A139710 -id:A139710 - 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]]]&]

A361566 a(n) is the denominator of the median of divisors of n.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 1

Stefano Spezia, Mar 15 2023

a(n) = 2 if n is in A139710, otherwise a(n) = 1. - Robert Israel, Mar 15 2023

			a(9) = 1 since the divisors of 9 are 1, 3, 9, and their median is 3.
a(12) = 2 since the divisors of 12 are 1, 2, 3, 4, 6, 12, and their median is 7/2.

Winston de Greef, Table of n, a(n) for n = 1..10000
Index entries for sequences related to divisors of numbers

Cf. A027750, A033676, A033677, A071090, A139710, A139711, A361565 (numerator).

a[n_]:=Denominator[Median[Divisors[n]]]; Array[a,88]

a(n) = my(d=divisors(n), m=#d+1); denominator((d[m\2] + d[m-m\2])/2); \\ Michel Marcus, Mar 16 2023

a(n) = denominator((A033676(n) + A033677(n))/2).

Except for p = 2 and k = 1, a(p^k) = 1, if p is a prime.

A360668 Numbers > 1 whose greatest prime index is not divisible by their number of prime factors (bigomega).

Original entry on oeis.org

4, 8, 10, 12, 15, 16, 18, 22, 24, 25, 27, 28, 32, 33, 34, 36, 40, 42, 44, 46, 48, 51, 54, 55, 60, 62, 63, 64, 66, 68, 69, 70, 72, 76, 77, 80, 81, 82, 85, 88, 90, 93, 94, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 116, 118, 119, 120, 121, 123, 124

Offset: 1

Gus Wiseman, Feb 17 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.

Also numbers > 1 whose first differences of 0-prepended prime indices have non-integer mean.

			The prime indices of 1617 are {2,4,4,5}, and 5 is not divisible by 4, so 1617 is in the sequence.

These partitions are counted by A200727.
The complement is A340610 (without 1), counted by A168659.
For median instead of mean we have A360557, counted by A360691.
Positions of terms > 1 in A360615 (numerator: A360614).
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A334201 adds up all prime indices except the greatest.
A348551 = numbers w/ non-integer mean of prime indices, complement A316413.
Cf. A072978, A078175, A139710, A307683, A359912, A360551, A360554, A360556, A360669.

Select[Range[2,100],!Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

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

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A361566 a(n) is the denominator of the median of divisors of n.

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A360668 Numbers > 1 whose greatest prime index is not divisible by their number of prime factors (bigomega).

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