A066417 -id:A066417 - cp's OEIS frontend

A213239 Numbers n such that sum of digits of n = sum of digits of anti-divisors of n.

5, 8, 64, 691, 1779, 2851, 6361, 9066, 9606, 9771, 10789, 10996, 18996, 21481, 22569, 27529, 27691, 31516, 36709, 36776, 42649, 48651, 53296, 56586, 58749, 60369, 64794, 72889, 76754, 78766, 79374, 79896, 80989, 86596, 90606, 90879, 92766, 96171, 98979, 108529

Offset: 1

Paolo P. Lava, Jun 07 2012

			Sum of digits of 1779 is 1+7+7+9=24.
Anti-divisors of 1779 are 2, 6, 1186 and their digits’ sum is 2+6+1+1+8+6=24.

Cf. A006753, A066417, A213240.

with(numtheory);
A213239:=proc(q)
local a,b,c,d,k,n;
for n from 1 to q do
  a:=0; b:=0;
  for k from 2 to n-1 do
    if abs((n mod k)-k/2)<1 then
       c:=k; while c>0 do b:=b+(c mod 10); c:=trunc(c/10); od; fi; od;
  c:=n; d:=0; while c>0 do d:=d+(c mod 10); c:=trunc(c/10); od;
  if b=d then print(n); fi; od; end:
A213239(100000);

[n for n in range(1,10**5) if sum([sum([int(x) for x in str(d)]) for d in range(2,n) if n % d and 2*n % d in [d-1,0,1]]) == sum([int(x) for x in str(n)])] # Chai Wah Wu, Aug 08 2014

A216982 Anti-Chowla's function: sum of anti-divisors of n except the largest.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 5, 3, 2, 7, 5, 5, 10, 7, 8, 3, 17, 16, 5, 11, 8, 21, 19, 7, 22, 7, 24, 27, 5, 16, 21, 37, 26, 7, 29, 8, 25, 45, 26, 28, 14, 38, 27, 11, 56, 27, 29, 24, 39, 47, 8, 59, 53, 16, 37, 19, 36, 57, 51, 67, 16, 37, 70, 3, 41, 42, 87, 67, 8, 55

Offset: 1

Juri-Stepan Gerasimov, Feb 19 2013

nonn

Numbers n such that Chowla's function(n) = a(n): 1, 2, 3, 10, 15, 28, 75, 88, 231, 284, 602,...
Places n where a(n) is zero: 1, 2, 3, 4, 6, 96,...
Fixed points of this sequence: 17, 53, 127, 217, 385, 2321,...
Places n where a(n) equals the largest anti-divisor: 1, 2, 7, 10, 31, 37, 39, 55, 78, 160, 482, 937, 1599, 2496,...
Numbers n such that n -/+ 1 and a(n -/+ 1) are all primes: 6, 18, 72, 102, 108, 198, 270, 432, 570, 882,...

			Anti-divisors of 7 are 2, 3, 5, so a(7) = 2 + 3 = 5.

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

Cf. A048050, A066272, A066417, A066481, A130799, A218767.

A216982 := proc(n)
        A066417(n)-A066481(n) ;
end proc: # R. J. Mathar, Feb 23 2013

a(n)=if(n<5,0,my(k=valuation(n, 2)); sigma(2*n+1)+sigma(2*n-1)+sigma(n>>k)<<(k+1)-2-20*n\/3) \\ Charles R Greathouse IV, Mar 05 2013

a(n) = A066417(n) - A066481(n).

A218767 Total number of divisors and anti-divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 5, 6, 5, 7, 5, 8, 6, 7, 7, 7, 7, 10, 5, 9, 7, 9, 7, 10, 8, 7, 9, 11, 5, 11, 7, 12, 9, 7, 9, 11, 7, 11, 9, 12, 6, 13, 7, 9, 13, 9, 7, 13, 9, 12, 7, 13, 9, 11, 9, 11, 9, 11, 9, 18, 6, 9, 13, 9, 9, 13, 11, 13, 7, 13, 7, 18, 9, 9, 11, 11, 13, 13, 5, 15, 11, 11, 9, 16, 12, 9

Offset: 1

Juri-Stepan Gerasimov, Feb 05 2013

nonn

Or tau(n) + anti-tau(n), where anti-tau = A066272.
Total sum of divisors and anti-divisors of n or sigma(n) + A066417(n): 1, 3, 6, 10, 11, 16, 18, 23, 21, 32, 24, 41, 33, 40, 42, 45, 46, 67, 38, 66, 54, 72, 58, 83, 70, 66, 82, 102, 54, 108,...
Numbers n such that sigma(n) = n + anti-sigma(n): A074751.
Numbers n such that Chowla's function(n) = anti-sigma(n): 1, 2, 16, 60, 72,...
Number of divisors of n minus number of anti-divisors of n or tau(n) - anti-tau(n): 1, 2, 1, 2, 0, 3, -1, 2, 1, 1, -1, 4, -2, 1, 1, 3, -3, 2, -1, 3, 1, -1, -3, 6, -2, 1, -1, 1, -1, 5, -3, 0, -1, 1, -1, 7, -3, -3, -1, 4, -2, 3, -3, 3, -1,...
Product of number of divisors of n and number of anti-divisors of n, or tau(n)*anti-tau(n): 0, 0, 2, 3, 4, 4, 6, 8, 6, 12, 6, 12, 8, 12, 12, 10, 10, 24, 6, 18, 12, 20, 10, 16, 15, 12, 20, 30, 6, 24,...
Number of ways to write n as k*(k - m) with k divisor and m anti-divisor of n: 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0,...
Numbers which are not of the form k*(k - m), k divisor, m anti-divisor (i.e., where the number of ways is zero): 1, 2, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 23, 24, 25, 26, 29,

Cf. A000005, A000203, A001065, A066272, A066417, A027750, A048050, A130799.

A218767 := proc(n)
        numtheory[tau](n)+A066272(n) ;
end proc: # R. J. Mathar, Feb 16 2013

a(n) = A000005(n) + A066272(n).

A229860 Let sigma_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma_m (n) = k*n; sequence gives the (2,k)-anti-perfect numbers.

Original entry on oeis.org

3, 5, 7, 8, 14, 16, 32, 41, 56, 92, 98, 114, 167, 507, 543, 946, 2524, 3433, 5186, 5566, 6596, 6707, 6874, 8104, 9615, 15690, 17386, 27024, 84026, 87667, 167786, 199282, 390982, 1023971, 1077378, 1336968, 1529394, 2054435, 2276640, 2667584, 3098834, 3978336

Offset: 1

Paolo P. Lava, Oct 01 2013

nonn

Tested up to n = 10^6.

			Anti-divisors of 92 are 3, 5, 8, 37, 61. Their sum is 114.
Again, anti-divisors of 114 are 4, 12, 76. Their sum is 92 and 92 / 92 = 1.

Cf. A066272, A066417, A019278, A019292, A019293, A192293, A214842, A229861, A229862.

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

Offset corrected and a(34)-a(42) from Donovan Johnson, Jan 09 2014

A229861 Let sigma_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma_m (n) = k*n; sequence gives the (3,k)-anti-perfect numbers.

Original entry on oeis.org

4, 5, 8, 32, 41, 54, 56, 68, 123, 946, 1494, 1856, 2056, 5186, 6874, 8104, 10419, 17386, 27024, 31100, 84026, 167786, 272089, 733253, 812600, 1188000, 1544579, 2667584, 4921776, 16360708, 21524990, 27914146

Offset: 1

Paolo P. Lava, Oct 01 2013

Tested up to n = 10^6.

			Anti-divisors of 54 are 4, 12, 36. Their sum is 52.
Again, anti-divisors of 52 are 3, 5, 7, 8, 15, 21, 35. Their sum is 94.
Finally, anti-divisors of 94 are 3, 4, 7, 9, 11, 17, 21, 27, 63. Their sum is 162 and 162 / 54 = 3.

Cf. A066272, A066417, A019278, A019292, A019293, A192293, A214842, A229860, A229862.

with(numtheory); P:=proc(q,h) local a,i,j,k,n;
for n from 4 to q do a:=n; for i from 1 to h do
k:=0; j:=a; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*a+1)+sigma(2*a-1)+sigma(a/2^k)*2^(k+1)-6*a-2; od;
if type(a/n,integer) then print(n); fi; od; end: P(10^6,3);

Offset corrected and a(26)-a(32) from Donovan Johnson, Jan 09 2014

A229862 Let sigma_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma_m (n) = k*n; sequence gives the (4,k)-anti-perfect numbers.

Original entry on oeis.org

5, 6, 7, 8, 14, 16, 41, 46, 56, 58, 64, 92, 96, 114, 946, 3307, 3325, 5186, 5566, 6596, 6874, 7982, 8104, 14621, 17386, 27024, 44217, 45970, 84026, 91282, 135592, 167786, 1077378, 1231058, 1529394, 2667584, 2873910, 3098834, 3978336, 4292594, 4921776, 27914146

Offset: 1

Paolo P. Lava, Oct 01 2013

nonn

Tested up to n = 10^6.

			Anti-divisors of 58 are 3, 4, 5, 9, 13, 23, 39. Their sum is 96.
The only anti-divisor of 96 is 64.
Again, anti-divisors of 64 are 3, 43. Their sum is 46. Finally, anti-divisors of 46 are 3, 4, 7, 13, 31. Their sum is 58 and 58 / 58 = 1.

Cf. A066272, A066417, A019278, A019292, A019293, A192293, A214842, A229860, A229861.

with(numtheory); P:=proc(q,h) local a,i,j,k,n;
for n from 5 to q do a:=n; for i from 1 to h do
k:=0; j:=a; while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*a+1)+sigma(2*a-1)+sigma(a/2^k)*2^(k+1)-6*a-2; od;
if type(a/n,integer) then print(n); fi; od; end: P(10^6,4);

Offset corrected and a(33)-a(42) from Donovan Johnson, Jan 09 2014

A274049 Numbers k such that sum of anti-divisors of k is palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 21, 40, 42, 53, 76, 84, 127, 137, 143, 150, 163, 173, 177, 199, 208, 211, 229, 236, 242, 249, 252, 255, 273, 277, 288, 289, 298, 316, 320, 321, 323, 329, 332, 334, 335, 336, 351, 372, 401, 411, 419, 431, 434, 437, 467, 475, 477, 485, 489, 497

Offset: 1

Paolo P. Lava, Jun 08 2016

A274028 is a subset of this sequence.

			Anti-divisors of 150 are 4, 7, 12, 13, 20, 23, 43, 60, 100 and their sum is 282 that is palindromic.

Paolo P. Lava, Table of n, a(n) for n = 1..1300

Cf. A066417, A073956, A274028.

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 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)=a then print(n); fi; od; end: P(10^12);

A373047 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z), where sigma*(k) is the sum of the anti-divisors of k.

Original entry on oeis.org

11, 33, 16, 20, 26, 37, 40, 19, 43, 46, 93, 91, 80, 76, 39, 78, 155, 103, 74, 135, 128, 152, 116, 117, 190, 104, 187, 138, 168, 147, 160, 223, 208, 403, 281, 173, 163, 170, 250, 243, 272, 257, 258, 232, 222, 278, 266, 245, 352, 253, 279, 256, 288, 295, 231, 291

Offset: 1

Paolo P. Lava, Aug 02 2024

			a(7) = 40 and 40 has 7 partitions of three numbers, x, y and
z, for which sigma*(65) = sigma*(x) + sigma*(y) + sigma*(z) = 55. In fact:
sigma*(1) + sigma*+(4) + sigma*(35) = 0 + 3 + 52 = 55;
sigma*(1) + sigma*(12) + sigma*(27) = 0 + 13 + 42 = 55;
sigma*(1) + sigma*(14) + sigma*(25) = 0 + 16 + 39 = 55;
sigma*(4) + sigma*(14) + sigma*(22) = 3 + 16 + 36 = 55;
sigma*(5) + sigma*(8) + sigma*(27) = 5 + 8 + 42 = 55;
sigma*(9) + sigma*(13) + sigma*(18) = 8 + 19 + 28 = 55;
sigma*(10) + sigma*(12) + sigma*(18) = 14 + 13 + 28 = 55;
Furthermore 40 is the minimum number to have this property.

Cf. A066417, A210732, A211223, A211224, A211225, A375154.

A378414 Sum of the integers from 1 to n that are not antidivisors of n.

Original entry on oeis.org

1, 3, 4, 7, 10, 17, 18, 28, 37, 41, 54, 65, 72, 89, 102, 122, 125, 143, 172, 186, 209, 217, 242, 277, 286, 327, 336, 360, 411, 429, 454, 470, 513, 565, 578, 634, 653, 671, 728, 765, 820, 837, 890, 950, 949, 1023, 1068, 1120, 1153, 1195, 1284, 1284, 1343, 1433

Offset: 1

Paolo P. Lava, Nov 25 2024

First two equal consecutive values for a(51) = a(52) = 1284.

			a(30) = 429 because 30*31/2 = 465, the antidivisors of 30 are 4, 12, 20 and 465 - 4 - 12 - 20 = 429.

Cf. A000217, A024816, A066417.

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

from sympy import divisor_sigma
def A378414(n): return 1 if n == 1 else (n*(n+13)>>1)+2-divisor_sigma((m:=n<<1)-1)-divisor_sigma(m+1)-(divisor_sigma(n>>(k:=(~n&n-1).bit_length()))<Chai Wah Wu, Dec 03 2024

a(n) = A000217(n) - A066417(n).

A073954 Numbers k such that the sum of the anti-divisors of k exceeds 2*k.

Original entry on oeis.org

143, 175, 203, 247, 248, 270, 280, 297, 315, 325, 333, 347, 357, 368, 410, 423, 462, 472, 473, 500, 518, 522, 553, 563, 567, 578, 585, 598, 630, 637, 675, 682, 693, 697, 725, 742, 760, 770, 787, 788, 808, 833, 850, 858, 878, 893, 913, 945, 963, 977, 990

Offset: 1

Jason Earls, Sep 03 2002

See A066272 for definition of anti-divisor.

			a(1) = 143 because A066417(143) = 292, which exceeds 2 * 143 = 286. (A066417 is the sum of anti-divisors of n).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..576 from Michael De Vlieger)

Cf. A066272 (number of anti-divisors of n), A066417 (sum of anti-divisors of n).

a066417[n_Integer] := Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]]; Select[Range[10000], If[a066417[#] > 2 #, True, False] &] (* _Michael De Vlieger, Aug 09 2014 *)

A073954 = [n for n in range(1,10**5) if sum([d for d in range(2,n,2) if n%d and not 2*n%d])+sum([d for d in range(3,n,2) if n%d and 2*n%d in [d-1,1]]) > 2*n] # Chai Wah Wu, Aug 09 2014

cp's OEIS Frontend

A213239 Numbers n such that sum of digits of n = sum of digits of anti-divisors of n.

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A216982 Anti-Chowla's function: sum of anti-divisors of n except the largest.

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A218767 Total number of divisors and anti-divisors of n.

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A229860 Let sigma*_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; sequence gives the (2,k)-anti-perfect numbers.

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A229861 Let sigma*_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; sequence gives the (3,k)-anti-perfect numbers.

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A229862 Let sigma*_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; sequence gives the (4,k)-anti-perfect numbers.

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A274049 Numbers k such that sum of anti-divisors of k is palindromic.

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A373047 Least k with exactly n partitions k = x + y + z satisfying sigma*(k) = sigma*(x) + sigma*(y) + sigma*(z), where sigma*(k) is the sum of the anti-divisors of k.

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A378414 Sum of the integers from 1 to n that are not antidivisors of n.

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A229860 Let sigma_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma_m (n) = k*n; sequence gives the (2,k)-anti-perfect numbers.

A229861 Let sigma_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma_m (n) = k*n; sequence gives the (3,k)-anti-perfect numbers.

A229862 Let sigma_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma_m (n) = k*n; sequence gives the (4,k)-anti-perfect numbers.

A373047 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z), where sigma*(k) is the sum of the anti-divisors of k.