A100367 -id:A100367 - 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]]]&]

A174164 Numbers n such that 1 = abs(sum{p-1|p is prime and divisor of n} - product{p-1|p is prime and divisor of n}).

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 68, 72, 74, 76, 80, 82, 86, 88, 90, 92, 94, 96, 98, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 134, 136, 142, 144, 146, 148, 150, 152, 158, 160, 162, 164, 166, 172

Offset: 1

Juri-Stepan Gerasimov, Mar 10 2010

nonn

			6 is a term because 6=2*3 and 1=abs((2-1)+(3-1)-(2-1)*(3-1)).
10 is a term because 10=2*5 and 1=abs((2-1)+(5-1)-(2-1)*(5-1)).

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

Cf. A055631, A173557.

Union of A100367 and A143207.

From R. J. Mathar, Apr 26 2010: (Start)
A055631 := proc(n) add(d-1, d= numtheory[factorset](n) ) ; end proc:
A173557 := proc(n) mul(d-1, d= numtheory[factorset](n) ) ; end proc:
isA174164 := proc(n) A055631(n)-A173557(n) ; abs(%) = 1 ; end proc:
for n from 2 to 200 do if isA174164(n) then printf("%d,",n) ; end if; end do: (End)

filterQ[n_] := With[{pp = FactorInteger[n][[All, 1]]}, 1 == Abs[Total[pp-1] - Times @@ (pp-1)]];
Select[Range[200], filterQ] (* Jean-François Alcover, Sep 17 2020 *)

Corrected (53 replaced by 52, 90 and 120 inserted) by R. J. Mathar, Apr 26 2010

A321193 Even numbers with no more than one odd prime factor, not counting multiplicity.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 62, 64, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 98, 100, 104, 106, 108, 112, 116, 118, 122, 124, 128, 134, 136, 142, 144, 146, 148, 152, 158, 160, 162, 164, 166, 172, 176, 178, 184, 188, 192, 194, 196

Offset: 1

Lei Zhou, Oct 29 2018

			18 = 2 * 3^2 is in the sequence because it has 1 odd prime factor (3 counts only once).
16 = 2^4 is in the sequence because it has no odd prime factors.
70 = 2 * 5 * 7 is not in the sequence because it has 2 odd prime factors.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A070776, A100367, A001221, A098902, A100368.

n = 0; Table[n = n + 2;
While[Length[FactorInteger[n]] > 2, n = n + 2]; n, {k, 1, 76}]

is(n) = n%2==0 && omega(n) <= 2 \\ Felix Fröhlich, Nov 01 2018

is(n)=my(o=valuation(n,2)); o && isprimepower(n>>o) \\ Charles R Greathouse IV, Dec 13 2021

list(lim)=my(v=List()); for(k=1,logint(lim\=1,2), listput(v,1<>k); for(e=2,logint(L,3), forprime(p=3, sqrtnint(L,e), listput(v,p^e<>k, listput(v,p<Charles R Greathouse IV, Dec 13 2021

Numbers of the form 2^k*p^h where k > 0, h >= 0 p is an odd prime.

a(n) = 2 * A070776(n-1) for n > 1. - Alois P. Heinz, Nov 20 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A174164 Numbers n such that 1 = abs(sum{p-1|p is prime and divisor of n} - product{p-1|p is prime and divisor of n}).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A321193 Even numbers with no more than one odd prime factor, not counting multiplicity.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula