A237287 -id:A237287 - cp's OEIS frontend

A238524 Numbers n such that the symmetric representation of sigma(n) is formed by two or more parts.

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92

Offset: 1

Omar E. Pol, Mar 06 2014

nonn

Complement of A174973.
First differs from A237046 at a(48).
First differs from A237287 at a(55).
For more information see A237270.
From Hartmut F. W. Hoft, Nov 27 2014: (Start)
Suppose n = 2^m * p1^e1 *...* pk^ek where p1 < ... < pk are the odd prime factors of n, m>=0 and all ej>0. Equivalent to the property of numbers in this sequence are:
(a) The number of 1's in odd positions equals the number of 1's in even positions in the n-th row of triangle A237048 through an index of the form 2^(m+1)*q where q is an odd divisor of n.
(b) There is one odd prime factor pj of n satisfying  pj > 2^(m+1) * product_{iAlso numbers n for which the n-th row in irregular triangle A249223 contains a zero.
(End)

			9 is in the sequence because the symmetric representation of sigma(9) = 13 is formed by three parts: [5, 3, 5], as shown below in the first quadrant:
.        5
.    _ _ _ _ _
.   |_ _ _ _ _|
.             |_ _ 3
.             |_  |       Sigma(9) = 5 + 3 + 5 = 13
.               |_|_ _
.                   | |
.                   | |
.                   | | 5
.                   | |
.                   |_|
.
From _Hartmut F. W. Hoft_, Nov 27 2014: (Start)
Number 78 = 2 * 3 * 13 has 1's in the 78th row of triangle A237048 at indices 1, 3, 4, 12 where 12 = 2^2*3 < 13. The symmetric representation of sigma(78) has two regions that meet at a point on the diagonal (width 0) and their third leg has width 2. Note also that 78 is the smallest number in this sequence for which width 0 occurs at an index that is not a power of 2.
(End)

Cf. A174973, A196020, A236104, A235791, A237046, A237287, A237591, A237593, A237270, A237271.

(* sequence of numbers k for m <= k <= n having two or more parts *)
(* Function a237270[] is defined in A237270 *)
a238524[m_, n_]:=Select[Range[m, n], Length[a237270[#]]>=2&]
a238524[1, 260] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)
(* function for the alternate description of the sequence *)
(* functions row[ ] & a237048[ ] are defined in A237048 *)
zero249223Q[n_] := Module[{i=2, bound=row[n], width=1}, While[width>=1 && i<=bound, width += (-1)^(i+1) * a237048[n, i]; i++]; width==0]
Select[Range[1, 100], zero249223Q] (* data *)
(* Hartmut F. W. Hoft, Nov 27 2014 *)

A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

Original entry on oeis.org

1, 6, 8, 28, 12, 78, 16, 120, 52, 144, 24, 406, 28, 192, 192, 496, 36, 780, 40, 903, 256, 288, 48, 1830, 124, 336, 320, 1596, 60, 2628, 64, 2016, 384, 432, 384, 4186, 76, 480, 448, 4095, 84, 4656, 88, 2688, 2184, 576, 96, 7750, 228, 2976, 576, 3136, 108, 7260

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 1 + 5 + 6 = 12 (each of the numbers 1, 5 and 6 is the sum of a subset of distinct divisors of 5).
The numbers n = 14 and 15 is an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237289(14) = A237289(15).
a(14) = 1+2+3+7+8+9+10+14+15+16+17+21+22+23+24 = a(15) = 1+3+4+5+6+8+9+15+16+18+19+20+21+23+24 = 192.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 200 terms from Vincenzo Librandi)
Jon Maiga, Computer-generated formulas for A237290, Sequence Machine.

Cf. A000203, A119348, A005153, A119347 (count of the same numbers), A184387, A229335, A237287, A237289.

isSumDist := proc(n,k)
    local dvs,s ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237290 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
end proc:
seq(A237290(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Plus @@ Union[Plus @@@ Subsets@ Divisors@ n]; Array[a, 54] (* Giovanni Resta, Mar 13 2014 *)

padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
a(n) = {vks = []; d = divisors(n); nbd = #d; for (i=1, 2^nbd-1, b = padbin(i, nbd); onek = sum(j=1, nbd, d[j]*b[j]); vks = Set(concat(vks, onek));); sum(i=1, #vks, vks[i]);} \\ Michel Marcus, Mar 09 2014

A237290(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); vecsum(c); }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024

from sympy import divisors
def A237290(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237289(n).
a(p) = 2(p+2) for odd primes p.
a(n) = A184387(n) for practical numbers n (A005153), a(n) < A184387(n) for numbers n that are not practical (A237287).
a(n) = A000203(n) * (A119347(n)+1) / 2. [Found by Sequence Machine and easily seen to be true. Compare for example to the formulas of A229335.] - Antti Karttunen, Nov 29 2024

A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

Original entry on oeis.org

0, 0, 2, 0, 9, 0, 20, 0, 39, 27, 54, 0, 77, 108, 108, 0, 135, 0, 170, 0, 272, 378, 252, 0, 372, 567, 500, 0, 405, 0, 464, 0, 792, 1053, 792, 0, 665, 1350, 1148, 0, 819, 0, 902, 882, 897, 2052, 1080, 0, 1425, 1395, 2052, 1715, 1377, 0, 2052, 0, 2600, 3375, 1710

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 2 + 3 + 4 = 9 (numbers 2, 3 and 4 are not a sum of any subset of distinct divisors of 5).
Numbers n = 14 and 15 are an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237290(14) = A237290(15).
a(14) = 4+5+6+11+12+13+18+19+20 = a(15) = 2+7+10+11+12+13+14+17+22 = 108.
a(6) = 0 as 6 is practical; the sums into distinct divisors from 1 through 12 are 1 = 1, 2 = 2, 3 = 3, 4 = 1 + 3, 5 = 2 + 3, 6 = 1 + 2 + 3, 7 through 12 are (1 through 6) + 6. So none are not a sum distinct divisors of 6. - _David A. Corneth_, Jul 22 2025

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A119348, A005153, A184387, A237287, A237290.

isSumDist := proc(n,k)
    local dvs ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237289 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if not isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
    a ;
end proc:
seq(A237289(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Block[{d = Divisors@n, s}, s = Plus @@ d; s*(s + 1)/2 - Plus @@ Union[Plus @@@ Subsets@d]]; m = Array[a, 59] (* Giovanni Resta, Mar 13 2014 *)

from sympy import divisors
def A237289(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return (s*(s+1)>>1)-sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237290(n).
a(p) = p(p - 1) / 2 - 1 for p = prime > 2.
a(n) = 0 for practical numbers (A005153), a(n) > 0 for numbers that are not practical (A237287).
a(n) = A184387(n) - A229335(n) for numbers n such that A119347(n) = A100587(n).

a(55) and a(57)-a(59) corrected by Giovanni Resta, Mar 13 2014

A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Mar 29 2019

T(n, k) > 0 for all values of k iff n is practical (A005153).

			Table begins as:
  1
  1,1,1
  1,0,1,1
  1,1,1,1,1,1,1
  1,0,0,0,1,1
  1,1,2,1,1,2,1,1,2,1,1,1
  1,0,0,0,0,0,1,1
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  1,0,1,1,0,0,0,0,1,1,0,1,1
  1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1

Amiram Eldar, Table of n, a(n) for n = 1..8299 (rows 1..100 flattened)

Cf. A000203 (row lengths), A307224 (row products).

Cf. A005153, A027750, A030057, A033630, A119348, A225561, A237287, A322860.

T[n_,k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; Table[T[n, k], {n,1,10}, {k, 1, DivisorSigma[1,n]}] // Flatten

T(n, n) = A033630(n).

T(n, A030057(n)) = 0 if there is a 0 in the n-th row, i.e. A030057(n) <= sigma(n) or n is not practical.

A378450 a(n) is the number of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 5, 0, 6, 3, 9, 0, 11, 9, 9, 0, 15, 0, 17, 0, 17, 21, 21, 0, 24, 27, 25, 0, 27, 0, 29, 0, 33, 39, 33, 0, 35, 45, 41, 0, 39, 0, 41, 21, 23, 57, 45, 0, 50, 30, 57, 35, 51, 0, 57, 0, 65, 75, 57, 0, 59, 81, 45, 0, 69, 0, 65, 63, 81, 2, 69, 0, 71, 99, 61, 77, 81, 0, 77, 0, 90, 111, 81, 0, 93, 117, 105

Offset: 1

Antti Karttunen, Nov 29 2024

nonn

Difference between the sum of divisors n and the number of distinct sums of distinct divisors of n.

			For n = 3, with divisors [1, 3] and sigma(3)=4, only 2 in range 1..4 cannot be represented as a sum of a subset of [1, 3], therefore a(3) = 1.
For n = 15, with divisors [1, 3, 5, 15] and sigma(15) = 24, the subset sums are 1, 3, 1+3, 5, 1+5, 3+5, 1+3+5, 15, 1+15, 3+15, 1+3+15, 5+15, 1+5+15, 3+5+15, 1+3+5+15 i.e., [1, 3, 4, 5, 6, 8, 9, 15, 16, 18, 19, 20, 21, 23, 24], which leaves 2, 7, 10, 11, 12, 13, 14, 17, 22 as unrepresented numbers, therefore a(15) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A119347, A237289 (gives the sums of unrepresented numbers), A322860.

Cf. A005153 (positions of 0's), A237287 (of nonzeros), A030057.

A119347(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; };
A378450(n) = (sigma(n)-A119347(n));

a(n) = A000203(n) - A119347(n).

A335030 Numbers m that are not practical and have an abundancy index sigma(m)/m which is larger than that of any smaller number that is not practical.

Original entry on oeis.org

3, 9, 10, 44, 70, 102, 350, 372, 1608, 3492, 6096, 10380, 44040, 100260, 180240, 425160, 1744560, 2425080, 5509980, 10048080, 23614920, 97639920, 396315360, 900229680, 2519017200, 3113704440, 12870562320, 52307529120

Offset: 1

Amiram Eldar, May 20 2020

None of the terms are superabundant (A004394) since all the superabundant numbers are practical numbers (A005153).
The least term m that is k-abundant (having sigma(m)/m > k) for k = 2, 3, ... is A005101(14) = 70, A068403(896) = 44040, A068404(792087) = 3113704440, ...
What is the least 5-abundant number (A215264) that is not practical?

			The first 5 numbers that are not practical are m = 3, 5, 7, 9, 10. Their abundancy indices sigma(m)/m are 1.333..., 1.2, 1.142..., 1.444..., 1.8. The record values occur at 3, 9 and 10.

Cf. A000203 (sigma), A004394, A005153, A237287, A330244, A335029.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; seq = {}; rm = 1; Do[fct = FactorInteger[n]; r = Times@@((First/@fct^ (1+Last/@ fct)-1)/(First/@fct-1))/n; If[r > rm && !pracQ[fct], rm = r; AppendTo[seq, n]], {n, 3, 10^5}]; seq

A378471 Numbers m whose symmetric representation of sigma(m), SRS(m), has at least 2 parts the first of which has width 1.

Original entry on oeis.org

Offset: 1

Hartmut F. W. Hoft, Nov 27 2024

nonn

Numbers m = 2^k * q, k >= 0 and q > 1 odd, without odd prime factors p < 2^(k+1).
This sequence is a proper subsequence of A238524. Numbers 78 = A370206(1) = A238524(55) and 102 = A237287(72) are not in this sequence since their width pattern (A341969) is 1210121.
A000079 is not a subsequence since SRS(2^k), k>=0, consists of a single part of width 1.
Let m = 2^k * q, k >= 0 and q > 1 odd, be a number in this sequence and s the size of the first part of SRS(m) which has width 1 and consists of 2^(k+1) - 1 legs of width 1. Therefore, s = Sum_{i=1..2^(k+1)-1} a237591(m, i) = a235791(m, 1) - a235791(m, 2^(k+1)) = ceiling((m+1)/1 - (1+1)/2) - ceiling((m+1)/2^(k+1) - (2^(k+1) + 1)/2) = (2^(k+1) - 1)(q+1)/2. In other words, point (m, s) is on the line s(m) = (2^(k+1) - 1)/2^(k+1) * m + (2^(k+1) - 1)/2.
For every odd number m in this sequence, the first part of SRS(m) has size (m+1)/2.
Let u = 2^k * Product_{i=1..PrimePi(2^(k+1)} p_i, where p_i is the i-th prime, and let v be the number of elements in this sequence that are in the set V = {m = 2^k * q | 1 < m <= u } then T(j + t*v, k) = T(j, k) + t*u, 1 <= j and 1 <= t, holds for the elements in column k.

			a(5) = 10 is in the sequence since SRS(10) = {9, 9} consists of 2 parts of width 1 and of sizes 9 = (2^2 - 1)(5+1)/2.
a(15) = 25 is in the sequence since the first part of SRS(25) = {13, 5, 13} has width 1 and has size 13 = (2^1 - 1)(25+1)/2.
a(28) = 44 is in the sequence since SRS(44) = {42, 42} has width 1 and has size 42 = (2^3 - 1)(11+1)/2.
The upper left hand 11 X 11 section of array T(j, k) shows the j-th number m in this sequence of the form m = 2^k * q with q odd. The first part of SRS(m) of every number in column k consists of 2^(k+1) - 1 legs of width 1.
j\k| 0   1   2    3    4     5     6      7      8       9       10  ...
------------------------------------------------------------------------
1  | 3   10  44   136  592   2144  8384   32896  133376  527872  2102272
2  | 5   14  52   152  656   2272  8768   33664  133888  528896  2112512
3  | 7   22  68   184  688   2336  8896   34432  138496  531968  2118656
4  | 9   26  76   232  752   2528  9536   34688  140032  537088  2130944
5  | 11  34  92   248  848   2656  9664   35456  142592  538112  2132992
6  | 13  38  116  296  944   2848  10048  35968  144128  543232  2137088
7  | 15  46  124  328  976   3104  10432  36224  145664  544256  2139136
8  | 17  50  148  344  1072  3232  10688  37504  146176  547328  2149376
9  | 19  58  164  376  1136  3296  11072  39296  147712  556544  2161664
10 | 21  62  172  424  1168  3424  11456  39808  150272  558592  2163712
11 | 23  70  188  472  1264  3488  11584  40064  151808  559616  2180096
...
Row 1 is A246956(n), n>=1.
Column 0 is A005408(n) with T(j + 1, 0) = T(j, 0) + 2, n>=1.
Column 1 is A091999(n) with T(j + 2, 1) = T(j, 1) + 12, n>=2.
Column 2 is A270298(n) with T(j + 48, 2) = T(j, 2) + 840, n>=1.
Column 3 is A270301(n) with T(j + 5760, 3) = T(j, 3) + 240240, n>=1.

Cf. A000079, A005408, A091999, A235791, A237287, A237591, A237593, A238524, A246956, A270298, A270301, A341969, A370206, A377654.

(* partsSRS[] and widthPattern[ ] are defined in A377654 *)
a378471[m_, n_] := Select[Range[m, n], Length[partsSRS[#]]>1&&widthPattern[#][[1;;2]]=={1, 0}&]
a378471[1, 105]

Showing 1-8 of 8 results.

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A335029 Numbers that are not practical (A237287) and have more divisors than any smaller number that is not practical.

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A238524 Numbers n such that the symmetric representation of sigma(n) is formed by two or more parts.

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A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

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A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

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A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).

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A378450 a(n) is the number of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

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A335030 Numbers m that are not practical and have an abundancy index sigma(m)/m which is larger than that of any smaller number that is not practical.

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A378471 Numbers m whose symmetric representation of sigma(m), SRS(m), has at least 2 parts the first of which has width 1.

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