A085790 -id:A085790 - cp's OEIS frontend

A206026 a(n) = smallest number m such that sigma(k) = m has at least n positive solutions k.

1, 12, 24, 72, 72, 168, 240, 336, 360, 504, 576, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 2880, 2880, 2880, 2880, 2880, 2880, 2880, 2880, 4320, 4320, 4320, 4320, 4320, 4320, 5760, 5760, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence of numbers from A145899.

			a(6) = 168 because 168 is the smallest value of sigma(k) for n = 6 positive integers k such that sigma(k) = 168 has solution; k = 60, 78, 92, 123, 143, 167.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A054973, A085790, A145899, A206027.

list(len) = {my(v = vector(len), k = 1, c = 0, i); while(c < len, i = invsigmaNum(k); for(j = 1, i, if(j <= len && v[j] == 0, v[j] = k; c++)); k++); v;} \\ Amiram Eldar, Dec 15 2024, using Max Alekseyev's invphi.gp

A329821 Largest k for which sigma(k) = A002191(n), where A002191 = range of sigma, the sum-of-divisors function A000203.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 9, 13, 8, 17, 19, 23, 12, 29, 25, 31, 22, 37, 18, 27, 41, 43, 47, 53, 39, 49, 59, 61, 32, 67, 71, 73, 45, 79, 83, 89, 36, 50, 77, 97, 101, 103, 107, 109, 91, 113, 95, 81, 75, 82, 64, 127, 131, 121, 137, 139, 119, 149, 151, 125

Offset: 1

M. F. Hasler, Nov 22 2019

nonn

			The possible values of sigma(x) are A002191 = {1, 3, 4, 6, 7, 8, 12, ...}.
The 7th value is 12 = sigma(x) for x = 6 or 11. Since 11 is the largest such value, a(7) = 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203 (sigma), A002191 (range of sigma), A085790 (table of pre-images of x in A002191), A054973 (number of solutions of sigma(x) = n).

Cf. A051444 (smallest k such that sigma(k) = n).

A329821(n)=vecmax(invsigma(A002191(n))) \\ see Alekseyev link for invsigma(). An invsigmaMax() function is announced.

a(n) = A085790(m,A054973(m)) with m = A002191(n).

A341879 a(n) is the largest d(k) such that sigma(k) = n, where d is the number of divisor function and sigma is the sum of divisors function.

Original entry on oeis.org

1, 0, 2, 2, 0, 2, 3, 2, 0, 0, 0, 4, 3, 2, 4, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 6, 0, 2, 5, 4, 0, 0, 0, 4, 0, 2, 6, 4, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 6, 3, 0, 0, 8, 0, 2, 6, 0, 0, 0, 0, 2, 0, 0, 0, 8, 0, 2, 0, 0, 0, 6, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 8, 9, 0, 6, 0, 0, 8, 0, 6, 0, 0, 0, 2, 0, 6, 0

Offset: 1

Seiichi Manyama, Feb 22 2021

			k that satisfies sigma(k) = 12 is 6 or 11. d(6) = 4 and d(11) = 2. So a(12) = 4.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A054973, A057637, A085790, A299762.

a[n_] := Module[{dmax = 0}, Do[If[DivisorSigma[1, k] == n && (d = DivisorSigma[0, k]) > dmax, dmax = d], {k, 1, n}]; dmax]; Array[a, 100] (* Amiram Eldar, Apr 28 2021 *)

a(n) = my(m=0); for(k=1, n, if(sigma(k)==n, m=max(m, numdiv(k)))); m;

A348711 Integers sorted by decreasing value of sigma(x)/x^2, where sigma is the sum of divisors.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 9, 14, 16, 18, 15, 20, 24, 11, 13, 30, 22, 21, 28, 36, 17, 26, 32, 40, 19, 27, 42, 48, 25, 34, 60, 23, 33, 44, 38, 54, 35, 45, 56, 72, 50, 39, 52, 29, 46, 31, 66, 84, 64, 70, 80, 90, 37, 51, 78, 96, 68, 58, 63, 120, 41, 62, 57, 76, 108, 55

Offset: 1

Michel Marcus, Oct 31 2021

nonn

Is it possible to find distinct x and y such that sigma(x)/x^2 = sigma(y)/y^2 ?

			For 1,2,3,4,6,5,... sigma(x)/x^2 gives: 1 > 3/4 > 4/9 > 7/16 > 1/3 > 6/25 > ...

Michel Marcus, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of sigma(x)/x^2 for x=1..240, highlighting the position of superabundant x (in A004394) in red.
Index entries for sequences that are permutations of the natural numbers

Cf. A000203, A004394, A085790.

Block[{nn = Function[s, FirstPosition[s, #][[1]] &@ Fold[Max, s]]@ Array[DivisorSigma[1, #]/# &, 125], s}, s = Array[DivisorSigma[1, #]/#^2 &, nn]; #[[1 ;; FirstPosition[#, nn][[1]]]] &@ Map[FirstPosition[s, #][[1]] &, ReverseSort[s]]] (* Michael De Vlieger, Oct 31 2021 *)

lista(nn) = Vec(vecsort(vector(10*nn, k, -sigma(k)/k^2), , 1), nn);

A256596 a(n) is the number of iterations of the map x->sigma(x) when starting from n before arriving at a number with more than one ancestor, with a(1)=0 and where sigma is the sum of divisors.

Original entry on oeis.org

0, 6, 5, 4, 2, 1, 3, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Michel Marcus, Apr 03 2015

nonn

That is, before arriving at a number x such that A054973(x) > 1.

			For n=2, the repeated map gives 2 -> 3 -> 4 -> 7 -> 8 -> 15 -> 24 where 24 is the first fork with sigma(15)=sigma(23)=24, so with 6 iterations starting from 2 we have a(2)=6, a(3)=5, a(4)=4, a(7)=3, a(8)=2, and a(15)=1.

G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A000203, A054973, A085790, A216200, A241954.

isfork(n) = {my(nba = 0); for (i=2, n-1, if (sigma(i) == n, nba++); if (nba > 1, return (1)););}
a(n) = {if (n==1, return (0)); my(nbit = 0); ok = 0; while (! ok, newn = sigma(n); nbit++; ok = isfork(newn); n = newn;); nbit;}

cp's OEIS Frontend

A206026 a(n) = smallest number m such that sigma(k) = m has at least n positive solutions k.

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A329821 Largest k for which sigma(k) = A002191(n), where A002191 = range of sigma, the sum-of-divisors function A000203.

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A341879 a(n) is the largest d(k) such that sigma(k) = n, where d is the number of divisor function and sigma is the sum of divisors function.

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Mathematica

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A348711 Integers sorted by decreasing value of sigma(x)/x^2, where sigma is the sum of divisors.

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Mathematica

PARI

A256596 a(n) is the number of iterations of the map x->sigma(x) when starting from n before arriving at a number with more than one ancestor, with a(1)=0 and where sigma is the sum of divisors.

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PARI