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A246353 If n = Sum 2^e_i, e_i distinct, then a(n) = Position of (product prime_{e_i+1}) among squarefree numbers (A005117).

%I A246353 #19 Feb 23 2025 02:42:10
%S A246353 1,2,3,5,4,7,11,19,6,10,14,28,23,44,65,129,8,15,21,41,34,69,101,203,
%T A246353 48,94,144,283,233,470,703,1405,9,17,26,49,40,80,120,236,57,111,168,
%U A246353 334,279,554,833,1661,89,176,261,521,438,873,1304,2610,609,1217,1827,3650,3046,6091,9131
%N A246353 If n = Sum 2^e_i, e_i distinct, then a(n) = Position of (product prime_{e_i+1}) among squarefree numbers (A005117).
%C A246353 This is an inverse function to A048672. Note the indexing: here the domain starts from 0, but the range starts from 1, while in A048672 it is the opposite.
%C A246353 Sequence is obtained when the range of A019565 is compacted so that it becomes surjective on N, thus the logarithmic scatter plots look very similar. (Same applies to A064273). Compare also to the plot of A005940.
%H A246353 Antti Karttunen, <a href="/A246353/b246353.txt">Table of n, a(n) for n = 0..478</a>
%H A246353 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A246353 a(n) = A013928(1+A019565(n)) = 1 + A013928(A019565(n)).
%F A246353 a(n) = A064273(n) + 1.
%F A246353 For all n >= 0, A048672(a(n)) = n.
%F A246353 For all n >= 1, a(A048672(n)) = n.
%o A246353 (PARI)
%o A246353 allocatemem(234567890);
%o A246353 default(primelimit, 2^22)
%o A246353 uplim_for_13928 = 13123111;
%o A246353 v013928 = vector(uplim_for_13928); A013928(n) = v013928[n];
%o A246353 v013928[1]=0; n=1; while((n < uplim_for_13928), if(issquarefree(n), v013928[n+1] = v013928[n]+1, v013928[n+1] = v013928[n]); n++);
%o A246353 A019565(n) = {factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from _M. F. Hasler_
%o A246353 A246353(n) = 1+A013928(A019565(n));
%o A246353 for(n=0, 478, write("b246353.txt", n, " ", A246353(n)));
%o A246353 (Scheme) (definec (A246353 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) (A013928 (+ 1 p))) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (* p (A000040 i)))) (else (loop (/ n 2) (+ 1 i) p)))))
%o A246353 (Python)
%o A246353 from math import prod, isqrt
%o A246353 from sympy import prime, mobius
%o A246353 def A246353(n):
%o A246353     m = prod(prime(i) for i,j in enumerate(bin(n)[-1:1:-1],1) if j=='1')
%o A246353     return int(sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))) # _Chai Wah Wu_, Feb 22 2025
%Y A246353 Cf. A000040, A005940, A013928, A019565, A048672, A064273.
%K A246353 nonn
%O A246353 0,2
%A A246353 _Antti Karttunen_, Aug 23 2014