A327476 -id:A327476 - cp's OEIS frontend

A126594 Floor of the average of the prime factors of n with multiplicity.

2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 4, 4, 2, 17, 2, 19, 3, 5, 6, 23, 2, 5, 7, 3, 3, 29, 3, 31, 2, 7, 9, 6, 2, 37, 10, 8, 2, 41, 4, 43, 5, 3, 12, 47, 2, 7, 4, 10, 5, 53, 2, 8, 3, 11, 15, 59, 3, 61, 16, 4, 2, 9, 5, 67, 7, 13, 4, 71, 2, 73, 19, 4, 7, 9, 6, 79, 2, 3, 21, 83, 3, 11, 22, 16, 4, 89, 3, 10

Offset: 2

Cino Hilliard, Jan 06 2007

Cf. A067629 (rounding instead of flooring), A076690.
This is the floor of A123528/A123529.
Without multiplicity we have A363895.
For prime indices instead of factors we have A363943, triangle A363945.
Positions of first appearances are A364037.
The ceiling is A364156.
Positions of 2's are A364157, for prime indices A363949.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, ranks A316413.
A078175 lists numbers with integer mean of prime factors.
Cf. A026905, A326567/A326568, A327473, A327476, A363944, A363948, A363950.

Table[Floor[(Plus@@Times@@@FactorInteger[n])/PrimeOmega[n]], {n, 2, 90}] (* Alonso del Arte, May 21 2012 *)

avg(n) = { local(x,j,ln) for(x=2,n,a=ifactor(x); ln=length(a); print1(floor(sum(j=1,ln,a[j])/ln)",")) } ifactor(n) = \The vector of the prime factors of n with multiplicity. { local(f,j,k,flist); flist=[]; f=Vec(factor(n)); for(j=1,length(f[1]), for(k = 1,f[2][j],flist = concat(flist,f[1][j]) ); ); return(flist) }

a(p^n)=p, p prime, n >= 1. - Philippe Deléham, Nov 23 2008

a(n) = floor(A001414(n)/A001222(n)). - Philippe Deléham, Nov 24 2008

A362980 Numbers whose multiset of prime factors (with multiplicity) has different median from maximum.

Original entry on oeis.org

6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 104, 105, 106, 110

Offset: 1

Gus Wiseman, May 12 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime factorization of 108 is 2*2*3*3*3, and the multiset {2,2,3,3,3} has median 3 and maximum 3, so 108 is not in the sequence.
The prime factorization of 2250 is 2*3*3*5*5*5, and the multiset {2,3,3,5,5,5} has median 4 and maximum 5, so 2250 is in the sequence.
The terms together with their prime indices begin:
     6: {1,2}        36: {1,1,2,2}      60: {1,1,2,3}
    10: {1,3}        38: {1,8}          62: {1,11}
    12: {1,1,2}      39: {2,6}          63: {2,2,4}
    14: {1,4}        40: {1,1,1,3}      65: {3,6}
    15: {2,3}        42: {1,2,4}        66: {1,2,5}
    20: {1,1,3}      44: {1,1,5}        68: {1,1,7}
    21: {2,4}        45: {2,2,3}        69: {2,9}
    22: {1,5}        46: {1,9}          70: {1,3,4}
    24: {1,1,1,2}    48: {1,1,1,1,2}    72: {1,1,1,2,2}
    26: {1,6}        51: {2,7}          74: {1,12}
    28: {1,1,4}      52: {1,1,6}        76: {1,1,8}
    30: {1,2,3}      55: {3,5}          77: {4,5}
    33: {2,5}        56: {1,1,1,4}      78: {1,2,6}
    34: {1,7}        57: {2,8}          80: {1,1,1,1,3}
    35: {3,4}        58: {1,10}         82: {1,13}

Partitions of this type are counted by A237821.
For mode instead of median we have A362620, counted by A240302.
The complement is A362621, counted by A053263.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
Cf. A000040, A002865, A171979, A237824, A237984, A327473, A327476, A359908, A362616, A362619, A362622.

Select[Range[100],(y=Flatten[Apply[ConstantArray,FactorInteger[#],{1}]];Max@@y!=Median[y])&]

A362981 Heinz numbers of integer partitions such that 2*(least part) >= greatest part.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 91, 96, 97, 101, 103, 105, 107, 108, 109, 113, 119, 121, 125

Offset: 1

Gus Wiseman, May 14 2023

nonn

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.

By conjugation, also Heinz numbers of partitions whose greatest part appears at a middle position, namely k/2, (k+1)/2, or (k+2)/2, where k is the number of parts. These partitions have ranks A362622.

			The terms together with their prime indices begin:
     1: {}         16: {1,1,1,1}      36: {1,1,2,2}
     2: {1}        17: {7}            37: {12}
     3: {2}        18: {1,2,2}        41: {13}
     4: {1,1}      19: {8}            43: {14}
     5: {3}        21: {2,4}          45: {2,2,3}
     6: {1,2}      23: {9}            47: {15}
     7: {4}        24: {1,1,1,2}      48: {1,1,1,1,2}
     8: {1,1,1}    25: {3,3}          49: {4,4}
     9: {2,2}      27: {2,2,2}        53: {16}
    11: {5}        29: {10}           54: {1,2,2,2}
    12: {1,1,2}    31: {11}           55: {3,5}
    13: {6}        32: {1,1,1,1,1}    59: {17}
    15: {2,3}      35: {3,4}          61: {18}

For prime factors instead of indices we have A081306.
Prime indices are listed by A112798, length A001222, sum A056239.
The complement is A362982, counted by A237820.
Partitions of this type are counted by A237824.
Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],2*Min@@prix[#]>=Max@@prix[#]&]

A362982 Heinz numbers of partitions such that 2*(least part) < greatest part.

Original entry on oeis.org

10, 14, 20, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 56, 57, 58, 60, 62, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 115, 116, 117, 118, 120, 122, 123, 124, 126

Offset: 1

Gus Wiseman, May 14 2023

nonn

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 together with their prime indices begin:
    10: {1,3}        44: {1,1,5}      70: {1,3,4}
    14: {1,4}        46: {1,9}        74: {1,12}
    20: {1,1,3}      50: {1,3,3}      76: {1,1,8}
    22: {1,5}        51: {2,7}        78: {1,2,6}
    26: {1,6}        52: {1,1,6}      80: {1,1,1,1,3}
    28: {1,1,4}      56: {1,1,1,4}    82: {1,13}
    30: {1,2,3}      57: {2,8}        84: {1,1,2,4}
    33: {2,5}        58: {1,10}       85: {3,7}
    34: {1,7}        60: {1,1,2,3}    86: {1,14}
    38: {1,8}        62: {1,11}       87: {2,10}
    39: {2,6}        66: {1,2,5}      88: {1,1,1,5}
    40: {1,1,1,3}    68: {1,1,7}      90: {1,2,2,3}
    42: {1,2,4}      69: {2,9}        92: {1,1,9}

For prime factors instead of indices we have A069900, complement A081306.
Prime indices are listed by A112798, length A001222, sum A056239.
Partitions of this type are counted by A237820.
The complement is A362981, counted by A237824.
Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],2*Min@@prix[#]

A363954 Numbers whose prime indices have low mean 2.

Original entry on oeis.org

3, 9, 10, 14, 15, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 70, 75, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 189, 196, 198, 204, 208, 210, 220, 225, 234, 243, 250, 252, 260, 264, 270, 272, 280, 294, 297, 300, 304, 308, 312, 315, 330, 350

Offset: 1

Gus Wiseman, Jul 05 2023

nonn

Extending the terminology of A124944, the "low mean" of a multiset is obtained by taking the mean and rounding down.

			The terms together with their prime indices begin:
     3: {2}
     9: {2,2}
    10: {1,3}
    14: {1,4}
    15: {2,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    42: {1,2,4}
    44: {1,1,5}
    45: {2,2,3}
    50: {1,3,3}
    52: {1,1,6}
    63: {2,2,4}
    66: {1,2,5}
    70: {1,3,4}
    75: {2,3,3}
    81: {2,2,2,2}
    84: {1,1,2,4}
    88: {1,1,1,5}
    90: {1,2,2,3}
   100: {1,1,3,3}

Partitions of this type are counted by A363745.
Positions of 2's in A363943 (high A363944), triangle A363945 (high A363946).
For mean 1 we have A363949.
The high version is A363950, counted by A026905.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A326567/A326568 gives mean of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
A363948 lists numbers whose prime indices have mean 1, counted by A363947.
Cf. A327473, A327476, A359889, A360013, A360015, A363488, A363951.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Floor[Mean[prix[#]]]==2&]

A364160 Numbers whose least prime factor has the greatest exponent.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 52, 53, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 84, 88, 89, 92, 96, 97, 99, 101, 103, 104, 107, 109, 112, 113, 116

Offset: 1

Gus Wiseman, Jul 14 2023

nonn

First differs from A334298 in having 600 and lacking 180.
Also numbers whose minimum part in prime factorization is a unique mode.
If k is a term, then so are all powers of k. - Robert Israel, Sep 17 2024

			The prime factorization of 600 is 2*2*2*3*5*5, and 3 > max(1,2), so 600 is in the sequence.
The prime factorization of 180 is 2*2*3*3*5, but 2 <= max(2,1), so 180 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           29: {10}              67: {19}
     2: {1}          31: {11}              68: {1,1,7}
     3: {2}          32: {1,1,1,1,1}       71: {20}
     4: {1,1}        37: {12}              72: {1,1,1,2,2}
     5: {3}          40: {1,1,1,3}         73: {21}
     7: {4}          41: {13}              76: {1,1,8}
     8: {1,1,1}      43: {14}              79: {22}
     9: {2,2}        44: {1,1,5}           80: {1,1,1,1,3}
    11: {5}          45: {2,2,3}           81: {2,2,2,2}
    12: {1,1,2}      47: {15}              83: {23}
    13: {6}          48: {1,1,1,1,2}       84: {1,1,2,4}
    16: {1,1,1,1}    49: {4,4}             88: {1,1,1,5}
    17: {7}          52: {1,1,6}           89: {24}
    19: {8}          53: {16}              92: {1,1,9}
    20: {1,1,3}      56: {1,1,1,4}         96: {1,1,1,1,1,2}
    23: {9}          59: {17}              97: {25}
    24: {1,1,1,2}    60: {1,1,2,3}         99: {2,2,5}
    25: {3,3}        61: {18}             101: {26}
    27: {2,2,2}      63: {2,2,4}          103: {27}
    28: {1,1,4}      64: {1,1,1,1,1,1}    104: {1,1,1,6}

Robert Israel, Table of n, a(n) for n = 1..10000

Allowing any unique mode gives A356862, complement A362605.
Allowing any unique co-mode gives A359178, complement A362606.
The even case is A360013, counted by A241131.
For greatest instead of least we have A362616, counted by A362612.
These partitions are counted by A364193.
A027746 lists prime factors (with multiplicity).
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Cf. A098859, A327473, A327476, A360014, A360015, A362610, A364061, A364062.

filter:= proc(n) local F,i;
  F:= ifactors(n)[2];
  if nops(F) = 1 then return true fi;
  i:= min[index](F[..,1]);
  andmap(t -> F[t,2] < F[i,2], {$1..nops(F)} minus {i})
end proc:
filter(1):= true:
select(filter, [$1..200]); # Robert Israel, Sep 17 2024

Select[Range[100],First[Last/@FactorInteger[#]] > Max@@Rest[Last/@FactorInteger[#]]&]

A363223 Numbers with bigomega equal to median prime index.

Original entry on oeis.org

2, 9, 10, 50, 70, 75, 105, 110, 125, 130, 165, 170, 175, 190, 195, 230, 255, 275, 285, 290, 310, 325, 345, 370, 410, 425, 430, 435, 465, 470, 475, 530, 555, 575, 590, 610, 615, 645, 670, 686, 705, 710, 725, 730, 775, 790, 795, 830, 885, 890, 915, 925, 970

Offset: 1

Gus Wiseman, May 29 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

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 terms together with their prime indices begin:
    2: {1}
    9: {2,2}
   10: {1,3}
   50: {1,3,3}
   70: {1,3,4}
   75: {2,3,3}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  130: {1,3,6}
  165: {2,3,5}
  170: {1,3,7}
  175: {3,3,4}

For maximum instead of median we have A106529, counted by A047993.
For minimum instead of median we have A324522, counted by A006141.
Partitions of this type are counted by A361800.
For twice median we have A362050, counted by A362049.
For maximum instead of length we have A362621, counted by A053263.
A000975 counts subsets with integer median.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A359908 lists numbers whose prime indices have integer median.
A360005 gives twice median of prime indices.
Cf. A000040, A013580, A079309, A240219, A327473, A327476, A361860, A362619, A362622, A362980.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],PrimeOmega[#]==Median[prix[#]]&]

2*A001222(a(n)) = A360005(a(n)).

A363489 Rounded mean of the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 1, 6, 2, 2, 1, 7, 2, 8, 2, 3, 3, 9, 1, 3, 4, 2, 2, 10, 2, 11, 1, 4, 4, 4, 2, 12, 4, 4, 2, 13, 2, 14, 2, 2, 5, 15, 1, 4, 2, 4, 3, 16, 2, 4, 2, 5, 6, 17, 2, 18, 6, 3, 1, 4, 3, 19, 3, 6, 3, 20, 1, 21, 6, 3, 3, 4, 3, 22, 1, 2, 7

Offset: 1

Gus Wiseman, Jul 07 2023

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.

We use the "rounding half to even" rule, see link.

			The prime indices of 180 are {1,1,2,2,3}, with mean 9/5, which rounds to 2, so a(180) = 2.

Wikipedia, Rounding.

Positions of first appearances are 1 and A000040.
Before rounding we had A326567/A326568.
For rounded-down: A363943, triangle A363945.
For rounded-up: A363944, triangle A363946.
Positions of 1's are A363948, complement A364059.
The triangle for this statistic (rounded mean) is A364060.
For prime factors instead of indices we have A364061.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
Cf. A327473, A327476, A327482, A359889, A360005, A363947, A363949, A363951.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Round[Mean[prix[n]]]],{n,100}]

A363261 The partial sums of the prime indices of n include half the sum of all prime indices of n.

Original entry on oeis.org

4, 9, 12, 16, 25, 30, 40, 48, 49, 63, 64, 70, 81, 84, 108, 112, 121, 144, 154, 160, 165, 169, 192, 198, 220, 256, 264, 270, 273, 286, 289, 325, 351, 352, 360, 361, 364, 390, 442, 448, 468, 480, 520, 529, 561, 567, 576, 595, 624, 625, 640, 646, 675, 714, 729

Offset: 1

Gus Wiseman, Jun 06 2023

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 terms together with their prime indices begin:
   4: {1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  25: {3,3}
  30: {1,2,3}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
  63: {2,2,4}
  64: {1,1,1,1,1,1}
  70: {1,3,4}
  81: {2,2,2,2}
  84: {1,1,2,4}

Partitions of this type are counted by A322439.
For parts instead of partial sums we have A344415, counted by A035363.
A025065 counts palindromic partitions, ranked by A265640.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A301987 lists numbers whose sum of prime indices equals their product.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
Cf. A001414, A106529, A327473, A327476, A334201, A340387, A360005.

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

A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, Jul 14 2023

nonn

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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The terms together with their prime indices begin:
     3: {2}        23: {9}          42: {1,2,4}
     5: {3}        25: {3,3}        43: {14}
     6: {1,2}      26: {1,6}        45: {2,2,3}
     7: {4}        27: {2,2,2}      46: {1,9}
     9: {2,2}      29: {10}         47: {15}
    10: {1,3}      30: {1,2,3}      49: {4,4}
    11: {5}        31: {11}         50: {1,3,3}
    13: {6}        33: {2,5}        51: {2,7}
    14: {1,4}      34: {1,7}        53: {16}
    15: {2,3}      35: {3,4}        54: {1,2,2,2}
    17: {7}        36: {1,1,2,2}    55: {3,5}
    18: {1,2,2}    37: {12}         57: {2,8}
    19: {8}        38: {1,8}        58: {1,10}
    21: {2,4}      39: {2,6}        59: {17}
    22: {1,5}      41: {13}         60: {1,1,2,3}

For mean instead of median we have A057716, counted by A000065.
These partitions are counted by A238495.
The complement is A364056, counted by A027336, low version A363488.
A000975 counts subsets with integer median, A051293 for mean.
A124943 counts partitions by low median, high version A124944.
A360005 gives twice the median of prime indices, A360459 for prime factors.
A359893 and A359901 count partitions by median.
Cf. A002865, A316413, A325347, A327473, A327476, A363727.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Median[prifacs[#]]>2&]

A360005(a(n)) > 1.

A360459(a(n)) > 2.

cp's OEIS Frontend

A126594 Floor of the average of the prime factors of n with multiplicity.

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PARI

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A362980 Numbers whose multiset of prime factors (with multiplicity) has different median from maximum.

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Mathematica

A362981 Heinz numbers of integer partitions such that 2*(least part) >= greatest part.

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Mathematica

A362982 Heinz numbers of partitions such that 2*(least part) < greatest part.

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A363954 Numbers whose prime indices have low mean 2.

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A364160 Numbers whose least prime factor has the greatest exponent.

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Maple

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A363223 Numbers with bigomega equal to median prime index.

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A363489 Rounded mean of the multiset of prime indices of n.

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A363261 The partial sums of the prime indices of n include half the sum of all prime indices of n.

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