A257348 -id:A257348 - cp's OEIS frontend

A007497 a(1) = 2, a(n) = sigma(a(n-1)).

2, 3, 4, 7, 8, 15, 24, 60, 168, 480, 1512, 4800, 15748, 28672, 65528, 122880, 393192, 1098240, 4124736, 15605760, 50328576, 149873152, 371226240, 1710858240, 7926750720, 33463001088, 109760857440, 384120963072, 1468475386560, 7157589626880, 33151875434496

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Note that a(32) = 125038913126400 = 11182080^2. - Zak Seidov, Aug 29 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 1..1500 (first 200 terms from T. D. Noe)
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.
Graeme L. Cohen, Herman J. J. te Riele, Iterating the Sum-of-Divisors Function, Experimental Mathematics, Vol. 5 (1996), No. 2, pp. 91-100.
R. G. Wilson, V, Notes, n.d.

Cf. A000203, A175877 (positions of odd terms), A175878 (odd terms).
See also the similarly defined A051572 which has a(1) = 5 instead.
See also A257348.

a007497 n = a007497_list !! (n-1)
a007497_list = iterate a000203 2  -- Reinhard Zumkeller, Feb 27 2014

A007497 := proc(n) options remember; if n <= 0 then RETURN(2) else numtheory[sigma](procname(n-1)); fi; end proc:

a[1] = 2; a[n_] := a[n] = DivisorSigma[1, a[n-1]]; Table[a[n], {n, 30}]
NestList[ DivisorSigma[1, # ] &, 2, 27] (* Robert G. Wilson v, Oct 08 2010 *)

normalize(M)={
    my(P=Set(M[,1]),f=concat(Mat(P),vector(#P))~);
    for(i=1,#M~,
        f[setsearch(P,M[i,1]),2] += M[i,2]
    );
    f
};
addhelp(normalize, "normalize(M): Given a factorization matrix M, combine all like factors and order.");
sf(f)=my(v=vector(#f~,i,(f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1)), g=factor(v[1])~);for(i=2,#v,g=concat(g,factor(v[i])~));normalize(g~)
v=vector(100);v[1]=2;f=factor(2);for(i=2,#v,print1(i" "); v[i]= factorback(f=sf(f))); v \\ Charles R Greathouse IV, Mar 27 2014

from itertools import accumulate, repeat # requires Python 3.2 or higher
from sympy import divisor_sigma
A007497_list = list(accumulate(repeat(2,100), lambda x, _: divisor_sigma(x)))
# Chai Wah Wu, May 02 2015

Conjecture: (1/2)*log(n) < a(n+1)/a(n) < 2*log(n). - Benoit Cloitre, May 08 2003
Conjecture: a(n) == 0 mod 9 for n > 34. - Ivan N. Ianakiev, Mar 27 2014
Checked up to n = 1000. Similar statements hold for other small primes. For example, a(n) seems to be divisible by 2^22 * 3^5 * 5 * 7 = 35672555520 for all n > 99. - Charles R Greathouse IV, Mar 27 2014

Changed the cross-reference from the tau to the sigma-function - R. J. Mathar, Feb 17 2010

A051572 a(1) = 5, a(n) = sigma(a(n-1)).

Original entry on oeis.org

5, 6, 12, 28, 56, 120, 360, 1170, 3276, 10192, 24738, 61440, 196584, 491520, 1572840, 5433480, 20180160, 94859856, 355532800, 1040179456, 2143289344, 4966055344, 10092086208, 31800637440, 137371852800, 641012414823

Offset: 1

Judson Neer

N. J. A. Sloane, Table of n, a(n) for n = 1..523
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.
Graeme L. Cohen and Herman J. J. te Riele, Iterating the Sum-of-Divisors Function, Experimental Mathematics, Vol. 5 (1996), No. 2, pp. 91-100.

Cf. A000203, A007497, A257348.

NestList[Plus@@Divisors[#] &, 5, 25] (* Alonso del Arte, Apr 28 2011 *)

a=[5];for(i=2,10,a=concat(a,sigma(a[#a]))); a \\ Charles R Greathouse IV, Oct 03 2011

from itertools import accumulate, repeat # requires Python 3.2 or higher
from sympy import divisor_sigma
A051572_list = list(accumulate(repeat(5,100), lambda x, _: divisor_sigma(x)))
# Chai Wah Wu, May 02 2015

A257670 Minimum term in the sigma(x) -> x subtree whose root is n.

Original entry on oeis.org

1, 2, 2, 2, 5, 5, 2, 2, 9, 10, 11, 5, 9, 9, 2, 16, 17, 10, 19, 19, 21, 22, 23, 2, 25, 26, 27, 5, 29, 29, 16, 16, 33, 34, 35, 22, 37, 37, 10, 27, 41, 19, 43, 43, 45, 46, 47, 33, 49, 50, 51, 52, 53, 34, 55, 5, 49, 58, 59, 2, 61, 61, 16, 64, 65, 66, 67, 67, 69

Offset: 1

Michel Marcus, May 03 2015

nonn

			We have the following trees (a <- b means sigma(a) = b):
  2 <-- 3 <-- 4 <-- 7 <-- 8 <-- 15 <-- 24 <-- 60 <-- ...
                    9 <-- 13 <-- 14 <-’
  5 <-- 6 <-- 12 <-- 28 <-- 56 <-- 120 <-- ...
        11 <-’             /
       10 <-- 18 <-- 39 <-’
The number 1 has strictly speaking an arrow to itself, so it is not part of a tree. (For all n > 1, sigma(n) > n, so no other fixed point or longer "cycle" can exist.) But actually we rather consider connected components, and let a(1) = 1 as the smallest element of this connected component.
a(2) = 2, since there is no smaller x such that sigma(x) = 2: the subtree with root 2 is reduced to a single node: 2. Similarly, a(m) = m for all m in A007369.
For n=3, since sigma(2) = 3, the tree whose root is 3 has 2 nodes: 2 and 3, and the smallest one is 2, hence a(3) = 2.
Similarly, although 24 occurs directly first at sigma(14), it is also reached from 15 which is in turn reached, via intermediate steps, from 2. Thus, the subtree with root 24 has as 2 as smallest element, whence a(24) = 2.

M. F. Hasler, Table of n, a(n) for n = 1..20000, Nov 19 2019 (first 1000 terms from Michel Marcus)
M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invsigma.gp, Oct. 2005
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.
R. J. Mathar, Illustrations

Cf. A000203 (sigma), A007369  (sigma(x) = n has no solution).
Cf. A216200 (number of disjoint trees), A257348 (minimal node of all trees).
Cf. A257669 (number of terms in current tree).

lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, my(s = i); while (s <= nn, if (v[s] == 0, v[s] = i); s = sigma(s););); for (i=1, nn, if (v[i] == 0, v[i] = i);); v;} \\ Michel Marcus, Nov 19 2019

A257670(n)=if(n>2,vecmin(concat(apply(self,invsigma(n)),n)),n) \\ See Alekseyev-link for invsigma(). - David A. Corneth and M. F. Hasler, Nov 20 2019

a(m) = m for m in A007369: sigma(x) = m has no solution. [Corrected by M. F. Hasler, Nov 19 2019]

a(A007497(n)) = 2; a(A051572(n)) = 5; a(A257349(n)) = 16. (These sequences being the trajectory of 2, 5 resp. 16 under iterations of sigma = A000203.)

Edited by M. F. Hasler, Nov 19 2019

A257349 a(1) = 16, a(n) = sigma(a(n-1)).

Original entry on oeis.org

16, 31, 32, 63, 104, 210, 576, 1651, 1792, 4088, 8880, 28272, 79360, 196416, 633984, 1827840, 7074432, 22032000, 86640840, 364989240, 1651141800, 7540142400, 33541980160, 90193969152, 334471118520, 1415960985600, 6118878991680, 29424972595200

Offset: 1

N. J. A. Sloane, May 01 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A000203, A007497, A051572, A257348.

NestList[DivisorSigma[1,#]&,16,27] (* Ivan N. Ianakiev, May 02 2015 *)

lista(nn) = {print1(v = 16, ", "); for (n=1, nn, v = sigma(v); print1(v, ", "););} \\ Michel Marcus, May 02 2015

from itertools import accumulate, repeat # requires Python 3.2 or higher
from sympy import divisor_sigma
A257349_list = list(accumulate(repeat(16,100), lambda x, _: divisor_sigma(x)))
# Chai Wah Wu, May 02 2015

A257669 Number of terms in the sigma(x) -> x subtree whose root is n.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 4, 5, 1, 1, 1, 4, 2, 3, 6, 1, 1, 3, 1, 2, 1, 1, 1, 11, 1, 1, 1, 5, 1, 2, 3, 5, 1, 1, 1, 2, 1, 2, 4, 2, 1, 5, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 10, 2, 1, 1, 15, 1, 2, 6, 1, 1, 1, 1, 2, 1, 1, 1, 7, 1, 2, 1, 1, 1, 2, 1, 4, 1, 1, 1, 5, 1, 1

Offset: 1

Michel Marcus, May 03 2015

nonn

For terms m of A007369, numbers m such that sigma(x) = m has no solution, as well as for m = 1, a(m) = 1.
See A257670 for more information, examples, etc. - M. F. Hasler, Nov 19 2019
Records are: a(1) = 1 = a(2), a(3) = 2, a(4) = 3, a(7) = 4, a(8) = 5, a(15) = 6, a(24) = 11, a(60) = 15, a(120) = 16, a(168) = 22 = a(336), a(360) = 26, a(480) = 39, a(1344) = 43, a(1512) = 54, a(1920) = 57, a(2016) = 65, a(2880) = 70, a(4800) = 80, a(5040) = 88, a(6552) = 93, a(8064) = 125, ... - M. F. Hasler, Nov 20 2019

			For n = 2, a(2) = 1, since there is no x such that sigma(x) = 2, so the subtree with root 2 is reduced to a single node: 2.
For n = 3, since sigma(2) = 3, the tree with root 3 has 2 nodes: 2 and 3, hence a(3) = 2.

M. F. Hasler, Table of n, a(n) for n = 1..10000, Nov 20 2019
M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invsigma.gp, Oct. 2005
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.

Cf. A007369 (sigma(x) = n has no solution).
Cf. A216200 (number of disjoint trees), A257348 (minimal nodes of all trees).
Cf. A257670 (minimal representative of current tree).

A257669_vec(N)={my(C=Map(),s,c); vector(N,n,mapput(C,s=sigma(n), if(mapisdefined(C,s), mapget(C,s))+ c=if(mapisdefined(C,n), mapget(C,n) + mapdelete(C,n))+1);c)} \\ M. F. Hasler, Nov 20 2019

apply( A257669(n)=if(n>1,vecsum(apply(self,invsigma(n))))+1, [1..99]) \\ See Alekseyev-link for invsigma(). - M. F. Hasler, Nov 20 2019, replacing earlier code from Michel Marcus

a(A007369(n)) = 1.

cp's OEIS Frontend

A007497 a(1) = 2, a(n) = sigma(a(n-1)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A051572 a(1) = 5, a(n) = sigma(a(n-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

A257670 Minimum term in the sigma(x) -> x subtree whose root is n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A257349 a(1) = 16, a(n) = sigma(a(n-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

A257669 Number of terms in the sigma(x) -> x subtree whose root is n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula