A355391 - OEIS

%I #21 Jul 10 2022 13:23:42

%S 1,4,8,16,32,24,128,256,512,48,2048,4096,8192,16384,96,65536,131072,

%T 262144,524288,240,192,4194304,8388608,16777216,33554432,67108864,

%U 134217728,384,536870912,1073741824,2147483648,4294967296,8589934592,17179869184,480,768,137438953472

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

%C The statistic omega = A001221 counts distinct prime factors (without multiplicity).

%C The statistic bigomega = A001222 counts prime factors with multiplicity.

%C We have A181591(2^k) = k, so the sequence is fully defined. Positions meeting this maximum are A185024, complement A006987.

%H Amiram Eldar, <a href="/A355391/b355391.txt">Table of n, a(n) for n = 1..168</a>

%F binomial(bigomega(a(n)), omega(a(n))) = n.

%e The terms together with their prime indices begin:

%e        1: {}

%e        4: {1,1}

%e        8: {1,1,1}

%e       16: {1,1,1,1}

%e       32: {1,1,1,1,1}

%e       24: {1,1,1,2}

%e      128: {1,1,1,1,1,1,1}

%e      256: {1,1,1,1,1,1,1,1}

%e      512: {1,1,1,1,1,1,1,1,1}

%e       48: {1,1,1,1,2}

%e     2048: {1,1,1,1,1,1,1,1,1,1,1}

%e     4096: {1,1,1,1,1,1,1,1,1,1,1,1}

%e     8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}

%e    16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}

%e       96: {1,1,1,1,1,2}

%e    65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

%e   131072: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

%e   262144: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

%e   524288: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

%e      240: {1,1,1,1,2,3}

%e      192: {1,1,1,1,1,1,2}

%t s=Table[Binomial[PrimeOmega[n],PrimeNu[n]],{n,1000}];

%t Table[Position[s,k][[1,1]],{k,Select[Union[s],SubsetQ[s,Range[#]]&]}]

%o (PARI) f(n) = binomial(bigomega(n), omega(n)); \\ A181591

%o a(n) = my(k=1); while (f(k) != n, k++); k; \\ _Michel Marcus_, Jul 10 2022

%Y Positions of powers of 2 are A185024, complement A006987.

%Y Counting multiplicity gives A355386.

%Y The sorted version is A355392.

%Y A000005 counts divisors.

%Y A001221 counts prime factors without multiplicity.

%Y A001222 count prime factors with multiplicity.

%Y A070175 gives representatives for bigomega and omega, triangle A303555.

%Y Cf. A022811, A056239 , A071625, A118914, A181819, A323014, A323023, A355383 (with multiplicity A339006), A355384.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 04 2022

%E a(22)-a(28) from _Michel Marcus_, Jul 10 2022

%E a(29)-a(37) from _Amiram Eldar_, Jul 10 2022