A057637 -id:A057637 - cp's OEIS frontend

A063974 Number of terms in inverse set of usigma = sum of unitary divisors = A034448.

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

Offset: 1

Labos Elemer, Sep 05 2001

nonn

			usigma(x) = 288, invusigma(288) = {138,154,165,168,213,235,248,253}, so a(288) = 8, the number of all terms in the inverse set.

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

Cf. A034444, A034448, A051444, A054973, A057637, A308041.

Size of set {x; usigma(x) = n}.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A308041. - Amiram Eldar, Dec 23 2024

A299762 Irregular triangle T(n,k) read by rows in which row n lists the positive integers whose sum of divisors is n, or 0 if no such integer exists.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 6, 11, 9, 13, 8, 0, 0, 10, 17, 0, 19, 0, 0, 0, 14, 15, 23, 0, 0, 0, 12, 0, 29, 16, 25, 21, 31, 0, 0, 0, 22, 0, 37, 18, 27, 0, 20, 26, 41, 0, 43, 0, 0, 0, 33, 35, 47, 0, 0, 0, 0, 0, 34, 53, 0, 28, 39, 49, 0, 0, 24, 38, 59, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 30, 46, 51, 55, 71, 0, 73

Offset: 1

Omar E. Pol, Mar 12 2018

Essentially the same as the triangle described in the example section of A085790, but with 0's added in empty rows.

Are the records the same as A008578?

			First 24 rows of triangle T(n,k):
-----------------------
. n / k:  1   2   3 ...
-----------------------
| 1|      1;
| 2|      0;
| 3|      2;
| 4|      3;
| 5|      0;
| 6|      5;
| 7|      4;
| 8|      7;
| 9|      0;
|10|      0;
|11|      0;
|12|      6, 11;
|13|      9;
|14|     13;
|15|      8;
|16|      0;
|17|      0;
|18|     10, 17;
|19|      0;
|20|     19;
|21|      0;
|22|      0;
|23|      0;
|24|     14, 15, 23;
...
For n = 23 there are no positive integers whose sum of divisors is 23, so T(23, 1) = 0, which is the only element in the 23rd row of the triangle.
For n = 24 there are three positive integers whose sum of divisors is 24; they are 14, 15 and 23, since sigma(14) = 1 + 2 + 7 + 14 = 24, sigma(15) = 1 + 3 + 5 + 15 = 24 and sigma(23) = 1 + 23 = 24, so the 24th row of the triangle is [14, 15, 23].

Row sums give A258913.
Column 1 gives A051444.
Right border gives A057637.
Positive terms give A085790.
Row n has A054973(n) positive integers.
Positive terms in the first column give A002192.
Indices of the rows that contain a zero give A007369.
Indices of the rows that contain positive terms give A002191.
Cf. A000203, A007368, A007609, A008578.

With[{nn = 74}, ReplacePart[ConstantArray[{0}, nn], PositionIndex@ Array[DivisorSigma[1, #] &, nn]]] // Flatten (* Michael De Vlieger, Mar 16 2018 *)

sigma(T(n,k)) = n, if T(n,k) >= 1.

A055489 Largest number x such that sum of divisors of x is n!.

Original entry on oeis.org

5, 23, 95, 719, 5039, 39917, 361657, 3624941, 39904153, 479001599, 6226862869, 87178291199, 1307672080867, 20922780738961, 355687390376431, 6402373545694717, 121645099711277873, 2432902005056589697, 51090942157413850441

Offset: 3

Labos Elemer, Jun 28 2000

nonn

For n = 1, a(1) = 1; for n = 2 there is no solution.

For n in A002982, a(n) = n!-1.

			For n = 6, the 15 solutions are as follows: {264, 270, 280, 354, 376, 406, 418, 435, 459, 478, 537, 623, 649, 667, 719}.

R. K. Guy (1981): Unsolved Problems In Number Theory, B39.

Ray Chandler, Table of n, a(n) for n=3..52
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.

Cf. A000203, A000142, A002982, A055486, A055488, A057637.

a(n) = {fn = n!; x = fn - 1; while(sigma(x) != fn, x--); x;} \\ Michel Marcus, Dec 17 2013

a(n) = Max{x; Sigma[x] = n!} = Max{x; A000203(x) = A000142(n)}
a(n) = A057637(A000142(n)). - Ray Chandler, Jan 15 2009
a(A002982(n)) = A000142(A002982(n)) - 1. - Ray Chandler, Jan 15 2009

More terms from Jud McCranie, Oct 09 2000
a(15) from Donovan Johnson, Aug 31 2008
a(16)-a(19) from Donovan Johnson, Nov 22 2008
a(20)-a(52) from Ray Chandler, Jan 15 2009

A153077 Largest number m such that sigma(m) = A002110(n) where A002110(n) is the product of the first n primes.

Original entry on oeis.org

5, 29, 116, 2309, 30029, 272264, 6715161, 154448901, 3696967556, 126321788241, 5984466237725, 304250263527209, 7475863618097156, 495878856926202725, 17521052944725830450, 1749278213298193453469, 65483587607609351045025

Offset: 2

Donovan Johnson, Dec 19 2008

nonn

			a(9) = 154448901. Sigma(154448901) = A002110(9) = 223092870 = 2*3*5*7*11*13*17*19*23.

Ray Chandler, Max Alekseyev, Table of n, a(n) for n = 2..24
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A000203, A002110, A153076, A153078.

a(n) = A057637(A002110(n)). - Chandler

a(A057704(n)) = A002110(A057704(n)) - 1. - Ray Chandler

Extended by Ray Chandler, Dec 28 2008

Terms a(22)-a(24) in b-file from Max Alekseyev, Jan 29 2012

A063972 a(n) is the smallest m such that usigma(m) = n (or 0 if no such m).

Original entry on oeis.org

1, 0, 2, 3, 4, 5, 0, 7, 8, 9, 0, 6, 0, 13, 0, 0, 16, 10, 0, 12, 0, 0, 0, 14, 0, 25, 0, 27, 0, 18, 0, 21, 32, 0, 0, 22, 0, 37, 0, 28, 0, 26, 0, 43, 0, 0, 0, 33, 0, 36, 0, 0, 0, 34, 0, 39, 0, 0, 0, 38, 0, 61, 0, 0, 64, 0, 0, 48, 0, 52, 0, 30, 0, 73, 0, 0, 0, 50, 0, 57, 0, 81, 0, 54, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Sep 05 2001

nonn

usigma(m) is the sum of unitary divisors of m, A034448.

			a(12)=6 because the unitary divisors of 6 are 1,2,3,6; they sum to 12.

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

Cf. A034444, A034448, A051444, A057637, A063973.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); m = 100; v = Table[0, {m}];  Do[u = usigma[n]; If[u <= m && v[[u]] == 0, v[[u]] = n], {n, 1, m}]; v (* Amiram Eldar, Sep 22 2019 *)

Edited by Don Reble, May 14 2006

A063973 a(n) is the largest m such that usigma(m) = n (or 0 if no such m).

Original entry on oeis.org

1, 0, 2, 3, 4, 5, 0, 7, 8, 9, 0, 11, 0, 13, 0, 0, 16, 17, 0, 19, 0, 0, 0, 23, 0, 25, 0, 27, 0, 29, 0, 31, 32, 0, 0, 24, 0, 37, 0, 28, 0, 41, 0, 43, 0, 0, 0, 47, 0, 49, 0, 0, 0, 53, 0, 39, 0, 0, 0, 59, 0, 61, 0, 0, 64, 0, 0, 67, 0, 52, 0, 71, 0, 73, 0, 0, 0, 50, 0, 79, 0, 81, 0, 83, 0, 0, 0, 0

Offset: 1

Labos Elemer, Sep 05 2001

nonn

usigma(m) is the sum of unitary divisors of m, A034448.

			a(12) = 11 because the unitary divisors of 11 are 1 and 11, and their sum is 12.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A034444, A034448, A051444, A057637, A063972.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); a[n_] := Module[{k = n}, While[k > 0 && usigma[k] != n, k--]; k]; Array[a, 100]  (* Amiram Eldar, Aug 27 2024 *)

Corrected by Don Reble, May 14 2006

A295043 a(n) is the largest number k such that sigma(k) = 2^n or 0 if no such k exists.

Original entry on oeis.org

1, 0, 3, 7, 0, 31, 0, 127, 217, 381, 889, 0, 3937, 8191, 11811, 27559, 57337, 131071, 253921, 524287, 1040257, 1777447, 4063201, 7281799, 16646017, 32247967, 66584449, 116522119, 225735769, 516026527, 1073602561, 2147483647, 4294434817, 7515217927, 15032385529

Offset: 0

Jaroslav Krizek, Nov 13 2017

nonn

If a(n) > 0, then it is a term of A046528 (numbers that are a product of distinct Mersenne primes).

			a(0) = 1 because 1 is the largest number k with sigma(k) = 1 = 2^0.
a(5) = 31 because 31 is the largest number k with sigma(k) = 32 = 2^5.
a(6) = 0 because there is no number k with sigma(k) = 64 = 2^6.

Amiram Eldar, Table of n, a(n) for n = 0..250
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000043, A000203, A046528, A048947, A057637, A180221.
Cf. A247956 (the smallest number k instead of the largest).
Cf. A078426 (no solution to the equation sigma(x)=2^n).
A000668 (Mersenne primes) is a subsequence.

a(n) = {local(r, k); r=0; for(k=1, 2^n, if(sigma(k) == 2^n, r=k)); return(r)}; \\ Michael B. Porter, Nov 14 2017

a(n) = forstep(k=2^n, 1, -1, if (sigma(k)==2^n, return (k))); return (0) \\ Rémy Sigrist, Jan 08 2018

a(n) = invsigmaMax(1<Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

a(A078426(n)) = 0.
a(A180221(n)) > 0.
a(n) <= 2^n - 1 with equality when n is a Mersenne exponent (A000043). - Michael B. Porter, Nov 14 2017

A063975 Smallest numbers such that the number of terms in inverse set usigma equals n; where usigma = A034448.

Original entry on oeis.org

1, 12, 24, 60, 120, 72, 216, 288, 1320, 480, 240, 840, 1296, 2700, 960, 1512, 1080, 720, 1728, 2016, 3840, 3240, 3456, 2520, 3360, 3024, 1440, 3600, 6912, 2160, 19152, 2880, 7920, 13680, 9072, 12600, 6048, 5040, 18000, 6480, 27216, 13440, 7200, 27648, 5760

Offset: 1

Labos Elemer, Sep 05 2001

nonn

			usigma(x) = 288, invusigma(288) = {138, 154, 165, 168, 213, 235, 248, 253}, so a(288) = 8, the number of all terms in the inverse set and all similar numbers are larger: {288, 648, 672, 900}.

Donovan Johnson, Table of n, a(n) for n = 1..3000

Cf. A034444, A034448, A051444, A054973, A057637.

with(numtheory): A034448 := proc(n) option remember: local ans,i: ans:=1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: return ans: end: for n from 1 to 5000 do m:=A034448(n): if(type(ct[m],integer))then ct[m]:=ct[m]+1: else ct[m]:=1: fi:od: for m from 1 to 28 do for n from 1 to 5000 do if(ct[n]=m)then printf("%d, ",n):break: fi: od:od: # Nathaniel Johnston, Apr 29 2011

a(9) - a(45) from Nathaniel Johnston, Apr 29 2011

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;

cp's OEIS Frontend

A063974 Number of terms in inverse set of usigma = sum of unitary divisors = A034448.

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A299762 Irregular triangle T(n,k) read by rows in which row n lists the positive integers whose sum of divisors is n, or 0 if no such integer exists.

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A055489 Largest number x such that sum of divisors of x is n!.

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A153077 Largest number m such that sigma(m) = A002110(n) where A002110(n) is the product of the first n primes.

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A063972 a(n) is the smallest m such that usigma(m) = n (or 0 if no such m).

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A063973 a(n) is the largest m such that usigma(m) = n (or 0 if no such m).

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A295043 a(n) is the largest number k such that sigma(k) = 2^n or 0 if no such k exists.

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A063975 Smallest numbers such that the number of terms in inverse set usigma equals n; where usigma = A034448.

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