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Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).
8

%I #30 May 24 2023 17:52:37

%S 1,2,6,12,30,60,210,360,420,1260,2310,2520,4620,13860,27720,30030,

%T 60060,75600,138600,180180,360360,510510,831600,900900,1021020,

%U 1801800,3063060,6126120,9699690,10810800,15315300,19399380,30630600,37837800

%N Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

%C Members m of A025487 such that A181819(m) is also a member of A025487.

%C If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A181818.

%C Also the least number with each sorted prime metasignature, where a number's metasignature is the sequence of multiplicities of exponents in its prime factorization. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}. - _Gus Wiseman_, May 21 2022

%H David A. Corneth, <a href="/A182863/b182863.txt">Table of n, a(n) for n = 1..10000</a> (first 1444 terms from Amiram Eldar)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConjugatePartition.html">Conjugate Partition</a>.

%e The prime signature of 360360 = 2^3*3^2*5*7*11*13 is (3,2,1,1,1,1). 2 appears as many times as 3 in 360360's prime signature, and 1 appears more times than 2. Since 360360 is also a member of A025487, it is a member of this sequence.

%e From _Gus Wiseman_, May 21 2022: (Start)

%e The terms together with their sorted prime signatures and sorted prime metasignatures begin:

%e 1: {} -> {} -> {}

%e 2: {1} -> {1} -> {1}

%e 6: {1,2} -> {1,1} -> {2}

%e 12: {1,1,2} -> {1,2} -> {1,1}

%e 30: {1,2,3} -> {1,1,1} -> {3}

%e 60: {1,1,2,3} -> {1,1,2} -> {1,2}

%e 210: {1,2,3,4} -> {1,1,1,1} -> {4}

%e 360: {1,1,1,2,2,3} -> {1,2,3} -> {1,1,1}

%e 420: {1,1,2,3,4} -> {1,1,1,2} -> {1,3}

%e 1260: {1,1,2,2,3,4} -> {1,1,2,2} -> {2,2}

%e 2310: {1,2,3,4,5} -> {1,1,1,1,1} -> {5}

%e 2520: {1,1,1,2,2,3,4} -> {1,1,2,3} -> {1,1,2}

%e 4620: {1,1,2,3,4,5} -> {1,1,1,1,2} -> {1,4}

%e 13860: {1,1,2,2,3,4,5} -> {1,1,1,2,2} -> {2,3}

%e 27720: {1,1,1,2,2,3,4,5} -> {1,1,1,2,3} -> {1,1,3}

%e 30030: {1,2,3,4,5,6} -> {1,1,1,1,1,1} -> {6}

%e 60060: {1,1,2,3,4,5,6} -> {1,1,1,1,1,2} -> {1,5}

%e (End)

%t nn=1000;

%t r=Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,nn}];

%t Select[Range[nn],!MemberQ[Take[r,#-1],r[[#]]]&] (* _Gus Wiseman_, May 21 2022 *)

%Y Intersection of A025487 and A179983.

%Y Subsequence of A129912 and A181826.

%Y Includes all members of A182862.

%Y Positions of first appearances in A353742, unordered version A238747.

%Y A001222 counts prime factors with multiplicity, distinct A001221.

%Y A003963 gives product of prime indices.

%Y A005361 gives product of prime signature, firsts A353500 (sorted A085629).

%Y A056239 adds up prime indices, row sums of A112798 and A296150.

%Y A124010 gives prime signature, sorted A118914.

%Y A130091 lists numbers with distinct prime exponents, counted by A098859.

%Y A181819 gives prime shadow, with an inverse A181821.

%Y A182850 gives frequency depth of prime indices, counted by A225485.

%Y A323014 gives adjusted frequency depth of prime indices, counted by A325280.

%Y Cf. A000040, A055932, A070175, A097318, A304678, A325238, A353507, A353745.

%K nonn

%O 1,2

%A _Matthew Vandermast_, Jan 14 2011