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 A323022 Fourth omega of n. Number of distinct multiplicities in the prime signature of n. 38
 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,60 COMMENTS The indices of terms greater than 1 are {60, 84, 90, 120, 126, 132, 140, 150, ...}. First term greater than 2 is a(1801800) = 3. In general, the first appearance of k is a(A182856(k)) = k. The prime signature of n (row n of A118914) is the multiset of prime multiplicities in n. We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first three omegas are A001222, A001221, A071625, and this sequence is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices). LINKS Antti Karttunen, Table of n, a(n) for n = 1..100000 FORMULA a(n) = A071625(A181819(n))      = A001221(A181819(A181819(n)))      = A001222(A181819(A181819(A181819(n))))      = A056239(A181819(A181819(A181819(A181819(n))))). EXAMPLE The prime signature of 1286485200 is {1, 1, 1, 2, 2, 3, 4}, in which 1 appears three times, two appears twice, and 3 and 4 both appear once, so there are 3 distinct multiplicities {1, 2, 3} and hence a(1286485200) = 3. MATHEMATICA red[n_]:=Times@@Prime/@Last/@If[n==1, {}, FactorInteger[n]]; Table[PrimeNu[red[red[n]]], {n, 200}] PROG (PARI) a(n) = my(e=factor(n)[, 2], s = Set(e), m=Map(), v=vector(#s)); for(i=1, #s, mapput(m, s[i], i)); for(i=1, #e, v[mapget(m, e[i])]++); #Set(v) \\ David A. Corneth, Jan 02 2019 (PARI) A071625(n) = #Set(factor(n)[, 2]); \\ From A071625 A181819(n) = factorback(apply(e->prime(e), (factor(n)[, 2]))); A323022(n) = A071625(A181819(n)); \\ Antti Karttunen, Jan 03 2019 CROSSREFS Cf. A001221, A001222, A006939, A025487, A056239, A059404, A062770, A071625, A118914, A181819, A182856, A182857, A304464, A304465, A323014, A323023. Sequence in context: A298481 A324872 A307608 * A284562 A087102 A194309 Adjacent sequences:  A323019 A323020 A323021 * A323023 A323024 A323025 KEYWORD nonn AUTHOR Gus Wiseman, Jan 02 2019 EXTENSIONS More terms from Antti Karttunen, Jan 03 2019 STATUS approved

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Last modified September 21 17:46 EDT 2019. Contains 327273 sequences. (Running on oeis4.)