A007626 -id:A007626 - cp's OEIS frontend

A051444 Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 6, 9, 13, 8, 0, 0, 10, 0, 19, 0, 0, 0, 14, 0, 0, 0, 12, 0, 29, 16, 21, 0, 0, 0, 22, 0, 37, 18, 27, 0, 20, 0, 43, 0, 0, 0, 33, 0, 0, 0, 0, 0, 34, 0, 28, 49, 0, 0, 24, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 30, 0, 73, 0, 0, 0, 45, 0, 57, 0, 0, 0, 44, 0, 0, 0, 0, 0

Offset: 1

Jud McCranie

Column 1 of A299762. - Omar E. Pol, Mar 14 2018

This is a right inverse of sigma = A000203 on A002191 = range(sigma): if n is in A002191, then there is some x with sigma(x) = n, and by definition a(n) is the smallest such x, so sigma(a(n)) = n. - M. F. Hasler, Nov 22 2019

			sigma(1) = 1, so a(1) = 1.
There is no k with sigma(k) = 2, since sigma(k) >= k + 1 for all k > 1 and sigma(1) = 1, so a(2) = 0.
sigma(4) = 7, and 4 is the smallest (since only) such number, so a(7) = 4.
6 and 12 are the only k with sigma(k) = 12, so 6 is the smallest and a(12) = 6.

R. K. Guy, Unsolved Problems Theory of Numbers, B1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005

Cf. A000203, A002192, A007626, A007369 (positions of zeros), A299762.

Do[ k = 1; While[ DivisorSigma[ 1, k ] != n && k < 10^4, k++ ]; If[ k != 10^4, Print[ k ], Print[ 0 ] ], {n, 1, 100} ]

a(n)=for(k=1,n,if(sigma(k)==n,return(k))); 0 \\ Charles R Greathouse IV, Mar 09 2014

A051444(n)=if(n=invsigma(n),vecmin(n)) \\ See Alekseyev link for invsigma(). An update including invsigmaMin = A051444 is planned. - M. F. Hasler, Nov 21 2019

Edited by M. F. Hasler, Nov 22 2019

A002192 Least integer with A000203(a(n)) = A002191(n), where A002191 = range of the sum-of-divisors function A000203.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 9, 13, 8, 10, 19, 14, 12, 29, 16, 21, 22, 37, 18, 27, 20, 43, 33, 34, 28, 49, 24, 61, 32, 67, 30, 73, 45, 57, 44, 40, 36, 50, 42, 52, 101, 63, 85, 109, 91, 74, 54

Offset: 1

N. J. A. Sloane

This is the least integer with the increasing sigma value A002191(n). For integers sorted on the ordered sigma values A007609(n), see A085790. - Lekraj Beedassy, Oct 08 2004

The sigma function (A000203) can't have a left nor a right inverse since it is neither injective nor surjective. The first column of the table A085790 (undefined when the row length A054973(n) = 0 <=> no x has sigma(x) = n) or A051444 (which has zeros filled in for these undefined values) are right-inverse of sigma on A002191 = range of sigma: one has A000203(A051444(n)) = A000203(A085790(n,1)) = n for all n in A002191 <=> A054973(n) > 0 <=> row A085790(n,.) nonempty <=> there is x with sigma(x) = n. Since sigma(6) = sigma(11) = 12, a hypothetical left inverse g must satisfy g(12) = 6 and g(12) = 11 which is impossible. Restricted to this list A002192 of smallest indices for the possible values of sigma, there exists a left inverse g such that g(sigma(x)) = x for all x in A002192. This equation defines the function g, i.e., g(A002191(n)) := a(n). A different left inverse exists on the set of largest pre-images for the possible values of sigma, {A085790(n,A054973(n)); n in A002191} = {1, 2, 3, 5, 4, 7, 11, 9, 13, 8, 17, 19, 23, 12, 29, 25, 31, 22, 37, 18, 27, 41, 43, ...}. - M. F. Hasler, Nov 21 2019

J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

A051444 is a better version of this sequence.

Cf. A000203, A002191, A007609, A007626, A054973, A085790.

m = 1000; Clear[f]; f[k_] := f[k] = Split[{DivisorSigma[1, #], #}& /@ Range[3k] // Sort, #1[[1]] == #2[[1]]&][[1 ;; m, 1]][[All, 2]]; f[k = m]; f[k = k+m]; While[f[k] != f[k, m], k = k+m]; A002192 = f[k] (* Jean-François Alcover, Oct 15 2015 *)

A349607 a(n) is the number of divisors of the n-th superabundant number (A004394).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 72, 80, 90, 96, 120, 144, 160, 180, 192, 216, 224, 240, 288, 320, 360, 384, 432, 448, 480, 576, 640, 720, 768, 864, 896, 960, 1152, 1280, 1344, 1440, 1440, 1536, 1728, 1792, 1920, 2304, 2688

Offset: 1

Amiram Eldar, Nov 23 2021

nonn

First differs from A002183 at n = 20.
The first position n where a(n+1) = a(n) is n = 49: a(49) = a(50) = 1440.
The first position n where a(n+1) < a(n) is n = 173: a(173) = 5308416 and a(174) = 5160960.

			a(1) = A000005(A004394(1)) = A000005(1) = 1.
a(10) = A000005(A004394(10)) = A000005(120) = 16.

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

Cf. A000005, A002183, A004394, A007626, A349608.

s = {}; rm = 0; Do[r = DivisorSigma[1, n]/n; If[r > rm, rm = r; AppendTo[s, DivisorSigma[0, n]]], {n, 1, 10^5}]; s

a(n) = A000005(A004394(n)).

A256259 Sum of divisors of the minimal numbers (A007416).

Original entry on oeis.org

1, 3, 7, 12, 28, 31, 60, 91, 124, 168, 127, 360, 403, 546, 508, 744, 1170, 1651, 2418, 2880, 2821, 3048, 2047, 4368, 3751, 5952, 9360, 9906, 8188, 12493, 8191, 19344, 15367, 22568, 22506, 24384, 28800, 26611, 39312, 32764, 51181, 59520, 49128, 79248, 99944, 92202, 112320, 116281, 106483, 160797

Offset: 1

Omar E. Pol, Apr 20 2015

nonn

Has a symmetric representation in the same way as A000203 and all its subsequences.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated from the b-file at A007416)

Cf. A000203, A005179, A007416, A007626, A053212, A063072, A081533, A237271, A237593.

(* The d-th element in list minDiv[n, b] is the smallest numbers k<=n with exactly d<=b divisors, otherwise it is zero. Computation stops as soon as either inequality fails. *)
minDiv[n_, b_] :=
Module[{list = Array[0 &, b], k = 1, d},
  While[k <= n, d = DivisorSigma[0, k];
   If[d <= b && list[[d]] == 0, list[[d]] = k];
   If[d <= b, k++, k = n + 2]]; list]
a256259[n_, b_] :=
Map[DivisorSigma[1, #] &, Sort[Select[minDiv[n, b], # != 0 &]]]
a256259[100000, 300] (* computes the first 60 elements of the sequence *)
(* Hartmut F. W. Hoft, Apr 27 2015 *)

a(n) = A000203(A007416(n)).

A182941 a(n) = sum of divisors of A094348(n).

Original entry on oeis.org

1, 3, 7, 12, 28, 60, 91, 124, 168, 195, 360, 546, 744, 1170, 1344, 2418, 2880, 4368, 5952, 9360, 19344, 28800, 39312, 59520, 79248, 99944, 112320, 120960, 180048, 203112, 232128, 345600, 471744, 714240, 950976, 1199328, 1451520, 1572480, 2160576, 2437344, 2926080

Offset: 1

Peter Luschny, Jan 03 2011

nonn

a(10) = 195 is the first term which is neither in A007626 nor in A063072.

Cf. A000203, A094348, A007626, A063072.

seq(numtheory[sigma](A094348[i]),i=1..42);

a(n) = A000203(A094348(n)). - Omar E. Pol, Dec 08 2019

a(37)-a(41) from Jinyuan Wang, Mar 02 2020

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A051444 Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.

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A002192 Least integer with A000203(a(n)) = A002191(n), where A002191 = range of the sum-of-divisors function A000203.

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A349607 a(n) is the number of divisors of the n-th superabundant number (A004394).

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A256259 Sum of divisors of the minimal numbers (A007416).

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A182941 a(n) = sum of divisors of A094348(n).

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