A178029 -id:A178029 - cp's OEIS frontend

A211185 Numbers whose number of proper divisors equals the number of their anti-divisors.

1, 3, 9, 10, 14, 15, 21, 26, 28, 34, 51, 69, 75, 76, 88, 92, 99, 102, 104, 106, 110, 124, 134, 135, 136, 138, 141, 146, 164, 170, 231, 232, 236, 256, 258, 261, 268, 285, 290, 309, 321, 328, 386, 394, 405, 411, 424, 429, 441, 484, 490, 525, 531, 574, 580, 590, 602, 608, 614, 615, 620, 628, 639, 645, 651, 656, 658

Offset: 1

Juri-Stepan Gerasimov, Feb 02 2013

nonn

See A066272 for definition of anti-divisor.
Numbers of divisors of n such that number of proper divisors of n equals the number of anti-divisors of n: 1, 2, 2, 3, 4, 4, 4, 4, 6, 4, 4, 4, 6, 6, 4, 4, 4, 12, 4, 6, 10, 4, 8, 8, 4, 12, 4, 6, 4, 12, 4, 4, 4,...
Primes p such that number of proper divisors of p - 1 equals the number of anti-divisors of p - 1 and number of proper divisors of p + 1 equals the number of anti-divisors of p + 1 : 2, 103, 137, 257,...
Numbers whose sum of proper divisors equals the sum of their anti-divisors: 1, 5, 41,...

			28 is here since it has 5 proper divisors {2, 4, 7, 14, 28} and 5 anti-divisors {3, 5, 8, 11, 19}.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A032741, A066272, A073694, A178029.

for n from 1 to 700 do
    if A032741(n) = A066272(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 03 2013

is(n)=numdiv(2*n+1)+numdiv(2*n-1)+numdiv(n>>valuation(n, 2))-numdiv(n)==4 || n==1 \\ Charles R Greathouse IV, Feb 04 2013

{n: A032741(n) = A066272(n)}.

Entries corrected by R. J. Mathar, Feb 03 2013

A216213 Numbers k such that sigma(k) = Sum_{j=anti-divisors of k} sigma(j), where sigma*(k) is the sum of the anti-divisors of k.

Original entry on oeis.org

1, 2, 11, 12, 15, 16, 22, 31, 76, 152, 309, 1576, 375479, 781314, 1114986, 3734218, 24311881, 68133239, 147881549

Offset: 1

Paolo P. Lava, Mar 13 2013

nonn

Tested up to k = 108122.

a(20) > 3*10^8. - Donovan Johnson, Mar 22 2013

			Anti-divisors of 76 are 3, 8, 9, 17 and 51 and their sum is 88.
Anti-divisor of 3 is 2 -> Sum is 2.
Anti-divisors of 8 are 3 and 5 -> Sum is 8.
Anti-divisors of 9 are 2 and 6 -> Sum is 8.
Anti-divisors of 17 are 2, 3, 5, 7 and 11 -> Sum is 28.
Anti-divisors of 51 are 2, 6 and 34 -> Sum is 42.
Finally, 2+8+8+28+42=88.

Cf. A066272, A178029, A191580, A191581, A192282-A192284.

A216213:= proc(q) local a,b,c,j,k,n;
for n from 1 to q do
  a:={}; b:=0; for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then b:=b+k; a:=a union {k}; fi; od;
  c:=0; for j from 1 to nops(a) do for k from 2 to a[j]-1 do if abs((a[j] mod k)-k/2)<1 then c:=c+k; fi; od; od; if b=c then print(n); fi; od; end:
A216213(10^10);

a(13)-a(19) from Donovan Johnson, Mar 22 2013

A248787 Numbers x such that sigma(x) = rev(sigma(x)), where sigma(x) is the sum of the divisors of x, sigma(x) the sum of the anti-divisors of x and rev(x) the reverse of x.

Original entry on oeis.org

20, 26, 36531, 42814, 4513010, 63033577

Offset: 1

Paolo P. Lava, Oct 14 2014

No further terms up to 10^6.

a(7) > 10^10. - Hiroaki Yamanouchi, Mar 18 2015

			Antidivisors of 20 are 3,8,13 and their sum is 24, while sigma(20) = 42.
Antidivisors of 26 are 3,4,17 and their sum is 24, while sigma(26) = 42.
Antidivisors of 36531 are 2, 6, 18, 22, 54, 66, 82, 162, 198, 246, 594, 738, 902, 1782, 2214, 2706, 6642, 8118, 24354 and their sum is sigma*(36531) = 48906, while sigma(36531) = 60984.

Diana Mecum, Anti-divisors from 3 to 500

Cf. A000203, A066417, A178029.

with(numtheory):T:=proc(w) local x,y,z; y:=w; z:=0;
for x from 1 to ilog10(w)+1 do z:=10*z+(y mod 10); y:=trunc(y/10); od; z; end:
P:=proc(q) local a,j,k,n; for n from 1 to q 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;
if T(a)=sigma(n) then print(n); fi; od; end: P(10^10);

rev(n) = subst(Polrev(digits(n)), x, 10);
sad(n) = k=valuation(n, 2); sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
isok(n) = sigma(n) == rev(sad(n)); \\ Michel Marcus, Dec 07 2014

a(5) from Chai Wah Wu, Dec 06 2014

a(6) from Hiroaki Yamanouchi, Mar 18 2015

A250067 Numbers n such that n = Rev(sigma(n)), where sigma(n) is the sum of the anti-divisors of n and Rev(n) is the reverse of n.

Original entry on oeis.org

5, 8, 64, 614, 47678, 4442395

Offset: 1

Paolo P. Lava, Nov 11 2014

			Anti-divisors of 5 are 2, 3 and 2 + 3 = 5 = Rev(5).
Anti-divisors of 614 are 3, 4, 409 and 3 + 4 + 409 = 416 = Rev(614).

Cf. A000203, A066417, A178029, A248787.

with(numtheory):T:=proc(w) local x,y,z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local j,k,n; for n from 1 to q do
k:=0; j:=n; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
if sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2=T(n);
then print(n); fi; od; end: P(10^9);

a(6) from Chai Wah Wu, Dec 06 2014

A273992 Numbers whose sum of anti-divisors is equal to the sum of its unitary divisors.

Original entry on oeis.org

11, 22, 33, 65, 82, 140, 218, 228, 483, 537, 616, 1184, 2889, 6430, 10216, 15849, 21541, 59620, 112590, 117818, 130356, 483153, 3028671, 3589646, 7231219, 8515767, 13050345, 36494625, 44498344, 50414595, 217728002, 459644211, 519061576, 1217532421, 1573368218

Offset: 1

Paolo P. Lava, Jun 06 2016

nonn

			Sum of anti-divisors of 11 is 12. Unitary divisors of 11 are 1, 11 and their sum is 12.

Cf. A034448, A066417, A074751, A178029.

with(numtheory): P:=proc(q) local a,b,c,j,k,n;
for n from 1 to q 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;
c:=ifactors(n)[2]; b:=mul(c[j][1]^c[j][2]+1,j=1..nops(c));
if a=b then print(n); fi; od; end: P(10^6);

Select[Range[5000], Function[n, Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]] == Plus @@ Select[Divisors@ n, GCD[#, n/#] == 1 &]]] (* _Michael De Vlieger, Jun 06 2016, after Robert G. Wilson v at A034448 and Harvey P. Dale at A066272 *)

sud(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d));
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);
isok(n) = sad(n) == sud(n); \\ Michel Marcus, Jun 12 2016

a(23)-a(26) from Michel Marcus, Jun 12 2016

a(27)-a(35) from Amiram Eldar, Jul 12 2022

A375471 Integers equal to the sum of divisors of their anti-divisors.

Original entry on oeis.org

3, 4, 16, 61, 1861, 4979, 90191, 58981439, 10024953661

Offset: 1

Paolo P. Lava, Aug 17 2024

Tested up to 2*10^10. - Giovanni Resta, Aug 19 2024

			Anti-divisors of 1861 are 2, 3, 17, 51, 61, 73, 219, 1241 and
sum of sigmas = 3 + 4 + 18 + 72 + 62 + 74 + 296 + 1332 = 1861.

Cf. A000203, A066417, A178029.

a(8)-a(9) from Giovanni Resta, Aug 19 2024

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A211185 Numbers whose number of proper divisors equals the number of their anti-divisors.

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A216213 Numbers k such that sigma*(k) = Sum_{j=anti-divisors of k} sigma*(j), where sigma*(k) is the sum of the anti-divisors of k.

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A248787 Numbers x such that sigma(x) = rev(sigma*(x)), where sigma(x) is the sum of the divisors of x, sigma*(x) the sum of the anti-divisors of x and rev(x) the reverse of x.

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A250067 Numbers n such that n = Rev(sigma*(n)), where sigma*(n) is the sum of the anti-divisors of n and Rev(n) is the reverse of n.

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A273992 Numbers whose sum of anti-divisors is equal to the sum of its unitary divisors.

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A375471 Integers equal to the sum of divisors of their anti-divisors.

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A216213 Numbers k such that sigma(k) = Sum_{j=anti-divisors of k} sigma(j), where sigma*(k) is the sum of the anti-divisors of k.

A248787 Numbers x such that sigma(x) = rev(sigma(x)), where sigma(x) is the sum of the divisors of x, sigma(x) the sum of the anti-divisors of x and rev(x) the reverse of x.

A250067 Numbers n such that n = Rev(sigma(n)), where sigma(n) is the sum of the anti-divisors of n and Rev(n) is the reverse of n.