A088530 -id:A088530 - cp's OEIS frontend

A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

1, 2, 6, 30, 42, 49, 60, 66, 70, 78, 84, 90, 102, 105, 114, 120, 126, 132, 138, 140, 150, 154, 156, 168, 174, 186, 198, 204, 210, 222, 228, 234, 246, 258, 264, 270, 276, 280, 282, 286, 294, 306, 308, 312, 315, 318, 330, 342, 348, 350, 354, 366, 372, 378, 385

Offset: 1

Gus Wiseman, Feb 19 2023

nonn

A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

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:
    1: {}
    2: {1}
    6: {1,2}
   30: {1,2,3}
   42: {1,2,4}
   49: {4,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with median 1. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with median 1/2. So 2760 is not in the sequence.

For distinct prime indices instead of 0-prepended differences: A360453.
For mean instead of median we have A360680.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Multisets with integer median:
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
Cf. A000975, A026424, A316413, A340610, A359904, A360006, A360248, A360558, A360687, A360688.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Median[Length/@Split[prix[#]]] == Median[Differences[Prepend[prix[#],0]]]&]

A360690 Number of integer partitions of n with non-integer median of multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465

Offset: 1

Gus Wiseman, Feb 22 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 a(1) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).

These partitions have ranks A360554.
The complement is counted by A360687, ranks A360553.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360069 = partitions with integer mean of multiplicities, ranks A067340.
Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

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}]

A289621 Compound filter (omega & bigomega): a(1) = 0, for n > 1, a(n) = P(A001221(n), A001222(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 4, 2, 5, 1, 8, 1, 5, 5, 7, 1, 8, 1, 8, 5, 5, 1, 12, 2, 5, 4, 8, 1, 13, 1, 11, 5, 5, 5, 12, 1, 5, 5, 12, 1, 13, 1, 8, 8, 5, 1, 17, 2, 8, 5, 8, 1, 12, 5, 12, 5, 5, 1, 18, 1, 5, 8, 16, 5, 13, 1, 8, 5, 13, 1, 17, 1, 5, 8, 8, 5, 13, 1, 17, 7, 5, 1, 18, 5, 5, 5, 12, 1, 18, 5, 8, 5, 5, 5, 23, 1, 8, 8

Offset: 1

Antti Karttunen, Jul 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000027, A046523.

Cf. A001221, A001222, A008966, A046660, A070012, A070013, A070014, A088529, A088530, A181591 (sequences with matching equivalence classes).

A289621(n) = if(1==n,0,(1/2)*(2 + ((omega(n)+bigomega(n))^2) - omega(n) - 3*bigomega(n)));

(define (A289621 n) (if (= 1 n) 0 (* (/ 1 2) (+ (expt (+ (A001221 n) (A001222 n)) 2) (- (A001221 n)) (- (* 3 (A001222 n))) 2))))

a(1) = 0, for n > 1, a(n) = (1/2)*(2 + ((A001221(n)+A001222(n))^2) - A001221(n) - 3*A001222(n)).

A360070 Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 128, 144, 147, 150, 160, 162, 169, 175, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 272

Offset: 1

Gus Wiseman, Jan 27 2023

nonn

Conjecture: No term > 1 is squarefree.

			A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in the sequence.

Positions of positive terms in A360068, ranked by A359903.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature (A124010).
A326567/A326568 gives mean of prime indices (A112798).
Cf. A005117, A067340, A067538, A240219, A316313, A349156, A359904, A359905, A360069.

Select[Range[30],Select[IntegerPartitions[#],Mean[#]==Mean[Length/@Split[#]]&]!={}&]

a(22)-a(58) from Alois P. Heinz, Jan 29 2023

A360680 Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 6, 30, 49, 152, 210, 513, 1444, 1776, 1952, 2310, 2375, 2664, 2760, 2960, 3249, 3864, 3996, 4140, 4144, 5796, 5994, 6072, 6210, 6440, 6512, 6517, 6900, 7176, 7400, 7696, 8694, 9025, 9108, 9384, 10064, 10120, 10350, 10488, 10764, 11248, 11960, 12167

Offset: 1

Gus Wiseman, Feb 19 2023

nonn

A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     49: {4,4}
    152: {1,1,1,8}
    210: {1,2,3,4}
    513: {2,2,2,8}
   1444: {1,1,8,8}
   1776: {1,1,1,1,2,12}
   1952: {1,1,1,1,1,18}
   2310: {1,2,3,4,5}
   2375: {3,3,3,8}
   2664: {1,1,1,2,2,12}
   2760: {1,1,1,2,3,9}
   2960: {1,1,1,1,3,12}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with mean 3/2. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with mean also 3/2. So 2760 is in the sequence.

For indices instead of 0-prepended differences: A359903, counted by A360068.
For median instead of mean we have A360681.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, mean A088529/A088530.
A316413 = numbers whose prime indices have integer mean, complement A348551.
A326619/A326620 gives mean of distinct prime indices.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A340610, A359904, A359905, A360008, A360460, A360555, A360556.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Mean[Length/@Split[prix[#]]] == Mean[Differences[Prepend[prix[#],0]]]&]

A382267 Denominator of the harmonic mean of the exponents in the prime factorization of n.

Original entry on oeis.org

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

Offset: 2

Ilya Gutkovskiy, Mar 19 2025

			1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4/3, 1, 1, 1, 4, 1, 4/3, 1, 4/3, 1, 1, 1, 3/2, 2, 1, 3, 4/3, ...

Index entries for sequences computed from exponents in factorization of n.

Cf. A070012, A088530, A250097, A382266 (numerators).

a:= n-> (l-> denom(nops(l)/add(1/i[2], i=l)))(ifactors(n)[2]):
seq(a(n), n=2..100);  # Alois P. Heinz, Mar 21 2025

Table[HarmonicMean[(#[[2]] & /@ FactorInteger[n])], {n, 2, 100}] // Denominator

cp's OEIS Frontend

A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

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A360690 Number of integer partitions of n with non-integer median of multiplicities.

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

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A289621 Compound filter (omega & bigomega): a(1) = 0, for n > 1, a(n) = P(A001221(n), A001222(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A360070 Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities.

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A360680 Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.

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A382267 Denominator of the harmonic mean of the exponents in the prime factorization of n.

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