A055020 -id:A055020 - cp's OEIS frontend

A055021 Smallest number k such that n iterations of sigma() are required for the result to be >= 2k.

6, 2, 9, 81

Offset: 1

Jud McCranie, May 31 2000

These are the first terms of A023196, A107912, A107913, A107914. - Jud McCranie, May 28 2005
a(5) > 4*10^9, if it exists. - Jud McCranie, May 28 2005
There are no more terms: sigma(2*k) is never prime if k is not a power of 2, so an even number needs at most two steps; sigma(k) is odd iff k is a square or twice a square. So A107914 (four recursive steps) contains only odd squares. Assume p prime so sigma(p^2) = p^2 + p + 1 = m^2 never meets the condition with p + 2k = m that (p + 2k)^2 = m^2. This implies the impossibility of a solution for numbers of the form p^(2i) and numbers of the form p^(2i)q^(2i). - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jun 06 2005
If k is a power of 2 then sigma(sigma(2*k)) = sigma(4*k - 1) >= 4*k and so the number of iterations is exactly 2. - David A. Corneth, Mar 18 2024

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

Cf. A000203, A055020, A107912, A107913, A107914, A060800.

isok(k, n) = my(kk=k); for (i=1, n, k = sigma(k); if ((i=2*kk), return(0))); k >= 2*kk;
a(n) = my(k=2); while (!isok(k, n), k++); k; \\ Michel Marcus, Mar 18 2024

seq() = {
	my(todo = Set([1,2,3,4]), res = vector(4));
	for(i = 2, oo,
		t = 1;
		s = sigma(i);
		while(s < 2*i,				
			s = sigma(s);
			t++
		);
		if(res[t] == 0,
			res[t] = i;
			todo = setminus(todo, Set(t));
			if(#todo == 0,
				return(res)
			)
		);	
	)
} \\ David A. Corneth, 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

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A055021 Smallest number k such that n iterations of sigma() are required for the result to be >= 2k.

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

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

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

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