=j/2; od;
b:=sigma(2*c+1)+sigma(2*c-1)+sigma(c/2^k)*2^(k+1)-6*c-2;
if sigma(a)=b then print(n); fi; od; end: P(10^6);

a(12)-a(30) from Giovanni Resta, Oct 23 2013

A230541 Numbers n such that the digits of sigma(n) are a permutation of those of sigma(n), where sigma(n) is the sum of anti-divisors of n (A066417).

Original entry on oeis.org

11, 20, 22, 26, 33, 65, 82, 117, 209, 218, 376, 417, 483, 508, 537, 561, 675, 758, 910, 1186, 1208, 1317, 1350, 1828, 2039, 2192, 2347, 2471, 2840, 2889, 4129, 4369, 4389, 4495, 4893, 5007, 6430, 7276, 7690, 8246, 8777, 9289, 10651, 11727, 11797, 12048, 12099

Offset: 1

Paolo P. Lava, Oct 23 2013

			Divisors of 376 are 1, 2, 4, 8, 47, 94, 376, 188 and sigma(376) = 720; anti-divisors of 376 are 3, 16, 251 and sigma*(376) = 270.
Therefore 376 is part of the sequence because the digits of 720 are a permutation of the digits of 270.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A000203, A066417, A115920.

with(numtheory); P:= proc(i) local a,b,c,j,k,n,ok,p;
for n from 3 to i do b:=[]; c:=[];
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;
while a>0 do b:=[op(b),a mod 10]; a:=trunc(a/10); od; a:=sigma(n);
while a>0 do c:=[op(c),a mod 10]; a:=trunc(a/10); od;
if nops(b)=nops(c) then b:=sort(b); c:=sort(c); b:=b-c; ok:=1;
for j from 1 to nops(b) do if b[j]<>0  then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end; P(10^6);

A229883 Numbers k such that Sum_{j=1..k} sigma_(j) == 0 (mod k), where sigma_(j) is the sum of the anti-divisors of j (A066417).

Original entry on oeis.org

1, 2, 5, 8, 11, 30, 34, 172, 311, 498, 562, 602, 630, 1742, 4608, 4842, 13664, 16386, 24659, 29150, 56357, 58185, 86267, 88114, 242156, 245325, 839756, 947942, 2524087, 2963552, 4218803, 18281326, 28292036, 30023108, 46376824, 52058844, 85990503, 139548984

Offset: 1

Paolo P. Lava, Oct 02 2013

nonn

Tested up to k = 10^6.

			The sum of the anti-divisors of the numbers from 1 to 8 is 0 + 0 + 2 + 3 + 5 + 4 + 10 + 8 = 32 and 32/8 = 4.

Cf. A056550, A066417.

with(numtheory); P:=proc(q) local a,b,j,k,n; b:=0;
for n from 1 to q do a:=0;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od;
b:=b+a; if b mod n=0 then print(n); fi; od; end: P(10^6);

a(29)-a(38) from Donovan Johnson, Oct 12 2013

A241004 Numbers n such that anti-phi(sigma(n)) = n, where anti-phi is A066452 and sigma is the sum of anti-divisors of n (A066417).

Original entry on oeis.org

90, 137, 162, 581, 714, 773, 3735, 4557, 71028

Offset: 1

Paolo P. Lava, Aug 07 2014

Like A001229 but using anti-phi(n) (A066452) and sigma*(n) (A066417).

			90 is in the sequence: Anti-divisors of 90 are 4, 12, 20, 36, 60 and their sum is 132. Anti-phi of 132 is 90.

Cf. A001229, A066417, A066452.

isA241004 := proc(n)
    simplify( n = A066452(A066417(n))) ;
end proc:
for n from 1 do
    if isA241004(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, Aug 07 2014

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);
antidiv(n) = {my(v = []); for (k=2, n-1, if (abs((n % k) - k/2) < 1, v = concat(v, k));); v;}
antiphi(n) = {my(vad = antidiv(n)); my(nbad = 0); for (j=1, n-1, isad = 1; for (k=1, #vad, if ((j % vad[k]) == 0, isad = 0; break); ); nbad += isad;); nbad;}
isok(n) = n == antiphi(sad(n)); \\ Michel Marcus, Feb 25 2016

a(9) from Michel Marcus, Feb 25 2016

A378481 Integers k such that A378414(k) == k (mod A066417(k)).

Original entry on oeis.org

33, 77, 153, 372, 1540, 2300, 2692, 2736, 7812, 8721, 12593, 26025, 26481, 27972, 39321, 64009, 104409, 175441, 325180, 335616, 422593, 455625, 564376, 575040, 756460, 800073, 1104521, 2180545, 2304332, 3502665, 3691968, 5130909, 5515121, 9331441, 9546265

Offset: 1

Paolo P. Lava, Nov 28 2024

Also integers k such that A000217(k) == k (mod A066417(k)).

So far, only 33 belongs both to A232538 and A378414.

			Antidivisors of 77 are 2, 3, 5, 9, 14, 17, 22, 31, 51 and their sum is 154.
Then 77*78/2 mod 154 = 3003 mod 154 = 77.

Cf. A000217, A066417, A232538, A378414.

with(numtheory): P:=proc(q) local j,k,n,v; v:=[];
for n from 3 to q do k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od;
if n*(n+1)/2 mod (sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2)=n
then v:=[op(v),n]; fi; od; op(v); end: P(10^5);

A066272 Number of anti-divisors of n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 31 2001

Anti-divisors are the numbers that do not divide a number by the largest possible margin. E.g. 20 has anti-divisors 3, 8 and 13. An alternative name for anti-divisor is unbiased non-divisors.
Definition: If an odd number i in the range 1 < i <= n divides N where N is any one of 2n-1, 2n or 2n+1 then d = N/i is called an anti-divisor of n. The numbers 1 and 2 have no anti-divisors.
Equivalently, an anti-divisor of n is a number d in the range [2,n-1] which does not divide n and is either a (necessarily odd) divisor of 2n-1 or 2n+1, or a (necessarily even) divisor of 2n.
Thus an anti-divisor of n is an integer d in [2,n-1] such that n == (d-1)/2, d/2, or (d+1)/2 (mod d), the class of d being -1, 0, or 1, respectively.
k is an anti-divisor of n if and only if 1 < k < n and | (n mod k) - k/2 | < 1. - Max Alekseyev, Jul 21 2007
The number of even anti-divisors of n is one less than the number of odd divisors of n; specifically, all but the largest odd divisor multiplied by the power of two dividing 2n (i.e., 2^A001511(n)). For example, the odd divisors of 18 are 1, 3, and 9, so the even anti-divisors of 18 are 1*4 = 4 and 3*4 = 12. - Franklin T. Adams-Watters, Sep 11 2009
2n-1 and 2n+1 are twin primes if and only if n has no odd anti-divisors (e.g., n=15 has no odd anti-divisors so 29 and 31 are twin primes). - Jon Perry, Sep 02 2012
Records are in A066464. - Robert G. Wilson v, Sep 03 2012

			For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.

Diana Mecum and T. D. Noe, Table of n, a(n) for n = 1..10000
Jon Perry, The Anti-divisor.
Jon Perry, The Anti-divisor: Even More Anti-Divisors.

Cf. A000005, A058838, A066464, A066241, A001227, A001511, A001620, A066417.

See A130799 for the anti-divisors.

antidivisors := proc(n)
    local a,k;
    a := {} ;
    for k from 2 to n-1 do
        if abs((n mod k)- k/2) < 1 then
            a := a union {k} ;
        end if;
     end do:
     a ;
end proc:
A066272 := proc(n)
    nops(antidivisors(n)) ;
end proc:
seq(A066272(n),n=1..120); # R. J. Mathar, May 24 2010

antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ], OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 & ], 2n / Select[ Divisors[ 2n ], OddQ[ # ] && # != 1 & ] ] ], # < n & ]; Table[ Length[ antid[ n ] ], {n, 1, 100} ]
f[n_] := Length@ Complement[ Sort@Join[ Select[ Union@ Flatten@ Divisors[{2 n - 1, 2 n + 1}], OddQ@ # && # < n &], Select[ Divisors[2 n], EvenQ@ # && # < n &]], Divisors@ n]; Array[f, 105] (* Robert G. Wilson v, Jul 17 2007 *)
nd[n_]:=Count[Range[2,n-1],?(Abs[Mod[n,#]-#/2]<1&)]; Array[nd,110] (* _Harvey P. Dale, Jul 11 2012 *)

al(n)=Vec(sum(k=1,n,(x^(3*k)+x*O(x^n))/(1-x^(2*k))+(x^(3*k+1)+x^(3*k+2)+x*O(x^n))/(1-x^(2*k+1)))) \\ Franklin T. Adams-Watters, Sep 11 2009

a(n) = if(n>1, numdiv(2*n+1) + numdiv(2*n-1) + numdiv(n/2^valuation(n,2)) - 5, 0) \\ Max Alekseyev, Apr 27 2010

antidivisors(n)=select(t->n%t && tCharles R Greathouse IV, May 12 2016

from sympy import divisors
def A066272(n):
    return len([d for d in divisors(2*n) if n > d >=2 and n%d]) +  len([d for d in divisors(2*n-1) if n > d >=2 and n%d]) +  len([d for d in divisors(2*n+1) if n > d >=2 and n%d]) # Chai Wah Wu, Aug 11 2014

G.f.: Sum_{k>0} x^(3k) / (1 - x^(2k)) + (x^(3k+1) + x^(3k+2)) / (1 - x^(2k+1)). - Franklin T. Adams-Watters, Sep 11 2009
a(n) = A000005(2*n-1) + A000005(2*n+1) + A001227(n) - 5. - Max Alekseyev, Apr 27 2010
a(n) = Sum_{i=3..n} (i mod 2) * (3 + floor((2n-1)/i) - ceiling((2n-1)/i) + floor(2n/i) - ceiling(2n/i) + floor((2n+1)/i) - ceiling((2n+1)/i)). - Wesley Ivan Hurt, Aug 10 2014
Sum_{k=1..n} a(k) ~ (n/2) * (3*log(n) + 6*gamma - 13 + 7*log(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 19 2024

More terms from Robert G. Wilson v, Jan 02 2002

More terms from Max Alekseyev, Apr 27 2010

A073930 Numbers that are equal to the sum of their anti-divisors.

Original entry on oeis.org

5, 8, 41, 56, 946, 5186, 6874, 8104, 17386, 27024, 84026, 167786, 2667584, 4921776, 27914146, 505235234, 3238952914, 73600829714, 455879783074, 528080296234, 673223621664, 4054397778846, 4437083907194, 4869434608274, 6904301600914, 7738291969456

Offset: 1

Jason Earls, Sep 03 2002

nonn

See A066272 for definition of anti-divisor.

			For k=5186, the anti-divisor sum: 3+4+11+23+41+253+451+943+3457 = 5186.

Jon Perry, Anti-perfects, anti-amicables and other records.

Cf. A066417, A192272.

sad(n) = vecsum(select(t->n%t && tA066417
isok(n) = sad(n) == n; \\ Michel Marcus, Oct 12 2019

from sympy import divisors
A073930 = [n for n in range(1,10**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]) == n] # Chai Wah Wu, Aug 14 2014

a(16)-a(17) from Lior Manor, Mar 03 2004
a(18) from Donovan Johnson, Jun 19 2010
a(19)-a(21) by Jud McCranie, Aug 31 2019
a(22)-a(26) by Jud McCranie, Oct 10 2019

A192268 Anti-abundant numbers.

Original entry on oeis.org

7, 10, 11, 12, 13, 14, 15, 17, 18, 20, 21, 22, 23, 25, 27, 28, 30, 31, 32, 33, 35, 37, 38, 39, 40, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 55, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113

Offset: 1

Paolo P. Lava, Jun 28 2011

An anti-abundant number is a number n for which sigma*(n) > n, where sigma*(n) is the sum of the anti-divisors of n. Like A005101 but using anti-divisors.

			25 is anti-abundant because its anti-divisors are 2, 3, 7, 10, 17 and their sum is 39 > 25.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Paolo P. Lava)

Cf. A066417, A005101, A066272, A192267.

isA192268 := proc(n) A066417(n) > n ; end proc:
for n from 1 to 500 do if isA192268(n) then printf("%d,",n); end if; end do: # R. J. Mathar, Jul 04 2011

antiAbQ[n_] := Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]] > n; Select[Range[120], antiAbQ] (* _Amiram Eldar, Jan 13 2022 after Michael De Vlieger at A066417 *)

from itertools import count, islice
from sympy import divisor_sigma, multiplicity
def A192268gen(): return filter(lambda n:divisor_sigma(2*n-1)+divisor_sigma(2*n+1)+divisor_sigma(n//2**(k:=multiplicity(2,n)))*2**(k+1)-7*n-2 > 0,count(2))
A192268_list = list(islice(A192268gen(),16)) # Chai Wah Wu, Dec 23 2021

A000027 = A073930 UNION A192267 UNION {this set}.

cp's OEIS Frontend

A372753 Integers m, with k digits, such that m = Sum_{i=1..k} A066417(m without its i-th digit).

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A066980 Numbers k such that antid(k) > antid(k+1) and santid(k) > santid(k+1), where antid(k) = A066272 and santid(k) = A066417.

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A230373 Numbers n such that sigma(sigma*(n)) = sigma*(sigma(n)), where sigma*(n) is the sum of anti-divisors of n (A066417).

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A230541 Numbers n such that the digits of sigma(n) are a permutation of those of sigma*(n), where sigma*(n) is the sum of anti-divisors of n (A066417).

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A229883 Numbers k such that Sum_{j=1..k} sigma_*(j) == 0 (mod k), where sigma_*(j) is the sum of the anti-divisors of j (A066417).

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A241004 Numbers n such that anti-phi(sigma*(n)) = n, where anti-phi is A066452 and sigma* is the sum of anti-divisors of n (A066417).

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A378481 Integers k such that A378414(k) == k (mod A066417(k)).

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A066272 Number of anti-divisors of n.

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A073930 Numbers that are equal to the sum of their anti-divisors.

A230373 Numbers n such that sigma(sigma(n)) = sigma(sigma(n)), where sigma*(n) is the sum of anti-divisors of n (A066417).

A230541 Numbers n such that the digits of sigma(n) are a permutation of those of sigma(n), where sigma(n) is the sum of anti-divisors of n (A066417).

A229883 Numbers k such that Sum_{j=1..k} sigma_(j) == 0 (mod k), where sigma_(j) is the sum of the anti-divisors of j (A066417).

A241004 Numbers n such that anti-phi(sigma(n)) = n, where anti-phi is A066452 and sigma is the sum of anti-divisors of n (A066417).