A048050 -id:A048050 - cp's OEIS frontend

A005276 Betrothed (or quasi-amicable) numbers.

48, 75, 140, 195, 1050, 1575, 1648, 1925, 2024, 2295, 5775, 6128, 8892, 9504, 16587, 20735, 62744, 75495, 186615, 196664, 199760, 206504, 219975, 266000, 309135, 312620, 507759, 526575, 544784, 549219, 573560, 587460, 817479, 1000824, 1057595, 1081184

Offset: 1

N. J. A. Sloane

Members of a pair (m,n) such that sigma(m) = sigma(n) = m+n+1, where sigma = A000203. - M. F. Hasler, Nov 04 2008
Also members of a pair (m,k) such that m = sum of nontrivial divisors of k and k = sum of nontrivial divisors of m. - Juri-Stepan Gerasimov, Sep 11 2009
Also numbers that are terms of cycles when iterating Chowla's function A048050. - Reinhard Zumkeller, Feb 09 2013
From Amiram Eldar, Mar 09 2024: (Start)
The first pair, (48, 75), was found by Nasir (1946).
Lehmer (1948) in a review of Nasir's paper, noted that "the pair (48, 75) behave like amicable numbers".
Makowski (1960) found the next 2 pairs, and called them "pairs of almost amicable numbers".
The next 6 pairs were found by independently by Garcia (1968), who named them "números casi amigos" and Lal and Forbes (1971), who named them "reduced amicable pairs".
Beck and Wajar (1971) found 6 more pairs, but missed the 15th and 16th pairs, (526575, 544784) and (573560, 817479).
Hagis and Lord (1977) found the first 46 pairs. They called them "quasi-amicable numbers", after Garcia (1968).
Beck and Wajar (1993) found the next 33 pairs.
According to Guy (2004; 1st ed., 1981), the name "betrothed numbers" was proposed by Rufus Isaacs. (End)

Mariano Garcia, Números Casi Amigos y Casi Sociables, Revista Annal, año 1, October 1968, Asociación Puertorriqueña de Maestros de Matemáticas.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B5, pp. 91-92.
D. H. Lehmer, Math. Rev., Vol. 8 (1948), p. 445.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..7819 (terms < 10^13, terms 1..100 from T. D. Noe, 101..1000 from Donovan Johnson)
Walter E. Beck and Rudolph M. Wajar, More reduced amicable pairs, Fibonacci Quarterly, Vol. 15, No. 4 (1977), pp. 331-332.
Walter E. Beck and Rudolph M. Wajar, Reduced and Augmented Amicable Pairs to 10^8, Fibonacci Quarterly, Vol. 31, No. 4 (1993), pp. 295-298.
Peter Hagis and Graham Lord, Quasi-amicable numbers, Math. Comp. 31 (1977), 608-611.
M. Lal and A. Forbes, A note on Chowla's function, Math. Comp., Vol. 25, No. 116 (1971), pp. 923-925.
A. Makowski, On Some Equations Involving Functions phi(n) and sigma(n), The American Mathematical Monthly, Vol. 67, No. 7 (1960), pp. 668-670.
Abdur Rahman Nasir, On a certain arithmetic function, Bull. Calcutta Math. Soc., Vol. 38 (1946), p. 140.
Paul Pollack, Quasi-Amicable Numbers are Rare, Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.2.
Eric Weisstein's World of Mathematics, Quasiamicable Pair..
Wikipedia, Betrothed numbers.

Cf. A003502, A003503, A048050, A328370.

Subsequence of A057533.

a005276 n = a005276_list !! (n-1)
a005276_list = filter p [1..] where
   p z = p' z [0, z] where
     p' x ts = if y `notElem` ts then p' y (y:ts) else y == z
               where y = a048050 x
-- Reinhard Zumkeller, Feb 09 2013

bnoQ[n_]:=Module[{dsn=DivisorSigma[1,n],m,dsm},m=dsn-n-1; dsm= DivisorSigma[ 1,m];dsm==dsn==n+m+1]; Select[Range[2,1100000],bnoQ] (* Harvey P. Dale, May 12 2012 *)

isA005276(n) = { local(s=sigma(n)); s>n+1 & sigma(s-n-1)==s }
for( n=1, 10^6, isA005276(n) & print1(n",")) \\ M. F. Hasler, Nov 04 2008

Equals A003502 union A003503. - M. F. Hasler, Nov 04 2008

Extended by T. D. Noe, Dec 29 2011

A065475 Natural numbers excluding 2.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Labos Elemer, Nov 16 2001

From the following 4 disjoint subsets of natural numbers A = {1}, B = {2}, OP = {odd primes}, C = {composites}, 16 sets are derivable: A000027 versus empty set, A002808 vs A008578, A065091 vs A065090, A000040 vs A018252, A006005 vs {{2} with A002808}, {1} vs {A000027 excluding 1}, {2} versus this sequence, {1, 2} versus Union[OP, C].
a(n) is the sum of the obvious divisors of n, which are 1 and n.
The natural numbers excluding 2 are the order numbers of magic squares. Order 2 magic squares do not exist. - William Walkington, Mar 12 2016
The numbers occurring at least twice in Pascal's triangle (A007318, A003016). - Rick L. Shepherd, Jun 05 2016
From Enrique Navarrete, Mar 03 2025: (Start)
a(n) is the number of binary strings of length n with at most one 0 and at least one 1. For example, the a(1)=1 string is 1 and the a(2)=3 strings are 01, 10, 11.
a(n) is also the number of ordered set partitions of an n-set into 2 sets such that the first set has at most one element and the second set has at least one element. (End)

Wikipedia, Magic square.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000027, A000040, A000203, A002808, A006005, A008578, A018252, A048050, A065090, A065091, A097330. A003016, A007318.

&cat[[1],[n : n in [3..100]]]; // Wesley Ivan Hurt, Mar 13 2016

printlevel := -1; a := [1]; T := x->LambertW(-x); f := series(((1+T(x)))/(1-T(x)), x, 77); for m from 3 to 77 do a := [op(a), op(2*m, f)] od; print(a); # Zerinvary Lajos, Mar 28 2009

Join[{1}, Range[3, 100]] (* Wesley Ivan Hurt, Mar 13 2016 *)
Drop[Range[100],{2}] (* Harvey P. Dale, Aug 11 2024 *)

a(n)=n+(n>1) \\ Charles R Greathouse IV, Sep 01 2015

x='x+O('x^99); Vec((1+x-x^2)/(1-x)^2) \\ Altug Alkan, Mar 26 2016

G.f.: x*(1+x-x^2)/(1-x)^2. - Paul Barry, Aug 05 2004
a(n) = A000203(n) - A048050(n).
a(n) = n+1 for n>1, a(n) = a(n-1)+1 for n>2. - Wesley Ivan Hurt, Mar 13 2016
E.g.f.: (x + 1)*(exp(x) - 1). - Ilya Gutkovskiy, Jun 05 2016
a(n) = n + [n>1], a(n) = 1+n-floor(1/n). - Alan Michael Gómez Calderón, May 12 2023

Incorrect formula removed by Charles R Greathouse IV, Mar 18 2010

A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 0, 4, 9, 0, 6, 8, 10, 15, 14, 21, 121, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 529, 95, 46, 69, 28, 115, 841, 32, 58, 45, 62, 93, 24, 155, 1369, 217, 44, 63, 30, 50, 82, 123, 52, 129, 2209, 75, 40, 141, 0, 235, 42, 36, 106, 99, 68, 265, 3481, 371, 118, 64, 56

Offset: 0

Labos Elemer, Apr 12 2002

For odd n >= 9, a(n) <= A073046((n-1)/2). - Max Alekseyev, Sep 01 2025

			For n=128: a(128)=16129, divisors={1,127,16129}, 1+127=sigma(n)-n=128 and 16129 is the smallest.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 9884 terms from Richard J. Mathar)
Mersenne Forum, Given sigma(n)-n, find the smallest possible n

Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.

See A359132 for another version.

f[x_] := DivisorSigma[1, x]-x; t=Table[0, {128}]; Do[c=f[n]; If[c<129&&t[[c]]==0, t[[c]]=n], {n, 1000000}]; t

a(n) = min(x, A001065(x)=n) or a(n)=0 if n is an untouchable number (i.e., if from A005114).

a(0)=1 prepended by Max Alekseyev, Sep 01 2025

A244051 Triangle read by rows in which row n lists the parts of the partitions of n into equal parts, in nonincreasing order.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 9, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 5, 5, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 08 2014

Row n has length sigma(n) = A000203(n).
Row sums give n*A000005(n) = A038040(n).
Column 1 is A000027.
Both columns 2 and 3 are A032742, n > 1.
For any k > 0 and t > 0, the sequence contains exactly one run of k consecutive t's. - Rémy Sigrist, Feb 11 2019
From Omar E. Pol, Dec 04 2019: (Start)
The number of parts congruent to 0 (mod m) in row m*n equals sigma(n) = A000203(n).
The number of parts greater than 1 in row n equals A001065(n), the sum of aliquot parts of n.
The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.
The number of partitions in row n equals A000005(n), the number of divisors of n.
The number of partitions in row n with an odd number of parts equals A001227(n).
The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).
The decreasing records in row n give the n-th row of A056538.
Row n has n 1's which are all at the end of the row.
First n rows contain A000217(n) 1's.
The number of k's in row n is A126988(n,k).
The number of odd parts in row n is A002131(n).
The k-th block in row n has A027750(n,k) parts.
Right border gives A000012. (End)
The r-th row of the triangle begins at index k = A160664(r-1). - Samuel Harkness, Jun 21 2022

			Triangle begins:
   [1];
   [2], [1,1];
   [3], [1,1,1];
   [4], [2,2], [1,1,1,1];
   [5], [1,1,1,1,1];
   [6], [3,3], [2,2,2], [1,1,1,1,1,1];
   [7], [1,1,1,1,1,1,1];
   [8], [4,4], [2,2,2,2], [1,1,1,1,1,1,1,1];
   [9], [3,3,3], [1,1,1,1,1,1,1,1,1];
  [10], [5,5], [2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1];
  [11], [1,1,1,1,1,1,1,1,1,1,1];
  [12], [6,6], [4,4,4], [3,3,3,3], [2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1];
  [13], [1,1,1,1,1,1,1,1,1,1,1,1,1];
  [14], [7,7], [2,2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [15], [5,5,5], [3,3,3,3,3], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [16], [8,8], [4,4,4,4], [2,2,2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  ...
For n = 6 the 11 partitions of 6 are [6], [3, 3], [4, 2], [2, 2, 2], [5, 1], [3, 2], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]. There are only four partitions of 6 that contain equal parts so the 6th row of triangle is [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The number of parts equals sigma(6) = A000203(6) = 12. The row sum is A038040(6) = 6*A000005(6) = 6*4 = 24.
From _Omar E. Pol_, Dec 04 2019: (Start)
The structure of the above triangle is as follows:
   1;
   2 11;
   3    111;
   4 22     1111;
   5             11111;
   6 33 222            111111;
   7                          1111111;
   8 44     2222                      11111111;
   9    333                                    111111111;
  ... (End)

Samuel Harkness, Table of n, a(n) for n = 1..10000
Samuel Harkness, Log-log Scatterplot of the first 1000000 terms

Cf. A000005, A000012, A000041, A000203, A000217, A001065, A001227, A002131, A027750, A032742, A038040, A048050, A056538, A126988, A160664, A237593, A245579, A299765, A328365, A309400 (mirror).

A244051row[n_]:=Flatten[Map[ConstantArray[#,n/#]&,Reverse[Divisors[n]]]];
Array[A244051row,10] (* Paolo Xausa, Oct 16 2023 *)

tabf(nn) = {for (n=1, nn, d = Vecrev(divisors(n)); for (i=1, #d, for (j=1, n/d[i], print1(d[i], ", "));); print(););} \\ Michel Marcus, Nov 08 2014

A048995 Numbers that are not the sum of the nontrivial factors (excluding 1 and n) of some natural number.

Original entry on oeis.org

1, 4, 51, 87, 95, 119, 123, 145, 161, 187, 205, 209, 215, 237, 245, 247, 261, 267, 275, 287, 289, 291, 303, 305, 321, 323, 325, 335, 341, 371, 405, 407, 425, 429, 447, 471, 473, 497, 515, 517, 519, 529, 539, 551, 555, 561, 575, 583, 611, 623, 625, 627, 657

Offset: 1

Bill Taylor (W.Taylor(AT)math.canterbury.ac.nz)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Complement of A048050.

Cf. A005114 (the same property with the sum of proper divisors) and A007369 (the same property with the sum of all divisors).

a048995[n_Integer] := Module[{t = Table[i, {i, n}], a048050, k},
  a048050[m_] := Plus @@ Take[Divisors[m], {2, -2}];
  Do[
   If[a048050[k] == 0 || a048050[k] > n, Null, t[[a048050[k]]] = 0],
   {k, 2, n^2}];
  Drop[DeleteDuplicates[t], {2}]
]; a048995[660] (* Michael De Vlieger, Nov 30 2014 *)

mx=8479; v=vector(mx); for(i=2, mx^2, ch=sigma(i)-i-1; if(ch<=mx, if(ch>0, v[ch]=1))); c=0; for(i=1, mx, if(v[i]==0, c++; write("b048995.txt", c " " i))) /* Donovan Johnson, Feb 11 2013 */

Corrected and extended by Jud McCranie

A054013 Chowla function of n read modulo n.

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 6, 3, 7, 0, 3, 0, 9, 8, 14, 0, 2, 0, 1, 10, 13, 0, 11, 5, 15, 12, 27, 0, 11, 0, 30, 14, 19, 12, 18, 0, 21, 16, 9, 0, 11, 0, 39, 32, 25, 0, 27, 7, 42, 20, 45, 0, 11, 16, 7, 22, 31, 0, 47, 0, 33, 40, 62, 18, 11, 0, 57, 26, 3, 0, 50, 0, 39, 48, 63, 18, 11, 0, 25, 39, 43, 0

Offset: 1

Asher Auel, Jan 17 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A048050, A054014, A054015.

with(numtheory): [seq((sigma(i) - i - 1) mod i, i=2..100)];

Table[Mod[DivisorSigma[1,n]-n-1,n],{n,90}] (* Harvey P. Dale, Dec 01 2011 *)

A054013(n) = ((sigma(n)-n-1) % n); \\ Antti Karttunen, Oct 30 2017

a(n) = A048050(n) mod n

A346878 Sum of the divisors, except for the largest, of the n-th positive even number.

Original entry on oeis.org

1, 3, 6, 7, 8, 16, 10, 15, 21, 22, 14, 36, 16, 28, 42, 31, 20, 55, 22, 50, 54, 40, 26, 76, 43, 46, 66, 64, 32, 108, 34, 63, 78, 58, 74, 123, 40, 64, 90, 106, 44, 140, 46, 92, 144, 76, 50, 156, 73, 117, 114, 106, 56, 172, 106, 136, 126, 94, 62, 240, 64, 100, 186, 127

Offset: 1

Omar E. Pol, Aug 20 2021

Sum of aliquot divisors (or aliquot parts) of the n-th positive even number.

a(n) has a symmetric representation.

			For n = 5 the 5th even number is 10 and the divisors of 10 are [1, 2, 5, 10] and the sum of the divisors of 10 except for the largest is 1 + 2 + 5 = 8, so a(5) = 8.

Bisection of A001065.
Partial sums give A347154.
Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346870, A346877, A346880, A347153.

a[n_] := DivisorSigma[1, 2*n] - 2*n; Array[a, 100] (* Amiram Eldar, Aug 20 2021 *)

a(n) = sigma(2*n) - 2*n; \\ Michel Marcus, Aug 20 2021

from sympy import divisors
def a(n): return sum(divisors(2*n)[:-1])
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Aug 20 2021

a(n) = A001065(2*n).
a(n) = 1 + A346880(n).
Sum_{k=1..n} a(k) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 17 2024

A346870 Sum of all divisors, except the smallest and the largest of every number, of the first n positive even numbers.

Original entry on oeis.org

0, 2, 7, 13, 20, 35, 44, 58, 78, 99, 112, 147, 162, 189, 230, 260, 279, 333, 354, 403, 456, 495, 520, 595, 637, 682, 747, 810, 841, 948, 981, 1043, 1120, 1177, 1250, 1372, 1411, 1474, 1563, 1668, 1711, 1850, 1895, 1986, 2129, 2204, 2253, 2408, 2480, 2596, 2709, 2814

Offset: 1

Omar E. Pol, Aug 18 2021

Partial sums of the even-indexed terms of Chowla's function A048050.

a(n) has a symmetric representation.

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

Partial sums of A346880.

Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346869.

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)+numtheory[sigma](2*n)-1-2*n)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 19 2021

s[n_] := DivisorSigma[1, 2*n] - 2*n - 1; Accumulate @ Array[s, 50] (* Amiram Eldar, Aug 19 2021 *)

from sympy import divisors
from itertools import accumulate
def A346880(n): return sum(divisors(2*n)[1:-1])
def aupton(nn): return list(accumulate(A346880(n) for n in range(1, nn+1)))
print(aupton(52)) # Michael S. Branicky, Aug 19 2021

from math import isqrt
def A346870(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-3*((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-n*(n+2) # Chai Wah Wu, Nov 02 2023

a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, May 15 2023

A346880 Sum of the divisors, except the smallest and the largest, of the n-th positive even number.

Original entry on oeis.org

0, 2, 5, 6, 7, 15, 9, 14, 20, 21, 13, 35, 15, 27, 41, 30, 19, 54, 21, 49, 53, 39, 25, 75, 42, 45, 65, 63, 31, 107, 33, 62, 77, 57, 73, 122, 39, 63, 89, 105, 43, 139, 45, 91, 143, 75, 49, 155, 72, 116, 113, 105, 55, 171, 105, 135, 125, 93, 61, 239, 63, 99, 185, 126, 121

Offset: 1

Omar E. Pol, Aug 18 2021

a(n) has a symmetric representation.

			For n = 5 the 5th even number is 10 and the divisors of 10 are [1, 2, 5, 10] and the sum of the divisors of 10 except the smaller and the largest is 2 + 5 = 7, so a(5) = 7.

Bisection of A048050.
Partial sums give A346870.
Cf. A000203, A005843, A062731, A237593, A245092, A244049, A326124, A346879.

a[n_] := DivisorSigma[1, 2*n] - 2*n - 1; Array[a, 100] (* Amiram Eldar, Aug 19 2021 *)

from sympy import divisors
def a(n): return sum(divisors(2*n)[1:-1])
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Aug 19 2021

a(n) = A048050(2*n).

Sum_{k=1..n} a(k) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 21 2024

A053813 Numbers which are an integral multiple of the sum of their proper divisors: prime and perfect numbers.

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 13, 17, 19, 23, 28, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Henry Bottomley, Mar 28 2000

nonn

Note that numbers which are an integral multiple of the sum of their proper divisors excluding 1 (A048050) are the squares of primes (A001248).

Subsequence of A166070. - Jaroslav Krizek, Oct 06 2009

Cf. A000040, A000396, A001065.
Cf. A048050, A001248.
Cf. A166070.

isA053813 := proc(n) add(1/j, j=numtheory:-divisors(n)): numer(%) = denom(%) + 1 end: select(isA053813, [$1..271]); # Peter Luschny, Oct 03 2018

Select[Range[300], PrimeQ[#] || DivisorSigma[1, #] == 2*#&] (* Jean-François Alcover, Jun 03 2019 *)

cp's OEIS Frontend

A005276 Betrothed (or quasi-amicable) numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A065475 Natural numbers excluding 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A244051 Triangle read by rows in which row n lists the parts of the partitions of n into equal parts, in nonincreasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A048995 Numbers that are not the sum of the nontrivial factors (excluding 1 and n) of some natural number.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A054013 Chowla function of n read modulo n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A346878 Sum of the divisors, except for the largest, of the n-th positive even number.

Views

Author

Keywords

Comments