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A355391 Position of first appearance of n in A181591 = binomial(bigomega(n), omega(n)).

%I A355391 #21 Jul 10 2022 13:23:42
%S A355391 1,4,8,16,32,24,128,256,512,48,2048,4096,8192,16384,96,65536,131072,
%T A355391 262144,524288,240,192,4194304,8388608,16777216,33554432,67108864,
%U A355391 134217728,384,536870912,1073741824,2147483648,4294967296,8589934592,17179869184,480,768,137438953472
%N A355391 Position of first appearance of n in A181591 = binomial(bigomega(n), omega(n)).
%C A355391 The statistic omega = A001221 counts distinct prime factors (without multiplicity).
%C A355391 The statistic bigomega = A001222 counts prime factors with multiplicity.
%C A355391 We have A181591(2^k) = k, so the sequence is fully defined. Positions meeting this maximum are A185024, complement A006987.
%H A355391 Amiram Eldar, <a href="/A355391/b355391.txt">Table of n, a(n) for n = 1..168</a>
%F A355391 binomial(bigomega(a(n)), omega(a(n))) = n.
%e A355391 The terms together with their prime indices begin:
%e A355391        1: {}
%e A355391        4: {1,1}
%e A355391        8: {1,1,1}
%e A355391       16: {1,1,1,1}
%e A355391       32: {1,1,1,1,1}
%e A355391       24: {1,1,1,2}
%e A355391      128: {1,1,1,1,1,1,1}
%e A355391      256: {1,1,1,1,1,1,1,1}
%e A355391      512: {1,1,1,1,1,1,1,1,1}
%e A355391       48: {1,1,1,1,2}
%e A355391     2048: {1,1,1,1,1,1,1,1,1,1,1}
%e A355391     4096: {1,1,1,1,1,1,1,1,1,1,1,1}
%e A355391     8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A355391    16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A355391       96: {1,1,1,1,1,2}
%e A355391    65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A355391   131072: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A355391   262144: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A355391   524288: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A355391      240: {1,1,1,1,2,3}
%e A355391      192: {1,1,1,1,1,1,2}
%t A355391 s=Table[Binomial[PrimeOmega[n],PrimeNu[n]],{n,1000}];
%t A355391 Table[Position[s,k][[1,1]],{k,Select[Union[s],SubsetQ[s,Range[#]]&]}]
%o A355391 (PARI) f(n) = binomial(bigomega(n), omega(n)); \\ A181591
%o A355391 a(n) = my(k=1); while (f(k) != n, k++); k; \\ _Michel Marcus_, Jul 10 2022
%Y A355391 Positions of powers of 2 are A185024, complement A006987.
%Y A355391 Counting multiplicity gives A355386.
%Y A355391 The sorted version is A355392.
%Y A355391 A000005 counts divisors.
%Y A355391 A001221 counts prime factors without multiplicity.
%Y A355391 A001222 count prime factors with multiplicity.
%Y A355391 A070175 gives representatives for bigomega and omega, triangle A303555.
%Y A355391 Cf. A022811, A056239 , A071625, A118914, A181819, A323014, A323023, A355383 (with multiplicity A339006), A355384.
%K A355391 nonn
%O A355391 1,2
%A A355391 _Gus Wiseman_, Jul 04 2022
%E A355391 a(22)-a(28) from _Michel Marcus_, Jul 10 2022
%E A355391 a(29)-a(37) from _Amiram Eldar_, Jul 10 2022