A329603 a(n) = A005940(1+(1+(3*A156552(n)))) = (1/2) * A005940(1+(3*A156552(2*n))).

2, 5, 8, 15, 18, 11, 50, 45, 20, 125, 98, 33, 242, 245, 32, 135, 338, 77, 578, 375, 72, 605, 722, 99, 42, 845, 60, 735, 1058, 17, 1682, 405, 200, 1445, 162, 231, 1922, 1805, 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

Offset: 1

Author

Antti Karttunen, Nov 21 2019

Keywords

• nonn

Comments

Function n -> 3n+1 (A016777) conjugated by A156552. - Antti Karttunen, Aug 21 2021

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Permutation of A329604.

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

A skewed diagonal of A341510.

Programs

PARI

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
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 };
A329603(n) = ((1/2)*A005940(1+(3*A156552(2*n))));

Python

from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime, primepi, factorint
def A329603(n): return prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(1+3*sum((1<Chai Wah Wu, Mar 11 2023

Formula

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

From Antti Karttunen, Feb 14 2021: (Start)

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

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

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

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

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

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

(End)

a(n) = A332449(2*n)/2, a(n) = Sum_{d|n} A347117(d). - Antti Karttunen, Aug 21 2021

Extensions

New primary definition added by Antti Karttunen, Feb 14 2021