A070039 -id:A070039 - cp's OEIS frontend

A337135 a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).

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

Offset: 1

Ilya Gutkovskiy, Nov 21 2020

nonn

From Gus Wiseman, Mar 05 2021: (Start)
This sequence counts all of the following essentially equivalent things:
1. Chains of distinct inferior divisors from n to 1, where a divisor d|n is inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.
2. Chains of divisors from n to 1 whose first-quotients (in analogy with first-differences) are term-wise greater than or equal to their decapitation (maximum element removed). For example, the divisor chain q = 60/4/2/1 has first-quotients (15,2,2), which are >= (4,2,1), so q is counted under a(60).
3. Chains of divisors from n to 1 such that x >= y^2 for all adjacent x, y.
4. Factorizations of n where each factor is greater than or equal to the product of all previous factors.
(End)

			From _Gus Wiseman_, Mar 05 2021: (Start)
The a(n) chains for n = 1, 2, 4, 12, 16, 24, 36, 60:
  1  2/1  4/1    12/1    16/1      24/1      36/1      60/1
          4/2/1  12/2/1  16/2/1    24/2/1    36/2/1    60/2/1
                 12/3/1  16/4/1    24/3/1    36/3/1    60/3/1
                         16/4/2/1  24/4/1    36/4/1    60/4/1
                                   24/4/2/1  36/6/1    60/5/1
                                             36/4/2/1  60/6/1
                                             36/6/2/1  60/4/2/1
                                                       60/6/2/1
The a(n) factorizations for n = 2, 4, 12, 16, 24, 36, 60:
    2  4    12   16     24     36     60
       2*2  2*6  2*8    3*8    4*9    2*30
            3*4  4*4    4*6    6*6    3*20
                 2*2*4  2*12   2*18   4*15
                        2*2*6  3*12   5*12
                               2*2*9  6*10
                               2*3*6  2*2*15
                                      2*3*10
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A002033, A008578 (positions of 1's), A068108.
The restriction to powers of 2 is A018819.
Not requiring inferiority gives A074206 (ordered factorizations).
The strictly inferior version is A342083.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A000929, or A342098 forbidding equality.
A000005 counts divisors, with sum A000203.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
A342086 counts strict factorizations of divisors.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341673.
Cf. A001248, A006530, A020639, A040039, A045690, A337105, A342087, A342094, A342095, A342096, A342097.

a:= proc(n) option remember; `if`(n=1, 1, add(
      `if`(d<=n/d, a(d), 0), d=numtheory[divisors](n)))
    end:
seq(a(n), n=1..128);  # Alois P. Heinz, Jun 24 2021

a[1] = 1; a[n_] := a[n] = DivisorSum[n, a[#] &, # <= Sqrt[n] &]; Table[a[n], {n, 95}]
(* second program *)
asc[n_]:=Prepend[#,n]&/@Prepend[Join@@Table[asc[d],{d,Select[Divisors[n],#Gus Wiseman, Mar 05 2021 *)

G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^k).

a(2^n) = A018819(n). - Gus Wiseman, Mar 08 2021

A341677 Number of strictly inferior prime-power divisors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 23 2021

nonn

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.

			The strictly inferior prime-power divisors of n!:
n = 1  2  6  24  120  720  5040  40320
    ----------------------------------
    .  .  2   2    2    2     2      2
              3    3    3     3      3
              4    4    4     4      4
                   5    5     5      5
                   8    8     7      7
                        9     8      8
                       16     9      9
                             16     16
                                    32
                                    64
                                   128

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

Positions of zeros are A166684.
The weakly inferior version is A333750.
The version for odd instead of prime-power divisors is A333805.
The version for prime instead of prime-power divisors is A333806.
The weakly superior version is A341593.
The version for squarefree instead of prime-power divisors is A341596.
The strictly superior version is A341644.
A000961 lists prime powers.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A005117 lists squarefree numbers.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A207375 lists central divisors.
- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A161908, A341591, A341592, A341676.
- Strictly Inferior: A060775, A070039, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341645, A341646, A341673.
Cf. A000005, A000203, A000430, A001248, A006530, A020639.

Table[Length[Select[Divisors[n],PrimePowerQ[#]&&#

a(n) = sumdiv(n, d, d^2 < n && isprimepower(d)); \\ Amiram Eldar, Nov 01 2024

A342085 Number of decreasing chains of distinct superior divisors starting with n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 6, 1, 5, 1, 4, 2, 2, 1, 11, 2, 2, 3, 4, 1, 7, 1, 10, 2, 2, 2, 15, 1, 2, 2, 10, 1, 6, 1, 4, 5, 2, 1, 26, 2, 5, 2, 4, 1, 11, 2, 10, 2, 2, 1, 21, 1, 2, 5, 20, 2, 6, 1, 4, 2, 7, 1, 39, 1, 2, 5, 4, 2, 6, 1, 23, 6, 2, 1

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.
These chains have first-quotients (in analogy with first-differences) that are term-wise less than or equal to their decapitation (maximum element removed). Equivalently, x <= y^2 for all adjacent x, y. For example, the divisor chain q = 24/8/4/2 has first-quotients (3,2,2), which are less than or equal to (8,4,2), so q is counted under a(24).
Also the number of ordered factorizations of n where each factor is less than or equal to the product of all previous factors.

			The a(n) chains for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
  2  4    8      12      16        20       24         30       32
     4/2  8/4    12/4    16/4      20/5     24/6       30/6     32/8
          8/4/2  12/6    16/8      20/10    24/8       30/10    32/16
                 12/4/2  16/4/2    20/10/5  24/12      30/15    32/8/4
                 12/6/3  16/8/4             24/6/3     30/6/3   32/16/4
                         16/8/4/2           24/8/4     30/10/5  32/16/8
                                            24/12/4    30/15/5  32/8/4/2
                                            24/12/6             32/16/4/2
                                            24/8/4/2            32/16/8/4
                                            24/12/4/2           32/16/8/4/2
                                            24/12/6/3
The a(n) ordered factorizations for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
  2  4    8      12     16       20     24       30     32
     2*2  4*2    4*3    4*4      5*4    6*4      6*5    8*4
          2*2*2  6*2    8*2      10*2   8*3      10*3   16*2
                 2*2*3  2*2*4    5*2*2  12*2     15*2   4*2*4
                 3*2*2  4*2*2           3*2*4    3*2*5  4*4*2
                        2*2*2*2         4*2*3    5*2*3  8*2*2
                                        4*3*2    5*3*2  2*2*2*4
                                        6*2*2           2*2*4*2
                                        2*2*2*3         4*2*2*2
                                        2*2*3*2         2*2*2*2*2
                                        3*2*2*2

Alois P. Heinz, Table of n, a(n) for n = 1..65536

The restriction to powers of 2 is A045690.
The inferior version is A337135.
The strictly inferior version is A342083.
The strictly superior version is A342084.
The additive version is A342094, with strict case A342095.
The additive version not allowing equality is A342098.
A001055 counts factorizations.
A003238 counts divisibility chains summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908, A341676.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A064052/A048098, A140271, A238535, A341673.
Cf. A000203, A001248, A005117, A006530, A020639, A057567, A057568, A112798, A169594, A337105, A342096, A342097.

a:= proc(n) option remember; 1+add(`if`(d>=n/d,
      a(d), 0), d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..128);  # Alois P. Heinz, Jun 24 2021

cmo[n_]:=Prepend[Prepend[#,n]&/@Join@@cmo/@Select[Most[Divisors[n]],#>=n/#&],{n}];
Table[Length[cmo[n]],{n,100}]

a(2^n) = A045690(n).

A333807 Sum of odd divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 8, 4, 1, 1, 9, 1, 1, 11, 1, 6, 4, 1, 1, 4, 13, 1, 4, 1, 1, 9, 1, 8, 4, 1, 6, 4, 1, 1, 11, 6, 1, 4, 1, 1, 18

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

nonn

Cf. A000593, A069289, A070039, A333805, A333808, A333810.

Table[DivisorSum[n, # &, # < Sqrt[n] && OddQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[(2 k - 1) x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} (2*k - 1) * x^(2*k*(2*k - 1)) / (1 - x^(2*k - 1)).

A333808 Sum of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 10, 0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8, 2, 0, 5, 0, 7, 3, 2, 0, 5, 5, 9, 3, 2, 0, 10, 0, 2, 10, 2, 5, 5, 0, 2, 3, 14, 0, 5, 0, 2, 8, 2, 7, 5, 0, 7, 3, 2, 0, 12, 5, 2, 3, 2, 0, 10

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

nonn

Cf. A008472, A070039, A097974, A333806, A333807.

Table[DivisorSum[n, # &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[Prime[k] x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k) * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

A333810 G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^(k*(k + 1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 2, 1, -1, 4, -1, 1, 2, 1, -5, 4, -1, 1, -2, 1, -1, 4, -5, 1, 7, 1, -5, 4, -1, 6, -2, 1, -1, 4, 0, 1, -4, 1, -5, 9, -1, 1, -8, 1, 4, 4, -5, 1, -4, 6, 2, 4, -1, 1, -3, 1, -1, 11, -5, 6, -4, 1, -5, 4, 11, 1, -16, 1, -1, 9, -5, 8, -4, 1, -8

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

sign

Excess of sum of odd divisors of n that are < sqrt(n) over sum of even divisors of n that are < sqrt(n).

Cf. A002129, A070039, A333782, A333807, A333809.

nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A334019 Sum of unitary divisors of n that are smaller than sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 4, 1, 1, 3, 1, 5, 4, 3, 1, 4, 1, 3, 1, 5, 1, 11, 1, 1, 4, 3, 6, 5, 1, 3, 4, 6, 1, 12, 1, 5, 6, 3, 1, 4, 1, 3, 4, 5, 1, 3, 6, 8, 4, 3, 1, 13, 1, 3, 8, 1, 6, 12, 1, 5, 4, 15, 1, 9, 1, 3, 4, 5, 8, 12, 1, 6, 1, 3, 1, 15, 6

Offset: 1

Amiram Eldar, Apr 12 2020

nonn

			The unitary divisors of 12 are {1, 3, 4, 12}, 1 and 3 are smaller than sqrt(12) and their sum is 1 + 3 = 4, hence a(12) = 4.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A034448, A070039, A077610, A334023.

a[n_] := DivisorSum[n, # &, #^2 < n && CoprimeQ[#, n/#] &]; Array[a, 100]

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1 && dJinyuan Wang, Apr 12 2020

a(n) = A070039(n) for squarefree numbers (A005117) or squares of primes (A001248).

A339353 G.f.: Sum_{k>=1} k^2 * x^(k*(k + 1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 5, 1, 5, 1, 14, 1, 5, 10, 5, 1, 14, 1, 21, 10, 5, 1, 30, 1, 5, 10, 21, 1, 39, 1, 21, 10, 5, 26, 30, 1, 5, 10, 46, 1, 50, 1, 21, 35, 5, 1, 66, 1, 30, 10, 21, 1, 50, 26, 70, 10, 5, 1, 91, 1, 5, 59, 21, 26, 50, 1, 21, 10, 79, 1, 130, 1, 5, 35, 21, 50, 50, 1, 110

Offset: 1

Ilya Gutkovskiy, Dec 01 2020

nonn

Sum of squares of divisors of n that are smaller than sqrt(n).

Index entries for sequences related to sums of divisors

Cf. A001157, A056924, A067558, A070039, A095118, A339354.

nmax = 80; CoefficientList[Series[Sum[k^2 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^2 &, # < Sqrt[n] &], {n, 80}]

a(n) = sumdiv(n, d, if (d^2 < n, d^2)); \\ Michel Marcus, Dec 02 2020

A348953 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d) * d.

Original entry on oeis.org

0, 1, -1, 1, -1, 3, -1, -1, -1, 3, -1, 2, -1, 3, -4, -1, -1, 6, -1, 3, -4, 3, -1, -2, -1, 3, -4, 3, -1, 11, -1, -5, -4, 3, -6, 6, -1, 3, -4, 0, -1, 12, -1, 3, -9, 3, -1, -8, -1, 8, -4, 3, -1, 12, -6, 2, -4, 3, -1, 5, -1, 3, -11, -5, -6, 12, -1, 3, -4, 15, -1, 0, -1, 3, -9, 3, -8, 12, -1, -8

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

sign

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

Cf. A000593, A046897, A070039, A109506, A333810, A348608, A348951, A348952, A348954, A348955, A348956, A037213.

Table[-DivisorSum[n, (-1)^(# + n/#) # &, # < Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[k x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A348953(n) = -sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

G.f.: Sum_{k>=1} k * x^(k*(k + 1)) / (1 + x^k).

a(n) = A037213(n) - A348608(n). - Ridouane Oudra, Aug 21 2025

A339354 G.f.: Sum_{k>=1} k^3 * x^(k*(k + 1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, 9, 1, 9, 1, 9, 1, 36, 1, 9, 28, 9, 1, 36, 1, 73, 28, 9, 1, 100, 1, 9, 28, 73, 1, 161, 1, 73, 28, 9, 126, 100, 1, 9, 28, 198, 1, 252, 1, 73, 153, 9, 1, 316, 1, 134, 28, 73, 1, 252, 126, 416, 28, 9, 1, 441, 1, 9, 371, 73, 126, 252, 1, 73, 28, 477, 1, 828, 1, 9, 153, 73, 344

Offset: 1

Ilya Gutkovskiy, Dec 01 2020

nonn

Sum of cubes of divisors of n that are smaller than sqrt(n).

Index entries for sequences related to sums of divisors

Cf. A001158, A056924, A070039, A276634, A280375, A339353.

nmax = 77; CoefficientList[Series[Sum[k^3 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^3 &, # < Sqrt[n] &], {n, 77}]

a(n) = sumdiv(n, d, if (d^2 < n, d^3)); \\ Michel Marcus, Dec 02 2020

cp's OEIS Frontend

A337135 a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).

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A341677 Number of strictly inferior prime-power divisors of n.

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A342085 Number of decreasing chains of distinct superior divisors starting with n.

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A333807 Sum of odd divisors of n that are < sqrt(n).

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A333808 Sum of distinct prime divisors of n that are < sqrt(n).

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A333810 G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^(k*(k + 1)) / (1 - x^k).

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A334019 Sum of unitary divisors of n that are smaller than sqrt(n).

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A339353 G.f.: Sum_{k>=1} k^2 * x^(k*(k + 1)) / (1 - x^k).

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