A322866 -id:A322866 - cp's OEIS frontend

A322867 a(n) = A000120(A122111(n)).

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 1, 3, 4, 2, 3, 2, 1, 4, 1, 3, 2, 2, 4, 3, 1, 2, 2, 3, 1, 4, 1, 2, 3, 2, 1, 3, 3, 4, 2, 2, 1, 3, 4, 3, 2, 2, 1, 3, 1, 2, 3, 3, 4, 4, 1, 2, 2, 4, 1, 2, 1, 2, 4, 2, 3, 4, 1, 3, 3, 2, 1, 3, 4, 2, 2, 3, 1, 3, 3, 2, 2, 2, 4, 3, 1, 4, 3, 6, 1, 4, 1, 3, 4

Offset: 1

Antti Karttunen, Dec 30 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A001222, A122111, A322863, A322865, A322866.

Array[DigitCount[#, 2, 1] &@ If[# < 3, 1, Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@Length@k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ #]] &, 105] (* Michael De Vlieger, Dec 31 2018, after JungHwan Min at A122111 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322867(n) = hammingweight(A122111(n));

a(n) = A000120(A122111(n)) = A000120(A322865(n)) = A001222(A322863(n)).

A322863 Permutation of natural numbers: a(0) = 1; for n >= 1, a(n) = A005940(1+A122111(n)).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 11, 6, 10, 25, 13, 15, 17, 49, 21, 8, 19, 16, 23, 35, 55, 121, 29, 27, 36, 169, 50, 77, 31, 81, 37, 12, 91, 289, 225, 30, 41, 361, 187, 125, 43, 625, 47, 143, 147, 529, 53, 45, 154, 90, 247, 221, 59, 28, 1225, 343, 391, 841, 61, 105, 67, 961, 605, 18, 5929, 2401, 71, 323, 551, 525, 73, 22, 79, 1369, 84, 437, 429, 14641, 83

Offset: 0

Antti Karttunen, Dec 30 2018

nonn

Note the indexing: the domain starts from 0, but the range excludes zero.

Inverse permutation: A322864.

Cf. A005940, A122111, A322866, A322867.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322863(n) = if(!n,1,A005940(1+A122111(n)));

a(0) = 1; for n >= 1, a(n) = A005940(1+A122111(n)).
For all n >= 1:
a(prime(n)) = prime(1+n).
A001222(a(n)) = A322867(n).

cp's OEIS Frontend

A322867 a(n) = A000120(A122111(n)).

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