A078175 -id:A078175 - cp's OEIS frontend

A174894 Numbers such that the arithmetic mean of their distinct prime factors and the arithmetic mean of all of their prime factors are both integers.

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 42, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 78, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 101

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

Subsequence of A078174 and A078175.

Complement of A176552. [From Jaroslav Krizek, Apr 21 2010]

			For a(11) = 16: 16 = 2^4; both (2+2+2+2)/4 and 2/1 are integers.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

mdmaQ[n_]:=With[{fi=FactorInteger[n]},AllTrue[{Mean[Flatten[Table[#[[1]],#[[2]]]&/@fi]],Mean[fi[[;;,1]]]},IntegerQ]]; Select[Range[ 2,110],mdmaQ] (* Harvey P. Dale, Nov 16 2024 *)

Definition clarified by Harvey P. Dale, Nov 16 2024

A200612 The arithmetic mean of the prime factors (with multiplicity) of n is 3.

Original entry on oeis.org

3, 9, 20, 27, 60, 81, 112, 180, 243, 336, 400, 540, 729, 1008, 1200, 1620, 2187, 2240, 2816, 3024, 3600, 4860, 6561, 6720, 8000, 8448, 9072, 10800, 12544, 13312, 14580, 19683, 20160, 24000, 25344, 27216, 32400, 37632, 39936, 43740, 44800, 56320, 59049, 60480

Offset: 1

Jeffrey Burch, Nov 19 2011

nonn

			20 is in the sequence because 20 = 2*2*5 and (2+2+5)/3 = 9/3 = 3.

Reinhard Zumkeller and Donovan Johnson, Table of n, a(n) for n = 1..500 (first 150 terms from Reinhard Zumkeller)

Subsequence of A078175.

a200612 n = a200612_list !! (n-1)
a200612_list = filter f [2..] where
   f x = r == 0 && x' == 3 where (x',r) = divMod (a001414 x) (a001222 x)
-- Reinhard Zumkeller, Nov 20 2011

for i from 2 to 35000 do: a:=ifactors(i): s:=sum((a[2][j][1]*a[2][j][2]),j=1..nops(a[2])): t:=sum((a[2][j][2]),j=1..nops(a[2])): if s/t=3 then print(i); fi od:

Select[Range[61000],Mean[Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[ #]]]==3&] (* Harvey P. Dale, Nov 08 2013 *)

isok(n) = my(f = factor(n)); (sum(k=1, #f~, f[k,1]*f[k,2]) / vecsum(f[,2])) == 3; \\ Michel Marcus, Feb 22 2016

A001414(a(n)) mod A001222(a(n)) = 0 and A001414(a(n))/A001222(a(n)) = 3. [Reinhard Zumkeller, Nov 20 2011]

A327900 Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).

Original entry on oeis.org

1, 15, 33, 51, 55, 85, 91, 93, 123, 155, 161, 165, 177, 187, 201, 203, 205, 249, 255, 295, 299, 301, 327, 329, 335, 341, 377, 381, 415, 451, 465, 471, 511, 527, 537, 545, 553, 559, 561, 573, 611, 615, 633, 635, 649, 667, 679, 697, 703, 707, 723, 737, 785, 831

Offset: 1

Gus Wiseman, Sep 30 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 sequence of terms together with their prime indices begins:
    1: {}
   15: {2,3}
   33: {2,5}
   51: {2,7}
   55: {3,5}
   85: {3,7}
   91: {4,6}
   93: {2,11}
  123: {2,13}
  155: {3,11}
  161: {4,9}
  165: {2,3,5}
  177: {2,17}
  187: {5,7}
  201: {2,19}
  203: {4,10}
  205: {3,13}
  249: {2,23}
  255: {2,3,7}
  295: {3,17}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The case including primes and nonsquarefree numbers is A320324.
The version for sum of prime indices is A327901.
The version for mean of prime indices is A327902.
Cf. A001222, A038041, A056239, A078175, A112798, A306017, A306021, A316413, A317583, A322794, A327908.

Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@PrimeOmega/@PrimePi/@First/@FactorInteger[#]&]

A327902 Nonprime squarefree numbers whose prime indices all have the same average of prime indices (A326567/A326568).

Original entry on oeis.org

1, 21, 57, 115, 133, 145, 159, 371, 393, 399, 515, 535, 565, 667, 803, 869, 917, 933, 1007, 1067, 1113, 1963, 2021, 2095, 2157, 2165, 2177, 2249, 2285, 2315, 2363, 2369, 2461, 2489, 2599, 2705, 2751, 2839, 2987, 3021, 3103, 3277, 3335, 3707, 3859, 4331, 4367

Offset: 1

Gus Wiseman, Sep 30 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 sequence of terms together with their prime indices begins:
     1: {}
    21: {2,4}
    57: {2,8}
   115: {3,9}
   133: {4,8}
   145: {3,10}
   159: {2,16}
   371: {4,16}
   393: {2,32}
   399: {2,4,8}
   515: {3,27}
   535: {3,28}
   565: {3,30}
   667: {9,10}
   803: {5,21}
   869: {5,22}
   917: {4,32}
   933: {2,64}
  1007: {8,16}
  1067: {5,25}

The version including primes and nonsquarefree numbers is A326536.
The version for number of prime indices is A327900.
The version for sum of prime indices is A327901.
Cf. A001222, A005117, A056239, A078175, A102627, A112798, A316413, A326567/A326568, A326621, A327908.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@Mean/@primeMS/@primeMS[#]&];

A327906 Numbers with only one factorization into factors > 1 with integer mean (namely, as a singleton).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 18, 19, 22, 23, 26, 29, 30, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 66, 67, 70, 71, 73, 74, 79, 82, 83, 86, 89, 90, 94, 97, 98, 101, 102, 103, 106, 107, 109, 113, 118, 122, 127, 130, 131, 134, 137, 138, 139, 142

Offset: 1

Gus Wiseman, Sep 30 2019

nonn

			There are 4 factorizations of 24 with integer mean, namely:
  (24)
  (4*6)
  (2*12)
  (2*3*4)
so 24 is not in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement of A327907.
Positions of 1's in A326622.
Cf. A001055, A051293, A067538, A078175, A123528/A123529, A316413, A326515, A326516, A326666, A326667, A326671.

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

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
isA327906(n) = (1==A326622(n)); \\ Antti Karttunen, Nov 10 2024

A339452 Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 1, 1, 5, 1, 1, 9, 7, 3, 1, 3, 1, 7, 11, 13, 1, 7, 1, 11, 35, 25, 31, 27, 5, 157, 1, 31, 131, 39, 31, 33, 37, 183, 179, 135, 157, 7, 265, 3, 871, 187, 865, 259, 879, 867, 179, 1593, 6073, 1593, 271, 5995, 149, 6661, 2411, 1509, 997, 1045, 5887

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(10) = 5 because we have [10], [9, 1], [1, 9], [8, 2] and [2, 8].

Eric W. Weisstein's World of Mathematics, Geometric Mean
Index entries for sequences related to compositions

For partitions we have A326625, non-strict A067539 (ranked by A326623).
The version for subsets is A326027.
For arithmetic mean we have A339175, non-strict A271654.
The non-strict case is counted by A357710, ranked by A357490.
A032020 counts strict compositions.
A067538 counts partitions with integer average.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A051293, A078174, A096199, A102627, A326622, A326624, A326028, A326641, A326645, A335405.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&IntegerQ[GeometricMean[#]]&]],{n,0,15}] (* Gus Wiseman, Oct 30 2022 *)

A303482 Numbers k such that the average of all distinct prime factors of all positive integers <= k is an integer.

Original entry on oeis.org

2, 5, 81, 10742, 10130527, 1041972864, 23292549600

Offset: 1

Ilya Gutkovskiy, Apr 24 2018

Numbers k such that A013939(k)|A024924(k).

			5 is in the sequence because the distinct prime factors of 2, 3, 4, and 5 are 2, 3, 2 and 5 respectively and their average (2 + 3 + 2 + 5) / 4 = 3 is an integer. - _David A. Corneth_, Apr 26 2018

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221, A008472, A013939, A024924, A078174, A078175, A226647, A284755, A303480.

s = t = 0; k = 2; lst = {}; While[k < 1000000000, p = #[[1]] & /@ FactorInteger@ k; s = s + Plus @@ p; t = t + Length@ p; If[ Mod[s, t] == 0, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Apr 26 2018 *)

a(5) from Daniel Suteu, Apr 24 2018

a(6)-a(7) from Giovanni Resta, Apr 26 2018

A327907 Numbers with more than one factorization into at factors > 1 with integer mean.

Original entry on oeis.org

4, 8, 9, 12, 15, 16, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 68, 69, 72, 75, 76, 77, 78, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 105, 108, 110, 111, 112, 114, 115, 116

Offset: 1

Gus Wiseman, Sep 30 2019

nonn

			There are 6 factorizations of 60 with integer mean, namely:
  (60)
  (2*30)
  (6*10)
  (3*4*5)
  (2*3*10)
  (2*2*3*5)
so 60 is in the sequence.

Complement of A327906.
Positions of terms > 1 in A326622.
Cf. A001055, A051293, A067538, A078175, A123528/A123529, A316413, A326515, A326516, A326666, A326667, A326671.

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

A357490 Numbers k such that the k-th composition in standard order has integer geometric mean.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 15, 16, 17, 24, 31, 32, 36, 42, 63, 64, 69, 70, 81, 88, 98, 104, 127, 128, 136, 170, 255, 256, 277, 278, 282, 292, 325, 326, 337, 344, 354, 360, 394, 418, 424, 511, 512, 513, 514, 515, 528, 547, 561, 568, 640, 682, 768, 769, 785, 792, 896

Offset: 1

Gus Wiseman, Oct 16 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  24: (1,4)
  31: (1,1,1,1,1)
  32: (6)
  36: (3,3)
  42: (2,2,2)
  63: (1,1,1,1,1,1)
  64: (7)
  69: (4,2,1)

For regular mean we have A096199, counted by A271654 (partitions A067538).
Subsets whose geometric mean is an integer are counted by A326027.
The unordered version (partitions) is A326623, counted by A067539.
The strict case is counted by A339452, partitions A326625.
These compositions are counted by A357710.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A051293, A078174, A102627, A301987, A326622, A326624, A326028, A326567/A326568, A326641, A326645, A335405, A357184.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],IntegerQ[GeometricMean[stc[#]]]&]

A358331 Number of integer partitions of n with arithmetic and geometric mean differing by one.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 3, 3, 0, 0, 2, 2, 0, 4, 0, 0, 5, 0, 0, 4, 5, 4, 3, 2, 0, 3, 3, 10, 4, 0, 0, 7, 0, 0, 16, 2, 4, 4, 0, 0, 5, 24, 0, 6, 0, 0, 9, 0, 27, 10, 0, 7, 7, 1, 0, 44

Offset: 0

Gus Wiseman, Nov 09 2022

nonn

The arithmetic and geometric mean from such partition is a positive integer. - David A. Corneth, Nov 11 2022

			The a(30) = 2 through a(36) = 3 partitions (C = 12, G = 16):
  (888222)      .  (99333311)  (G2222222111)  .  (C9662)    (G884)
  (8844111111)                                   (C9833)    (888222111111)
                                                 (8884421)  (G42222221111)

Chai Wah Wu, Table of n, a(n) for n = 0..143
Wikipedia, Geometric mean

The version for subsets seems to be close to A178832.
These partitions are ranked by A358332.
A000041 counts partitions.
A067538 counts partitions with integer average, ranked by A316413.
A067539 counts partitions with integer geometric mean, ranked by A326623.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A078174, A102627, A271654, A326027, A326624, A326625, A326028, A326641, A326645, A357710.

Table[Length[Select[IntegerPartitions[n],Mean[#]==1+GeometricMean[#]&]],{n,0,30}]

a(n) = if (n, my(nb=0,vp); forpart(p=n, vp=Vec(p); if (vecsum(vp)/#p == 1 + sqrtn(vecprod(vp), #p), nb++)); nb, 0); \\ Michel Marcus, Nov 11 2022

from math import prod
from sympy import divisors, integer_nthroot
from sympy.utilities.iterables import partitions
def A358331(n):
    divs = {d:n//d-1 for d in divisors(n,generator=True)}
    return sum(1 for s,p in partitions(n,m=max(divs,default=0),size=True) if s in divs and (t:=integer_nthroot(prod(a**b for a, b in p.items()),s))[1] and divs[s]==t[0]) # Chai Wah Wu, Sep 24 2023

a(61)-a(80) from Giorgos Kalogeropoulos, Nov 11 2022

a(81)-a(84) from Chai Wah Wu, Sep 24 2023

cp's OEIS Frontend

A174894 Numbers such that the arithmetic mean of their distinct prime factors and the arithmetic mean of all of their prime factors are both integers.

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A200612 The arithmetic mean of the prime factors (with multiplicity) of n is 3.

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A327900 Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).

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A327902 Nonprime squarefree numbers whose prime indices all have the same average of prime indices (A326567/A326568).

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A327906 Numbers with only one factorization into factors > 1 with integer mean (namely, as a singleton).

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A339452 Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.

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Mathematica

A303482 Numbers k such that the average of all distinct prime factors of all positive integers <= k is an integer.

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Mathematica

Extensions

A327907 Numbers with more than one factorization into at factors > 1 with integer mean.

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