A055021 -id:A055021 - cp's OEIS frontend

A055020 Number of iterations of sigma() required until 2n (or greater) is reached.

2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 4, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 1, 2, 1, 2, 1

Offset: 2

Jud McCranie, May 31 2000

nonn

			sigma(sigma(sigma(9))) = 24 >= 2*9, so a(9) = 3.

Amiram Eldar, Table of n, a(n) for n = 2..10001

Cf. A000203, A055021.

a[n_] := -1 + Length@ NestWhileList[DivisorSigma[1, #] &, n, # < 2*n &]; Array[a, 100, 2] (* Amiram Eldar, Mar 18 2024 *)

a(n) = {my(m = n, nn = 2*n, c = 0); while(m < nn, m = sigma(m); c++); c;} \\ Amiram Eldar, Mar 18 2024

A107912 Numbers k that require two iterations of the sigma function to be >= 2*k.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 10, 11, 14, 15, 16, 17, 19, 21, 22, 23, 26, 27, 29, 31, 32, 33, 34, 35, 38, 39, 41, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 62, 63, 64, 65, 68, 69, 71, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 98, 99, 101, 103, 105, 106, 107

Offset: 1

Jud McCranie, May 27 2005

nonn

			sigma(8) = 15 but sigma(sigma(8)) = 25, so 8 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A055020, A055021, A107913, A107914.

ds2Q[n_]:=Module[{ds1=DivisorSigma[1,n]},ds1<2n<=DivisorSigma[1,ds1]]; Select[Range[120],ds2Q]  (* Harvey P. Dale, Feb 13 2011 *)

is(n) = {my(m = n, nn = 2*n, c = 0); while(m < nn, m = sigma(m); c++); c == 2;} \\ Amiram Eldar, Mar 18 2024

Offset 1 by Michel Marcus, Apr 25 2020

A107913 Numbers k that require three iterations of the sigma function to be >= 2*k.

Original entry on oeis.org

9, 13, 25, 37, 43, 49, 61, 67, 73, 97, 109, 121, 151, 157, 163, 169, 181, 193, 211, 225, 229, 241, 277, 283, 289, 313, 331, 337, 361, 373, 397, 409, 421, 433, 457, 487, 523, 529, 541, 547, 577, 601, 613, 625, 631, 661, 673, 691, 709, 729, 733, 751, 757, 787

Offset: 1

Jud McCranie, May 27 2005

nonn

			sigma(sigma(9)) = 14 but sigma(sigma(sigma(9))) = 24, so 9 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000203, A055020, A055021, A107912, A107914.

is3Q[n_]:=Boole[#>=2n&/@NestList[DivisorSigma[1,#]&,n,3]]=={0,0,0,1}; Select[Range[ 800],is3Q] (* Harvey P. Dale, Jul 09 2023 *)

is(n) = {my(m = n, nn = 2*n, c = 0); while(m < nn, m = sigma(m); c++); c == 3;} \\ Amiram Eldar, Mar 18 2024

Offset 1 by Michel Marcus, Apr 25 2020

A107914 Numbers k that require four iterations of the sigma function to be >= 2*k.

Original entry on oeis.org

81, 1681, 3481, 5041, 7921, 17161, 27889, 29929, 83521, 146689, 279841, 491401, 552049, 579121, 635209, 683929, 703921, 707281, 829921, 1190281, 1203409, 1352569, 1481089, 1885129, 2036329, 2211169, 2430481, 2505889, 3279721, 3411409, 3523129, 3568321, 3728761

Offset: 1

Jud McCranie, May 27 2005

nonn

Most of the early terms are the square of a prime or the fourth power of a prime.

			sigma(sigma(sigma(81))) = 160 but sigma(sigma(sigma(sigma(81)))) = 378, so 81 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000203, A055020, A055021, A107912, A107913.

is(n) = {my(m = n, nn = 2*n, c = 0); while(m < nn, m = sigma(m); c++); c == 4;} \\ Amiram Eldar, Mar 18 2024

Offset 1 by Michel Marcus, Apr 25 2020

More terms from Amiram Eldar, Mar 18 2024

A066919 a(n) = least number of applications of f to n to reach 1, where f is defined by f(n) = phi(n) if n is even; = sigma(n) if n is odd.

Original entry on oeis.org

0, 1, 3, 2, 3, 2, 4, 3, 5, 3, 4, 3, 4, 3, 5, 4, 4, 3, 5, 4, 6, 4, 5, 4, 7, 4, 6, 4, 5, 4, 6, 5, 6, 5, 6, 4, 5, 4, 6, 5, 5, 4, 6, 5, 6, 5, 6, 5, 8, 5, 6, 5, 5, 4, 6, 5, 7, 5, 6, 5, 6, 5, 7, 6, 6, 5, 7, 6, 7, 5, 6, 5, 6, 5, 7, 5, 7, 5, 7, 6, 10, 6, 6, 5, 6, 5, 7, 6, 6, 5, 7, 6, 8, 6, 7, 6, 6, 5, 7, 6, 7, 6, 7, 6

Offset: 1

Joseph L. Pe, Jan 23 2002

nonn

a(n) is in [0,19] for n < 10^5. Conjecture: a(n) exists for all n, i.e. repeated application of f to n eventually yields 1, for any n. The only way this could fail is if n, f(n), f(f(n)), ... are all odd squares.

The conjecture is true, since sequence A055021 (smallest x such that n iterations of sigma() are required for the result to be >= 2x) is complete. - Vim Wenders, Apr 07 2008

			f(f(f(f(7)))) = f(f(f(8))) = f(f(4)) = f(2) = 1 and 4 applications of f are required to achieve this. Therefore a(7) = 4.

f[n_] := If[EvenQ[n], EulerPhi[n], DivisorSigma[1, n]]; a[n_] := Module[{b=n, k=0}, While[b>1, b=f[b]; k++ ]; k]; Table[a[i], {i, 1, 105}]
Table[Length[NestWhileList[If[EvenQ[#],EulerPhi[#],DivisorSigma[1,#]]&,n,#!=1&]],{n,110}]-1 (* Harvey P. Dale, Jun 16 2018 *)

Edited by Dean Hickerson, Oct 26 2002

cp's OEIS Frontend

A055020 Number of iterations of sigma() required until 2n (or greater) is reached.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A107912 Numbers k that require two iterations of the sigma function to be >= 2*k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A107913 Numbers k that require three iterations of the sigma function to be >= 2*k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A107914 Numbers k that require four iterations of the sigma function to be >= 2*k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A066919 a(n) = least number of applications of f to n to reach 1, where f is defined by f(n) = phi(n) if n is even; = sigma(n) if n is odd.

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Extensions