A019293 -id:A019293 - cp's OEIS frontend

A019292 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (3,k)-perfect numbers.

1, 12, 14, 24, 52, 98, 156, 294, 684, 910, 1368, 1440, 4480, 4788, 5460, 5840, 6882, 7616, 9114, 14592, 18288, 22848, 32704, 40880, 52416, 53760, 54864, 56448, 60960, 65472, 94860, 120960, 122640, 169164, 185535, 186368, 194432, 196137, 201872, 208026, 286160

Offset: 1

N. J. A. Sloane

nonn

Currently, up to k=50, the least integers to be (3,k)-perfect numbers are: 1, ?, ?, ?, 52, 98, ?, ?, ?, 12, ?, 14, ?, 5840, 7616, 294, ?, 201872, 169164, 24, 684, ?, ?, 910, ?, 40880, 60960, 4480, ?, 4788, 316160, 185535, 3138192, 1440, 186368, 5460, ?, 208026, 194432, 1454544, 481057305600, 26873600, 13225790247247872, 1937376, 10905024, ?, ?, 94860, ?, 683956224. - Michel Marcus, Jun 04 2017

			14 is a term because applying sigma three times we see that 14 -> 24 -> 60 -> 168, and 168 = 12*14. So 14 is a (3,12)-perfect number. - _N. J. A. Sloane_, May 29 2017

Michel Marcus, Table of n, a(n) for n = 1..131
Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Michel Marcus, Unexhaustive list of terms

Cf. A019278 ((2,k)-perfect numbers), A019293.

isok(n) = denominator(sigma(sigma(sigma(n)))/n) == 1; \\ Michel Marcus, Jan 02 2017

More terms from Michel Marcus, Jan 02 2017

A129076 a(n) = sigma(sigma(sigma(sigma(n)))), where sigma(n) = sum of divisors of n.

Original entry on oeis.org

1, 8, 15, 24, 56, 120, 60, 168, 60, 120, 120, 360, 168, 480, 480, 104, 120, 360, 252, 728, 210, 248, 480, 1512, 104, 728, 546, 1170, 336, 992, 210, 576, 504, 1170, 504, 480, 480, 1512, 1170, 1344, 728, 1680, 504, 1560, 1512, 992, 504, 1560, 384, 432, 992, 588

Offset: 1

Jonathan Vos Post, May 27 2007

a(n) = A000203(A000203(A051027(n))) = A051027(A000203(A000203(n))) = A000203(A051027(A000203(n))) = A000203(A066971(n)) = A066971(A000203(n)) = A051027(A051027(n)). - Jaroslav Krizek, Jul 19 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A051027, A066971.

Cf. A019293.

[ SumOfDivisors(SumOfDivisors(SumOfDivisors(SumOfDivisors(n)))) : n in [1..100]];

Nest[DivisorSigma[1,#]&,Range[60],4] (* Harvey P. Dale, Oct 05 2011 *)

a(n) = sigma(sigma(sigma(sigma(n)))); \\ Michel Marcus, Apr 29 2017

a(n) = A000203(A000203(A000203(A000203(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

cp's OEIS Frontend

A019292 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (3,k)-perfect numbers.

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A129076 a(n) = sigma(sigma(sigma(sigma(n)))), where sigma(n) = sum of 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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Maple

Extensions

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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Author

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Maple

Extensions

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.