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

1,1

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

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

Prog

(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<<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

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.

Sequence in context: A282444 A103077 A090980 * A309546 A261543 A307190

Adjacent sequences: A329600 A329601 A329602 * A329604 A329605 A329606

Keyword

nonn

Author

Antti Karttunen, Nov 21 2019

Extensions

New primary definition added by Antti Karttunen, Feb 14 2021

Status

approved