A078174 -id:A078174 - cp's OEIS frontend

A351847 Arithmetic mean of the distinct prime factors of A078174(n).

2, 3, 2, 5, 7, 2, 3, 11, 13, 4, 2, 17, 19, 5, 23, 5, 3, 29, 31, 2, 7, 6, 37, 8, 41, 4, 43, 4, 47, 7, 10, 53, 8, 11, 59, 61, 5, 2, 9, 67, 13, 71, 73, 4, 9, 6, 79, 3, 83, 4, 11, 16, 89, 10, 17, 12, 97, 7, 101, 103, 5, 107, 109, 6, 20, 113, 8, 14, 8, 12, 11, 22, 5, 4, 127, 2

Offset: 1

Wesley Ivan Hurt, Feb 21 2022

nonn

			a(10) = 4; sopf(A078174(10))/omega(A078174(10)) = sopf(15)/omega(15) = (3+5)/(2) = 4.

Cf. A001221 (omega), A008472 (sopf), A078174.

a(n) = sopf(A078174(n))/omega(A078174(n)).

A326622 Number of factorizations of n into factors > 1 with integer average.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 3, 2, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 5, 2, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 3, 3, 1, 1, 6, 2, 2, 2, 2, 1, 2, 2, 4, 2, 1, 1, 6, 1, 1, 3, 7, 2, 1, 1, 3, 2, 1, 1, 6, 1, 1, 3, 2, 2, 2, 1, 7, 5, 1, 1, 4, 2, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 8, 1, 1, 3, 3, 1, 1, 1, 4, 5, 1, 1, 6

Offset: 1

Gus Wiseman, Jul 14 2019

nonn

			The a(80) = 7 factorizations:
  (2*2*2*10)
  (2*2*20)
  (2*5*8)
  (2*40)
  (4*20)
  (8*10)
  (80)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Partitions with integer average are A067538.
Strict partitions with integer average are A102627.
Heinz numbers of partitions with integer average are A316413.
Factorizations with integer geometric mean are A326028.
Cf. A001055, A051293, A078174, A078175, A326514, A326515, A326567/A326568, A326621, A326623, A326667 (= a(2^n)), A327906 (positions of 1's), A327907 (of terms > 1).

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],IntegerQ[Mean[#]]&]],{n,2,100}]

A326622(n, m=n, facsum=0, facnum=0) = if(1==n,facnum > 0 && 1==denominator(facsum/facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326622(n/d, d, facsum+d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024

Data section extended up to a(108), with missing term a(1)=0 also added (thus correcting the offset) - Antti Karttunen, Nov 10 2024

A078175 Numbers with an integer arithmetic mean of all prime factors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 42, 43, 44, 47, 49, 50, 51, 53, 55, 57, 59, 60, 61, 64, 65, 67, 68, 69, 71, 73, 77, 78, 79, 81, 83, 85, 87, 89, 91, 92, 93, 95, 97, 101, 103, 105, 107, 109, 110, 111, 112, 113

Offset: 1

Reinhard Zumkeller, Nov 20 2002

nonn

A001414(a(n)) == 0 modulo A001222(a(n)).

			2100=2*2*3*5*5*7: (2+2+3+5+5+7)/6=4, therefore 2100 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A001414, A078174.

Subsequences: A000040, A000079, A200612.

a078175 n = a078175_list !! (n-1)
a078175_list = filter (\x -> (a001414 x) `mod` (a001222 x) == 0) [2..]
-- Reinhard Zumkeller, Nov 20 2011

sopfr[n_] := If[n == 1, 0, Total[Times @@@ FactorInteger[n]]];
filterQ[n_] := Divisible[sopfr[n], PrimeOmega[n]];
Select[Range[2, 1000], filterQ] (* Jean-François Alcover, Apr 06 2021 *)
iamQ[n_]:=IntegerQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]]]; Select[Range[2,150],iamQ] (* Harvey P. Dale, Jun 23 2021 *)

A349156 Number of integer partitions of n whose mean is not an integer.

Original entry on oeis.org

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752

Offset: 0

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

By conjugation, also the number of integer partitions of n with greatest part not dividing n.

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

Table[Length[Select[IntegerPartitions[n],!IntegerQ[Mean[#]]&]],{n,0,30}]

a(n > 0) = A000041(n) - A067538(n).

A328966 Number of strict factorizations of n with integer average.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 1, 5, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 5, 1, 1, 3, 3, 2, 1, 1, 2, 2, 1, 1, 5, 1, 1, 3, 2, 2, 2, 1, 5, 2, 1, 1, 4, 2, 1, 2, 3

Offset: 2

Gus Wiseman, Nov 16 2019

nonn

			The a(n) factorizations for n = 2, 8, 24, 48, 96:
  (2)  (8)    (24)     (32)    (48)     (96)
       (2*4)  (4*6)    (4*8)   (6*8)    (2*48)
              (2*12)   (2*16)  (2*24)   (4*24)
              (2*3*4)          (4*12)   (6*16)
                               (2*4*6)  (8*12)
                                        (3*4*8)
                                        (2*3*16)
                                        (2*4*12)

The non-strict version is A326622.
Partitions with integer average are A067538.
Strict partitions with integer average are A102627.
Heinz numbers of partitions with integer average are A316413.
Factorizations with integer geometric mean are A326028.
Cf. A001055, A051293, A078174, A078175, A326515, A326567/A326568, A326619/A326620, A326621, A326625.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],UnsameQ@@#&&IntegerQ[Mean[#]]&]],{n,2,100}]

A326027 Number of nonempty subsets of {1..n} whose geometric mean is an integer.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 12, 19, 20, 21, 28, 29, 30, 31, 40, 41, 70, 71, 74, 75, 76, 77, 108, 123, 124, 211, 214, 215, 216, 217, 332, 333, 334, 335, 592, 593, 594, 595, 612, 613, 614, 615, 618, 639, 640, 641, 1160, 1183, 1324, 1325, 1328, 1329, 2176, 2177, 2196, 2197, 2198, 2199, 2414, 2415, 2416, 2443, 4000, 4001, 4002, 4003, 4006, 4007, 4008, 4009, 6626, 6627, 6628, 9753, 9756, 9757, 9758, 9759, 11136

Offset: 0

Gus Wiseman, Jul 14 2019

nonn

			The a(1) = 1 through a(9) = 19 subsets:
  {1}  {1}  {1}  {1}      {1}      {1}      {1}      {1}      {1}
       {2}  {2}  {2}      {2}      {2}      {2}      {2}      {2}
            {3}  {3}      {3}      {3}      {3}      {3}      {3}
                 {4}      {4}      {4}      {4}      {4}      {4}
                 {1,4}    {5}      {5}      {5}      {5}      {5}
                 {1,2,4}  {1,4}    {6}      {6}      {6}      {6}
                          {1,2,4}  {1,4}    {7}      {7}      {7}
                                   {1,2,4}  {1,4}    {8}      {8}
                                            {1,2,4}  {1,4}    {9}
                                                     {2,8}    {1,4}
                                                     {1,2,4}  {1,9}
                                                     {2,4,8}  {2,8}
                                                              {4,9}
                                                              {1,2,4}
                                                              {1,3,9}
                                                              {2,4,8}
                                                              {3,8,9}
                                                              {4,6,9}
                                                              {3,6,8,9}

Max Alekseyev, Table of n, a(n) for n = 0..255
Wikipedia, Geometric mean

First differences are A082553.
Partitions whose geometric mean is an integer are A067539.
Strict partitions whose geometric mean is an integer are A326625.
Subsets whose average is an integer are A051293.
Cf. A078174, A078175, A102627, A326567/A326568, A326622, A326623, A326624.

Table[Length[Select[Subsets[Range[n]],IntegerQ[GeometricMean[#]]&]],{n,0,10}]

a(n) = A357413(n) + A357414(n). For a squarefree n, a(n) = a(n-1) + 1. - Max Alekseyev, Mar 01 2025

Terms a(57) onward from Max Alekseyev, Mar 01 2025

A326620 Denominator of the average of the set of distinct prime indices of n.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Jul 14 2019

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.

			The distinct prime indices of 12 are {1,2}, with average 3/2, so a(12) = 2.
The sequence of fractions begins: 1, 2, 1, 3, 3/2, 4, 1, 2, 2, 5, 3/2, 6, 5/2, 5/2, 1, 7, 3/2, 8, 2, 3, 3, 9, 3/2, 3, 7/2, 2, 5/2, 10, 2.

Antti Karttunen, Table of n, a(n) for n = 2..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of 1's are A326621.
The average of the multiset of prime indices is A326567/A326568.
The average of the multiset of prime factors is A123528/A123529.
The average of the set of distinct prime indices is A326619/A326620.
The average of the set of distinct prime factors is A323171/A323172.
Cf. A001221, A067629, A078174, A078175, A112798, A316413.

Table[Denominator[Mean[PrimePi/@First/@FactorInteger[n]]],{n,2,100}]

A326620(n) = if(1==n,0,denominator(vecsum(apply(primepi,factor(n)[,1]))/omega(n))); \\ Antti Karttunen, Jan 28 2025

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142

Offset: 1

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms and their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  42: {1,2,4}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}

A version counting nonempty subsets is A000079 - A051293.
A version counting factorizations is A001055 - A326622.
A version counting compositions is A011782 - A271654.
A version for prime factors is A175352, complement A078175.
A version for distinct prime factors A176587, complement A078174.
The complement is A316413, counted by A067538, strict A102627.
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A018818 counts partitions into divisors, ranked by A326841.
A143773 counts partitions into multiples of the length, ranked by A316428.
A236634 counts unbalanced partitions.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement A237984.
Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[];  # Alois P. Heinz, Nov 19 2021

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!IntegerQ[Mean[primeMS[#]]]&]

A326619 Numerator of the average of the set of distinct prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 3, 4, 1, 2, 2, 5, 3, 6, 5, 5, 1, 7, 3, 8, 2, 3, 3, 9, 3, 3, 7, 2, 5, 10, 2, 11, 1, 7, 4, 7, 3, 12, 9, 4, 2, 13, 7, 14, 3, 5, 5, 15, 3, 4, 2, 9, 7, 16, 3, 4, 5, 5, 11, 17, 2, 18, 6, 3, 1, 9, 8, 19, 4, 11, 8, 20, 3, 21, 13, 5, 9, 9, 3, 22, 2, 2, 7

Offset: 2

Gus Wiseman, Jul 14 2019

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.

			The distinct prime indices of 12 are {1,2}, with average 3/2, so a(12) = 3.
The sequence of fractions begins: 1, 2, 1, 3, 3/2, 4, 1, 2, 2, 5, 3/2, 6, 5/2, 5/2, 1, 7, 3/2, 8, 2, 3, 3, 9, 3/2, 3, 7/2, 2, 5/2, 10, 2.

The average of the multiset of prime indices is A326567/A326568.
The average of the multiset of prime factors is A123528/A123529.
The average of the set of distinct prime indices is A326619/A326620.
The average of the set of distinct prime factors is A323171/A323172.
Cf. A001221, A067629, A078174, A078175, A112798, A316413, A326621.

Table[Numerator[Mean[PrimePi/@First/@FactorInteger[n]]],{n,2,100}]

A326621 Numbers n such that the average of the set of distinct prime indices of n is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 60, 61, 62, 63, 64, 67, 68, 71, 73, 78, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 94, 97, 100, 101, 103, 105

Offset: 1

Gus Wiseman, Jul 14 2019

nonn

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose distinct parts have an integer average.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}

Positions of 1's in A326620.

Cf. A051293, A056239, A067538, A078174, A078175, A102627, A112798, A326567/A326568, A326619/A326620, A326622.

Select[Range[2,100],IntegerQ[Mean[PrimePi/@First/@FactorInteger[#]]]&]

cp's OEIS Frontend

A351847 Arithmetic mean of the distinct prime factors of A078174(n).

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A326622 Number of factorizations of n into factors > 1 with integer average.

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A078175 Numbers with an integer arithmetic mean of all prime factors.

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A349156 Number of integer partitions of n whose mean is not an integer.

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A328966 Number of strict factorizations of n with integer average.

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Mathematica

A326027 Number of nonempty subsets of {1..n} whose geometric mean is an integer.

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A326620 Denominator of the average of the set of distinct prime indices of n.

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A348551 Heinz numbers of integer partitions whose mean is not an integer.

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Maple