A055212 -id:A055212 - cp's OEIS frontend

A344713 a(n) is the number of iterations needed for n to reach 0 under the mapping x -> A055212(x).

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

Offset: 1

Timothy L. Tiffin, May 26 2021

nonn

Since x > A055212(x) for all positive integers x, and the smallest value of A055212(x) is 0, every trajectory under iteration of the mapping x -> A055212(x) will end at 0.
If n = 1 or n is prime, then 0 will be reached in just one iteration of the mapping. Moreover, a(1), a(2), and a(3) form the only run of three consecutive 1's. All other 1's are isolated according to the prime numbers greater than 3.
If n is a composite number, then its trajectory under the mapping consists of a first step n -> A055212(n) followed by a(A055212(n)) steps to reach 0. So, a(n) = a(A055212(n)) + 1.

			a(1) = 1, since 1 -> 0.
a(p) = 1, since p -> 0 for any prime p.
a(4) = 2, since 4 -> 1 -> 0.
a(30) = 3, since 30 -> 4 -> 1 -> 0.
a(1440) = 4, since 1440 -> 32 -> 4 -> 1 -> 0.

Cf. A000005, A036459, A055212.

f[0] = 0; f[n_] := DivisorSigma[0, n] - PrimeNu[n] - 1; a[n_] := -2 + Length @ FixedPointList[f, n]; Array[a, 100] (* Amiram Eldar, Jun 03 2021 *)

A083399 Number of divisors of n that are not divisors of other divisors of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jun 12 2003

a(n) <= tau(n); a(n) = tau(n) iff n is prime or n=1 (A008578, A000040); a(n)=tau(n)-1 iff n is semiprime (A001358).
Number of noncomposite divisors of n, (cf. A008578). - Jaroslav Krizek, Nov 25 2009
From Wilf A. Wilson, Jul 21 2017: (Start)
a(n) is the number of maximal subsemigroups of the annular Jones monoid of degree n.
a(n) is the number of maximal subsemigroups of the monoid of orientation-preserving mappings on a set with n elements.
a(n) + 1 is the number of maximal subsemigroups of the monoid of orientation-preserving partial mappings on a set with n elements.
(End)
This is the restricted growth sequence transform of A001221 (and thus also of A007875, A034444, A082476, A292586 and many other sequences). This follows from the formula a(n) = 1+A001221(n), and from the fact that for any n, A001221(n) <= 1+A001221(k) for all k = 1..(n-1). A067003 gives the ordinal transform of A001221. See also A292582, A292583, A292585. - Antti Karttunen, Sep 25 2017

			{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2, 3, 4 and 6 divide not only 24, but also 8 or 12, therefore a(24) = 3.
{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2 and 3 are noncomposites, therefore a(24) = 3. - _Jaroslav Krizek_, Nov 25 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]

Cf. A000005, A001221, A067003, A055212, A077761.

a083399 = (+ 1) . a001221  -- Reinhard Zumkeller, Sep 14 2015

[(#(PrimeDivisors(n)))+1: n in [1..100]]; // Vincenzo Librandi, Feb 15 2015

A083399 := proc(n)
    1+nops(numtheory[factorset](n)) ;
end proc:
seq(A083399(n),n=1..100) ; # R. J. Mathar, Sep 26 2017

A083399[n_Integer]:=1+PrimeNu[n]; (* Enrique Pérez Herrero, Jun 25 2010 *)
Rest@ CoefficientList[ Series[(1/(1 - x)) + Sum[1/(1 - x^Prime[j]), {j, 200}], {x, 0, 111}], x] (* Robert G. Wilson v, Aug 16 2011 *)

a(n)=#factor(n)~+1 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = omega(n) + 1, where omega = A001221.
a(n) = tau(n) - A055212(n) = A000005(n)-A055212(n).
a(n) = A000005(n) - A033273(n) + 1. - Jaroslav Krizek, Nov 25 2009
a(n) = A010553(A007947(n)) = A000005(A000005(A007947(n))) = tau_2(tau_2(rad(n))). - Enrique Pérez Herrero, Jun 25 2010
G.f.: x/(1 - x) + Sum_{k>=1} x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) = n * (log(log(n)) + B + 1) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 29 2024

A364531 Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299702 (knapsack) in having 525: {2,3,3,4}.
First differs from A325778 in lacking 462: {1,2,4,5}.
These are the Heinz numbers of partitions whose parts are disjoint from their own non-singleton subset-sums.

Partitions of this type are counted by A237667, strict A364349.
The binary version is A364462, complement A364461.
The complement is A364532, counted by A237668.
A000005 counts divisors, nonprime A033273, composite A055212.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, counted by A108917, complement A299729.
A363260 counts partitions disjoint from differences, complement A364467.
Cf. A002865, A236912, A237113, A320340, A326083, A363225, A364347.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]=={}&]

A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325, 330

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299729 (non-knapsack) in lacking 525: {2,3,3,4}.
First differs from A325777 in having 462: {1,2,4,5} and lacking 675:{2,2,2,3,3}.
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.
These are the Heinz numbers of partitions containing the sum of some non-singleton submultiset.

			The terms together with their prime indices begin:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Partitions not of this type are counted by A237667, strict A364349.
Partitions of this type are counted by A237668, strict A364272.
The binary complement is A364461, re-usable A364347 (counted by A364345).
The binary version is A364462, re-usable A364348 (counted by A363225).
The complement is A364531.
Subsets of this type are counted by A364534, complement A151897.
A000005 counts divisors, nonprime A033273, composite A055212.
A001222 counts prime indices.
A108917 counts knapsack partitions, strict A275972, for subsets A325864.
A112798 lists prime indices, sum A056239.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, complement A299729.
Cf. A085489, A088809, A093971, A236912, A237113, A301900, A326083, A364350.

Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]!={}&]

A325863 Number of integer partitions of n such that every distinct non-singleton submultiset has a different sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 24, 29, 31, 41, 51, 58, 67, 84, 91, 117, 117

Offset: 0

Gus Wiseman, May 31 2019

A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1).

			The partition (2,1,1,1) has non-singleton submultisets {1,2} and {1,1,1} with the same sum, so (2,1,1,1) is not counted under a(5).
The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (311)    (222)     (322)      (71)
                            (11111)  (321)     (331)      (332)
                                     (411)     (421)      (422)
                                     (3111)    (511)      (431)
                                     (111111)  (2221)     (521)
                                               (4111)     (611)
                                               (1111111)  (2222)
                                                          (3311)
                                                          (5111)
                                                          (41111)
                                                          (11111111)
The 10 non-knapsack partitions counted under a(12):
  (7,6,1)
  (7,5,2)
  (7,4,3)
  (7,5,1,1)
  (7,4,2,1)
  (7,3,3,1)
  (7,3,2,2)
  (7,4,1,1,1)
  (7,2,2,2,1)
  (7,1,1,1,1,1,1,1)

Dominates A108917.

Cf. A002033, A055212, A143823, A196723, A276024, A299702, A325856, A325862, A325864, A325865, A325866, A325867, A325877.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Union[Subsets[#,{2,Length[#]}]]&]],{n,0,15}]

A137944 Numbers such that the number of composite divisors is a multiple of the number of prime divisors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 168, 169, 173, 179, 181, 191, 193

Offset: 1

Reinhard Zumkeller, Feb 24 2008

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A001221, A055212.

Disjoint union of A000961 and A137945.

aQ[n_] := Divisible[DivisorSigma[0, n] - 1,  PrimeNu[n]]; Select[Range[2, 193], aQ] (* Amiram Eldar, Aug 31 2019 *)

isok(k) = if(k == 1, 0, my(f = factor(k)); !((numdiv(f)-1) % omega(f))); \\ Amiram Eldar, Apr 18 2025

A055212(a(n)) mod A001221(a(n)) = 0.

A137945 Non-prime-powers such that the number of composite divisors is a multiple of the number of prime divisors.

Original entry on oeis.org

36, 100, 120, 144, 168, 196, 225, 264, 270, 280, 312, 324, 378, 400, 408, 440, 441, 456, 484, 520, 552, 576, 594, 616, 676, 680, 696, 702, 728, 744, 750, 760, 784, 888, 918, 920, 945, 952, 960, 984, 1026, 1032, 1064, 1089, 1128, 1144, 1156, 1160, 1225, 1240

Offset: 1

Reinhard Zumkeller, Feb 24 2008

nonn

			A055212(120) = #{4,6,8,10,12,15,20,24,30,40,60,120} = 12 = 4*A001221(120) = 4*#{2,3,5} = 12, therefore 120 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Divisor Function

Cf. A001221, A055212.

Intersection of A024619 and A137944.

aQ[n_] := (omega = PrimeNu[n]) > 1 && Divisible[DivisorSigma[0, n] - 1, omega]; Select[Range[2, 1240], aQ] (* Amiram Eldar, Aug 31 2019 *)

A055212(a(n)) mod A001221(a(n)) = 0.

A261609 Number of composite divisors of n^2+1.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 3, 1, 1, 0, 1, 1, 4, 0, 1, 0, 4, 3, 1, 0, 4, 1, 4, 0, 1, 0, 4, 1, 1, 1, 4, 3, 4, 1, 1, 0, 4, 3, 1, 0, 3, 1, 8, 1, 1, 1, 11, 1, 1, 1, 1, 1, 4, 0, 4, 0, 12, 1, 1, 1, 1, 1, 4, 1, 1, 0, 4, 5, 1, 3, 1, 4, 11, 0, 4, 1, 4, 1, 1, 1, 4, 3, 11, 0, 1, 1

Offset: 1

Michel Lagneau, Aug 26 2015

			a(7) = 3 because the composite divisors of 7^2+1 are 10, 25, 50.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A002522, A055212, A128428, A193330, A193432.

Table[ Count[ PrimeQ[ Divisors[n^2+1] ], False] - 1, {n, 1, 105} ]
Table[Count[Divisors[n^2+1],?CompositeQ],{n,100}] (* _Harvey P. Dale, Dec 16 2024 *)

a(n) = A055212(A002522(n)).

A306345 Absolute difference between the number of prime divisors and the number of composite divisors of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 0, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 5, 1, 1, 1, 3

Offset: 1

Felix Fröhlich, Feb 08 2019

nonn

Conjecture: a(n) = 0 iff n is a term of A280076 = union of A001248 and {1}.

Conjecture is true, since having an n with k distinct prime factors such that a(n) = 0 requires that 2k+1 can be factored into k parts > 1, and 1 is the only positive k for which this is possible. - Charlie Neder, Feb 12 2019

			For n = 24: The set of divisors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}. The prime divisors are {2, 3} and the composite divisors are {4, 6, 8, 12, 24}. The cardinalities of the sets are 2 and 5, respectively, and abs(2-5) = 3, so a(24) = 3.

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

Cf. A000005, A001221, A001248, A055212, A280076.

Array[Abs[2 PrimeNu@ # - DivisorSigma[0, #] + 1] &, 105] (* Michael De Vlieger, Feb 17 2019 *)

a(n) = my(d=divisors(n), p=0, c=0); for(k=2, #d, if(ispseudoprime(d[k]), p++, c++)); abs(p-c)

a(n) = abs(2*omega(n) - numdiv(n) + 1); \\ Michel Marcus, Feb 12 2019

a(n) = abs(A001221(n) - A055212(n)).

a(n) = abs(2*A001221(n) - A000005(n) + 1). - Michel Marcus, Feb 12 2019

a(1)=0 prepended by David A. Corneth, Feb 12 2019

cp's OEIS Frontend

A344713 a(n) is the number of iterations needed for n to reach 0 under the mapping x -> A055212(x).

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A083399 Number of divisors of n that are not divisors of other divisors of n.

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A364531 Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.

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A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

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A325863 Number of integer partitions of n such that every distinct non-singleton submultiset has a different sum.

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A137944 Numbers such that the number of composite divisors is a multiple of the number of prime divisors.

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A137945 Non-prime-powers such that the number of composite divisors is a multiple of the number of prime divisors.

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A261609 Number of composite divisors of n^2+1.

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