A329603 - OEIS

%I #38 Mar 12 2023 04:16:07

%S 2,5,8,15,18,11,50,45,20,125,98,33,242,245,32,135,338,77,578,375,72,

%T 605,722,99,42,845,60,735,1058,17,1682,405,200,1445,162,231,1922,1805,

%U 392,1125,2738,1331,3362,1815,44,2645,3698,297,110,275,968,2535,4418,539,450,2205,1352,4205,5618,51,6962,4805,500,1215,882,1859,7442,4335,2312

%N a(n) = A005940(1+(1+(3*A156552(n)))) = (1/2) * A005940(1+(3*A156552(2*n))).

%C Function n -> 3n+1 (A016777) conjugated by A156552. - _Antti Karttunen_, Aug 21 2021

%H Antti Karttunen, <a href="/A329603/b329603.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = (1/2) * A005940(1+(3*A156552(2*n))).

%F From _Antti Karttunen_, Feb 14 2021: (Start)

%F A156552(2*a(n)) = 3*A156552(2*n) = 3*(1+2*A156552(n)) = 3 + 6*A156552(n).

%F a(n) = A341510(n,2n) = A005940(1+A156552(n)+A156552(2n)) = A005940(1+(1+(3*A156552(n)))).

%F a(n) = A005940(1+A016777(A156552(n))).

%F For all n >= 1, A329903(a(n)) = A332814(a(n)) = A332823(A332461(a(n))) = 1.

%F For all n >= 1, A341354(a(n)) > 0.

%F For all n >= 1, A000035(a(n)) = 1 - A000035(n). [Flips the parity of n]

%F (End)

%F a(n) = A332449(2*n)/2, a(n) = Sum_{d|n} A347117(d). - _Antti Karttunen_, Aug 21 2021

%o (PARI)

%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };

%o A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };

%o A329603(n) = ((1/2)*A005940(1+(3*A156552(2*n))));

%o (Python)

%o from math import prod

%o from itertools import accumulate

%o from collections import Counter

%o from sympy import prime, primepi, factorint

%o def A329603(n): return prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(1+3*sum((1<<primepi(p)-1)<<i for i, p in enumerate(factorint(n,multiple=True))))[2:].split('1')[:0:-1])).items()) # _Chai Wah Wu_, Mar 11 2023

%Y Permutation of A329604.

%Y Cf. A000035, A005940, A016777, A064989, A156552, A329903, A332449, A332461, A332814, A332823, A341354, A341510, A341515, A341516, A347117 (Möbius transform).

%Y A skewed diagonal of A341510.

%K nonn

%O 1,1

%A _Antti Karttunen_, Nov 21 2019

%E New primary definition added by _Antti Karttunen_, Feb 14 2021