A019292 -id:A019292 - cp's OEIS frontend

A019294 Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.

1, 2, 4, 2, 5, 1, 5, 2, 7, 4, 15, 3, 13, 3, 2, 2, 13, 4, 12, 5, 2, 13, 16, 2, 17, 4, 9, 1, 78, 7, 10, 4, 17, 11, 6, 5, 28, 22, 4, 7, 39, 2, 16, 16, 16, 10, 32, 5, 13, 17, 9, 3, 58, 11, 19, 5, 13, 67, 97, 2, 23, 5, 16, 2, 4, 8, 101, 21, 19, 11, 50, 4, 20, 20, 23, 14, 21, 10, 36, 5, 15

Offset: 1

N. J. A. Sloane

Let sigma^m(n) be result of applying sum-of-divisors function m times to n; sequence gives m(n) = min m such that n divides sigma^m(n).
Perfect numbers require one iteration.
It is conjectured that the sequence is finite for all n.
See also the Cohen-te Riele links under A019276.
a(A111227(n)) > A111227(n). - Reinhard Zumkeller, Aug 02 2012
a(659) > 870. - Michel Marcus, Jan 04 2017

			If n = 9 the iteration sequence is s(9) = {9, 13, 14, 24, 60, 168, 480, 1512, 4800, 15748, 28672} and Mod[s(9), 9] = {0, 4, 5, 6, 6, 6, 3, 0, 3, 7, 7}. The first iterate which is a multiple of 9 is the 7th = 1512, so a(9) = 7. For n = 67, the 101st iterate is the first, so a(67) = 101. Usually several iterates are divisible by the initial value. E.g., if n = 6, then 91 of the first 100 iterates are divisible by 6.
A difficult term to compute: a(461) = 557. - _Don Reble_, Jun 23 2005

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.

Don Reble, Table of n, a(n) for n = 1..1578 (Terms a(1..400) from T. D. Noe, Nov 2007; a(401..659) from Michel Marcus, Jan 02 2017), Feb 20 2022.
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100. See Table 2, p. 95.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Carl Pomerance, On the composition of the functions sigma and phi, Colloq. Math., 59 (1989), 11-15.
Wikipedia, Iterated function, as of Jan 02 2020.
Zeraoulia Rafik, On congruence of the iterated form sigma^k(m) = 0 mod m, arXiv:2102.09941 [math.NT], 2021.

Cf. A019295 (ratio sigma^m(n)/n), A019276 (indices of records), A019277 (records), A000396.

a019294 n = snd $ until ((== 0) . (`mod` n) . fst)
                        (\(x, i) -> (a000203 x, i + 1)) (a000203 n, 1)
-- Reinhard Zumkeller, Aug 02 2012

a:=[]; f:=func; for n in [1..81] do k:=n; s:=1; while f(k) mod n ne 0 do k:=f(k); s:=s+1; end while; Append(~a,s); end for; a; // Marius A. Burtea, Jan 11 2020

A019294 := proc(n)
    local a,nitr ;
    a := 1 ;
    nitr := numtheory[sigma](n);
    while modp(nitr,n) <> 0 do
        nitr := numtheory[sigma](nitr) ;
        a := a+1 ;
    end do:
    return a;
end proc: # R. J. Mathar, Aug 22 2016

f[n_, m_] := Block[{d = DivisorSigma[1, n]}, If[ Mod[d, m] == 0, 0, d]]; Table[ Length[ NestWhileList[ f[ #, n] &, n, # != 0 &]] - 1, {n, 84}] (* Robert G. Wilson v, Jun 24 2005 *)
Table[Length[NestWhileList[DivisorSigma[1,#]&,DivisorSigma[1,n], !Divisible[ #,n]&]],{n,90}] (* Harvey P. Dale, Mar 04 2015 *)

a(n)=if(n<0,0,c=1; s=n; while(sigma(s)%n>0,s=sigma(s); c++); c)

apply( A019294(n,s=n)=for(k=1,oo,(s=sigma(s))%n||return(k)), [1..99]) \\ M. F. Hasler, Jan 07 2020

Conjecture: lim_{n -> oo} log(Sum_{k=1..n} a(k))/log(n) = C = 1.6... - Benoit Cloitre, Aug 24 2002
From Michel Marcus, Jan 02 2017: (Start)
a(n) = 1 for n in A007691.
a(n) = 2 for n in A019278 unless it belongs to A007691.
a(n) = 3 for n in A019292 unless it belongs to A007691 or A019278. (End)

Additional comments from Labos Elemer, Jun 20 2001

Edited by M. F. Hasler, Jan 07 2020

A019293 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 (4,k)-perfect numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, 336, 455, 512, 672, 896, 960, 992, 1023, 1280, 1536, 1848, 2040, 2688, 4092, 5472, 5920, 7808, 7936, 10416, 11934, 16352, 16380, 18720, 20384, 21824, 23424, 24564, 29127, 30240, 33792, 36720, 41440

Offset: 1

N. J. A. Sloane

nonn

Similarly to A019278, 2 and sigma(2) are both terms, and this can happen also for further iterations of sigma on known terms, thus providing new terms outside the current known range. - Michel Marcus, May 01 2017

Also it occurs that many divisors of A019278 are terms of this sequence. - Michel Marcus, May 28 2017

			10 is a term because applying sigma four times we see that 10 -> 18 -> 39 -> 168 -> 120, and 120 = 12*10. So 10 is a (4,12)-perfect number.

Michel Marcus, Table of n, a(n) for n = 1..320
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. A019276, A019278, A019292, A129076.

isok(n) = denominator(sigma(sigma(sigma(sigma(n))))/n) == 1; \\ Michel Marcus, Apr 29 2017

Corrected by Michel Marcus, Apr 29 2017

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

A019294 Number (> 0) of iterations of sigma (A000203) required to reach a multiple of n when starting with n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A019293 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 (4,k)-perfect numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

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.