A166684 -id:A166684 - cp's OEIS frontend

A046951 a(n) is the number of squares dividing n.

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

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Rediscovered by the HR automatic theory formation program.
a(n) depends only on prime signature of n (cf. A025487, A046523). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
First differences of A013936. Average value tends towards Pi^2/6 = 1.644934... (A013661, A013679). - Henry Bottomley, Aug 16 2001
We have a(n) = A159631(n) for all n < 125, but a(125) = 2 < 3 = A159631(125). - Steven Finch, Apr 22 2009
Number of 2-generated Abelian groups of order n, if n > 1. - Álvar Ibeas, Dec 22 2014 [In other words, number of order-n abelian groups with rank <= 2. Proof: let b(n) be such number. A finite abelian group is the inner direct product of all Sylow-p subgroups, so {b(n)} is multiplicative. Obviously b(p^e) = floor(e/2)+1 (corresponding to the groups C_(p^r) X C_(p^(e-r)) for 0 <= r <= floor(e/2)), hence b(n) = a(n) for all n. - Jianing Song, Nov 05 2022]
Number of ways of writing n = r*s such that r|s. - Eric M. Schmidt, Jan 08 2015
The number of divisors of the square root of the largest square dividing n. - Amiram Eldar, Jul 07 2020
The number of unordered factorizations of n into cubefree powers of primes (1, primes and squares of primes, A166684). - Amiram Eldar, Jun 12 2025

			a(16) = 3 because the squares 1, 4, and 16 divide 16.
G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Antonio Amariti, Claudius Klare, Domenico Orlando and Susanne Reffert, The M-theory origin of global properties of gauge theories, Nuclear Physics B, Vol. 901 (2015), pp. 318-337, arXiv preprint, arXiv:1507.04743 [hep-th], 2015 (see (A.13)).
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics
Ian G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34. See delta(n).
Andrew V. Lelechenko, Average number of squares dividing mn, arXiv preprint arXiv:1407.1222 [math.NT], 2014.
Werner Georg Nowak and László Tóth, On the average number of subgroups of the group Z_m X Z_n, International Journal of Number Theory, Vol. 10, No. 2 (2014), pp. 363-374, arXiv preprint, arXiv:1307.1414 [math.NT], 2013.
N. J. A. Sloane, Transforms.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000188, A004101, A005117 (positions of 1's), A008619, A008833, A013936 (partial sums), A038538, A046952, A052304, A056595, A159631, A007814, A010052, A027750, A239930, A007862, A046523, A064989, A065704, A130279, A156552, A166684, A278161, A307445 (Dirichlet inverse).
One more than A071325.
Differs from A096309 for the first time at n=32, where a(32) = 3, while A096309(32) = 2 (and also A185102(32) = 2).
Sum of the k-th powers of the square divisors of n for k=0..10: this sequence (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: this sequence (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).
Cf. A082293 (a(n)==2), A082294 (a(n)==3).

a046951 = sum . map a010052 . a027750_row
-- Reinhard Zumkeller, Dec 16 2013

[#[d: d in Divisors(n)|IsSquare(d)]:n in [1..120]]; // Marius A. Burtea, Jan 21 2020

A046951 := proc(n)
    local a,s;
    a := 1 ;
    for p in ifactors(n)[2] do
        a := a*(1+floor(op(2,p)/2)) ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 17 2012
# Alternatively:
isbidivisible := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d:
a := n -> nops(select(k -> isbidivisible(n, k), [seq(1..n)])): # Peter Luschny, Jun 13 2025

a[n_] := Length[ Select[ Divisors[n], IntegerQ[Sqrt[#]]& ] ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 26 2012 *)
Table[Length[Intersection[Divisors[n], Range[10]^2]], {n, 100}] (* Alonso del Arte, Dec 10 2012 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ DivisorSigma[ 0, d], 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (Quotient[ #[[2]], 2] + 1 & /@ FactorInteger @ n)]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / (1 - x^k^2), {k, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 13 2014 *)
f[p_, e_] := 1 + Floor[e/2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n)=my(f=factor(n));for(i=1,#f[,1],f[i,2]\=2);numdiv(factorback(f)) \\ Charles R Greathouse IV, Dec 11 2012

a(n) = direuler(p=2, n, 1/((1-X^2)*(1-X)))[n]; \\ Michel Marcus, Mar 08 2015

a(n)=factorback(apply(e->e\2+1, factor(n)[,2])) \\ Charles R Greathouse IV, Sep 17 2015

from math import prod
from sympy import factorint
def A046951(n): return prod((e>>1)+1 for e in factorint(n).values()) # Chai Wah Wu, Aug 04 2024

def is_bidivisible(n, d) -> bool: return gcd(n, d) == d and gcd(n//d, d) == d
def aList(n) -> list[int]: return [k for k in range(1, n+1) if is_bidivisible(n, k)]
print([len(aList(n)) for n in range(1, 126)])  # Peter Luschny, Jun 13 2025

(definec (A046951 n) (if (= 1 n) 1 (* (A008619 (A007814 n)) (A046951 (A064989 n)))))
(define (A008619 n) (+ 1 (/ (- n (modulo n 2)) 2)))
;; Antti Karttunen, Nov 14 2016

a(p^k) = A008619(k) = [k/2] + 1. a(A002110(n)) = 1 for all n. (This is true for any squarefree number, A005117). - Original notes clarified by Antti Karttunen, Nov 14 2016
a(n) = |{(i, j) : i*j = n AND i|j}| = |{(i, j) : i*j^2 = n}|. Also tau(A000188(n)), where tau = A000005.
Multiplicative with p^e --> floor(e/2) + 1, p prime. - Reinhard Zumkeller, May 20 2007
a(A130279(n)) = n and a(m) <> n for m < A130279(n); A008966(n)=0^(a(n) - 1). - Reinhard Zumkeller, May 20 2007
Inverse Moebius transform of characteristic function of squares (A010052). Dirichlet g.f.: zeta(s)*zeta(2s).
G.f.: Sum_{k > 0} x^(k^2)/(1 - x^(k^2)). - Vladeta Jovovic, Dec 13 2002
a(n) = Sum_{k=1..A000005(n)} A010052(A027750(n,k)). - Reinhard Zumkeller, Dec 16 2013
a(n) = Sum_{k = 1..n} ( floor(n/k^2) - floor((n-1)/k^2) ). - Peter Bala, Feb 17 2014
From Antti Karttunen, Nov 14 2016: (Start)
a(1) = 1; for n > 1, a(n) = A008619(A007814(n)) * a(A064989(n)).
a(n) = A278161(A156552(n)). (End)
G.f.: Sum_{k>0}(theta(q^k)-1)/2, where theta(q)=1+2q+2q^4+2q^9+2q^16+... - Mamuka Jibladze, Dec 04 2016
From  Antti Karttunen, Nov 12 2017: (Start)
a(n) = A000005(n) - A056595(n).
a(n) = 1 + A071325(n).
a(n) = 1 + A001222(A293515(n)). (End)
L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A000005(d) * A008836(n/d). - Torlach Rush, Jan 21 2020
a(n) = A000005(sqrt(A008833(n))). - Amiram Eldar, Jul 07 2020
a(n) = Sum_{d divides n} mu(core(d)^2), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

Data section filled up to 125 terms and wrong claim deleted from Crossrefs section by Antti Karttunen, Nov 14 2016

A381870 Numbers whose prime indices have a unique multiset partition into sets with distinct sums.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Gus Wiseman, Mar 12 2025

nonn

First differs from A212166 in lacking 360.
First differs from A293511 in having 600.
Also numbers with a unique factorization into squarefree numbers with distinct sums of prime indices (A056239).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			For n = 600 the unique multiset partition is {{1},{1,3},{1,2,3}}. The unique factorization is 2*10*30.

Without distinct block-sums we have A000961, ones in A050320.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
For distinct blocks instead of sums we have A293511, ones in A050326.
These are the positions of ones in A381633, see A381634, A381806, A381990.
Normal multiset partitions of this type are counted by A381718, see A279785.
For constant instead of strict blocks we have A381991, ones in A381635.
A001055 counts multiset partitions of prime indices, strict A045778.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A317141 counts coarsenings of prime indices, refinements A300383.
A321469 counts factorizations with distinct sums of prime indices, ones A166684.
Cf. A000720, A001222, A005117, A066328, A293243, A299202, A300385.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[100],Length[Select[sfacs[#],UnsameQ@@hwt/@#&]]==1&]

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

A050345 Number of ways to factor n into distinct factors with one level of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 15, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 12, 3, 12, 1, 6, 3, 12, 1, 37, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3, 1, 31, 3, 3, 3, 13, 1, 31, 3, 6, 3, 3

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

First differs from A296120 at a(36) = 15, A296120(36) = 14. - Gus Wiseman, Apr 27 2025
Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2)*(2).
From _Gus Wiseman_, Apr 26 2025: (Start)
This is the number of ways to partition a factorization of n (counted by A001055) into a set of sets. For example, the a(12) = 6 choices are:
  {{2},{2,3}}
  {{2},{6}}
  {{3},{4}}
  {{2,6}}
  {{3,4}}
  {{12}}
(End)

R. J. Mathar, Table of n, a(n) for n = 1..2519

Cf. A050346-A050350. a(A002110)=A000258.
For multisets of multisets we have A050336.
For integer partitions we have a(p^k) = A050342(k), see A001970, A089259, A261049.
For normal multiset partitions see A116539, A292432, A292444, A381996, A382214, A382216.
The case of a unique choice (positions of 1) is A166684.
Twice-partitions of this type are counted by A358914, see A270995, A281113, A294788.
For sets of multisets we have A383310 (distinct products A296118).
For multisets of sets we have we have A383311, see A296119.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A302494 gives MM-numbers of sets of sets.
A382077 counts partitions that can be partitioned into a sets of sets, ranks A382200.
A382078 counts partitions that cannot be partitioned into a sets of sets, ranks A293243.
Cf. A000009, A002865, A005117, A045782, A279785, A293511, A296120, A316439, A381441, A382201.

facs[n_]:=If[n<=1,{{}}, Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d, Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}] := Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort /@ (#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[y], UnsameQ@@#&&And@@UnsameQ@@@#&]], {y,facs[n]}],{n,30}] (* Gus Wiseman, Apr 26 2025 *)

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A045778(n).
a(n) = A050346(A025487^(-1)(A046523(n))), where A025487^(-1) is the inverse with A025487^(-1)(A025487(n))=n. - R. J. Mathar, May 25 2017
a(n) = A050346(A101296(n)). - Antti Karttunen, May 25 2017

A265393 a(n) = the smallest number k such that floor(Sum_{d|k} 1/tau(d)) = n.

Original entry on oeis.org

1, 6, 24, 60, 180, 420, 840, 2520, 4620, 9240, 13860, 27720, 60060, 55440, 110880, 166320, 180180, 480480, 360360, 900900, 720720, 1441440, 1801800, 2162160, 3063060, 4084080, 7207200, 12612600, 6126120, 27027000, 12252240, 18378360, 43243200, 24504480

Offset: 1

Jaroslav Krizek, Dec 08 2015

nonn

Further known terms: a(29) = 6126120, a(31) = 12252240.
Are there numbers n > 1 such that Sum_{d|n} 1/tau(d) is an integer?
Sequences of numbers n such that floor(Sum_{d|n} 1/tau(d)) = k for k = 1..6:
k=1: 1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, ... (A166684);
k=2: 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, ...;
k=3: 24, 30, 36, 40, 42, 48, 54, 56, 66, 70, 72, 78, 80, 88, 96, 100, ...;
k=4: 60, 84, 90, 120, 126, 132, 140, 144, 150, 156, 168, 198, 204, 216, ...;
k=5: 180, 210, 240, 252, 300, 330, 336, 360, 390, 396, 450, 462, 468, ...;
k=6: 420, 630, 660, 720, 780, 900, 924, 990, 1008, 1020, 1050, 1080, ....
Values of function F = Sum_{d|n} 1/tau(d) for some numbers according to their prime signature: F{} = 1; F{1} = 3/2; F{2} = 11/6; F{1, 1} = 9/4; F{3} = 25/12; F{2, 1} = 11/4; F{4} = 137/60; F{3, 1} = 25/8, ...

			For n = 2; a(2) = 6 because 6 is the smallest number with floor(Sum_{d|6} 1/tau(d)) = floor(1/1 + 1/2 + 1/2 + 1/4) = floor(9/4) = 2.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..5000

Cf. A000005, A253139, A265390, A265391, A265392.

Cf. A237350 (a(n) = the smallest number k such that Sum_{d|k} 1/tau(d) >= n).

a:=1; S:=[a]; for n in [2..14] do k:=0; flag:= true; while flag do k+:=1; if Floor(&+[1/NumberOfDivisors(d): d in Divisors(k)]) eq n then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;

Table[k = 1; While[Floor@ Sum[1/DivisorSigma[0, d], {d, Divisors@ k}] != n, k++]; k, {n, 17}] (* Michael De Vlieger, Dec 09 2015 *)

a(n) = {k=1; while(k, if(floor(sumdiv(k, d, 1/numdiv(d))) == n, return(k)); k++)} \\ Altug Alkan, Dec 09 2015

More terms from Michel Marcus, Dec 23 2015

a(33)-a(34) from Hiroaki Yamanouchi, Dec 31 2015

A229964 Number of pairs of integers q1, q2 with 1 < q1 < q2 < n such that if we randomly pick an integer in {1, ..., n}, the event of being divisible by q1 is independent of being divisible by q2.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 3, 2, 1, 0, 4, 0, 5, 1, 3, 0, 8, 0, 4, 3, 4, 0, 10, 0, 7, 3, 5, 2, 9, 0, 6, 4, 9, 0, 13, 0, 12, 6, 6, 0, 16, 0, 9, 6, 9, 0, 14, 1, 12, 3, 8, 0, 25, 0, 12, 10, 11, 4, 17, 0, 12, 7, 17, 0, 25, 0, 14, 12, 14, 2, 21, 0, 21, 5

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

			If n = 12, then q1 = 2 and q2 = 5 satisfy the condition as the probability of an integer in {1, ..., 12} being divisible by 2 is 1/2, by 5 is 1/6, and by both 2 and 5 is 1/12.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

The n such that a(n) = m for various m are given by: m=0, A166684; m=1, A229965; m=2, A082663; m=3, A229966; m=4, A229967.

from math import lcm
from sympy import divisors
def A229964(n): return sum(1 for d in divisors(n,generator=True) for e in range(d+1,n) if 1Chai Wah Wu, Aug 09 2024

def A229964(n) : return sum(sum(dprob(q1, n) * dprob(q2, n) == dprob(lcm(q1,q2), n) for q2 in range(q1+1, n)) for q1 in n.divisors() if q1 not in [1,n])
def dprob(q, n) : return (n // q)/n

A229966 Numbers n such that A229964(n) = 3.

Original entry on oeis.org

12, 14, 22, 27, 33, 57, 85, 161, 203, 533, 689, 901, 1121, 1633, 2581, 4181, 5513, 5633, 7439, 10561, 18023, 18881, 20833, 21389, 23941, 25043, 28421, 32033, 37733, 48641, 58241, 64643, 66901, 77423, 80033, 84001, 90133, 106439, 116821, 119201, 149189, 155041

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {12, 14, 22, 27, 57} UNION {pq | p, q prime, q = 3p+2 or (p >= 5 and q = 4p+1)}.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229965, A229967, A023208, A023212.

[p * (3*p+2) for p in prime_range(10000) if (3*p+2).is_prime()] + [p * (4*p+1) for p in prime_range(5, 10000) if (4*p+1).is_prime()] + [12, 14, 22, 27, 57]

A229967 Numbers n such that A229964(n) = 4.

Original entry on oeis.org

18, 26, 28, 39, 65, 115, 119, 133, 319, 341, 377, 403, 481, 517, 629, 697, 731, 779, 799, 817, 893, 1007, 1207, 1219, 1357, 1403, 1541, 1769, 1943, 2059, 2077, 2117, 2201, 2263, 2291, 2407, 2449, 2573, 2759, 2923, 3071, 3293, 3589, 3649, 3737, 3811, 3827, 3959

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {18, 26, 28, 39} UNION {pq | p, q prime, p >= 5 and (2p+3 <= q <= 3p-2 or (p == 2 (mod 3) and q = 4p+3))}.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229965, A229966.

sum(([p * q for q in prime_range(2*p+3, 3*p-1)] for p in prime_range(5, 10000)), []) + [p * (4*p + 3) for p in prime_range(5, 10000) if (4*p+3).is_prime() and p%3==2] + [18, 26, 28, 39]

A229965 Numbers n such that A229964(n) = 1.

Original entry on oeis.org

6, 8, 10, 16, 21, 55, 253, 1081, 1711, 3403, 5671, 13861, 15931, 25651, 34453, 60031, 64261, 73153, 108811, 114481, 126253, 158203, 171991, 258121, 351541, 371953, 392941, 482653, 518671, 703891, 822403, 853471, 869221, 933661, 1034641, 1104841, 1159003

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {6, 8, 16} UNION A156592.

Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229966, A229967.

A266226 a(n) = floor(Sum_{d|n} 1 / tau(d)).

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Dec 24 2015

a(n) = floor(Sum_{d|n} 1 / A000005(d)).
Sequences of numbers n such that floor(Sum_{d|n} 1/tau(d)) = k for k = 1..6:
k=1: 1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, ... (A166684);
k=2: 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, ...;
k=3: 24, 30, 36, 40, 42, 48, 54, 56, 66, 70, 72, 78, 80, 88, 96, 100, ...;
k=4: 60, 84, 90, 120, 126, 132, 140, 144, 150, 156, 168, 198, 204, 216, ...;
k=5: 180, 210, 240, 252, 300, 330, 336, 360, 390, 396, 450, 462, 468, ...;
k=6: 420, 630, 660, 720, 780, 900, 924, 990, 1008, 1020, 1050, 1080, ....
See A265393 - the smallest number n such that a(n) = k for k>= 1.

			For n = 6; a(6) = floor(Sum_{d|6} 1/tau(d)) = floor(1/1 + 1/2 + 1/2 + 1/4) = floor(9/4) = 2.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000005, A237350, A253139, A265390, A265391, A265392, A265393

[Floor(&+[1/NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]];

Table[Floor[Sum[1/DivisorSigma[0, d], {d, Divisors[ n]}]], {n, 1, 100}] (* G. C. Greubel, Dec 24 2015 *)

cp's OEIS Frontend

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

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A050345 Number of ways to factor n into distinct factors with one level of parentheses.

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A265393 a(n) = the smallest number k such that floor(Sum_{d|k} 1/tau(d)) = n.

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A229964 Number of pairs of integers q1, q2 with 1 < q1 < q2 < n such that if we randomly pick an integer in {1, ..., n}, the event of being divisible by q1 is independent of being divisible by q2.

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A229966 Numbers n such that A229964(n) = 3.

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