A324112 -id:A324112 - cp's OEIS frontend

A324054 a(n) = A000203(A005940(1+n)).

1, 3, 4, 7, 6, 12, 13, 15, 8, 18, 24, 28, 31, 39, 40, 31, 12, 24, 32, 42, 48, 72, 78, 60, 57, 93, 124, 91, 156, 120, 121, 63, 14, 36, 48, 56, 72, 96, 104, 90, 96, 144, 192, 168, 248, 234, 240, 124, 133, 171, 228, 217, 342, 372, 403, 195, 400, 468, 624, 280, 781, 363, 364, 127, 18, 42, 56, 84, 84, 144, 156, 120, 112, 216, 288, 224

Offset: 0

Antti Karttunen, Feb 14 2019

nonn

As noted by David A. Corneth, the function f(n) = a(n-1) [that is, the offset-1 version of this sequence] seems to be "almost multiplicative". Sequence A324109 gives the positions n where f(n) satisfies the multiplicativity in a sense that f(n) = f(p(1)^e(1)) * ... * f(p(k)^e(k)), when n = p(1)^e(1) * ... * p(k)^e(k), and A324110 the positions where this does not hold.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327
Index entries for sequences related to binary expansion of n
Michael De Vlieger, Annotated fan-style binary tree diagram showing 16 levels, where blue represents A336834(n) = 1 and yellow A336834(n) = 0. Labels are a(n) for the 8 smallest levels.

Cf. A000203, A005940, A038712, A324055, A324056, A324057, A324058, A324108, A324109, A324110, A324111, A324112, A356321.

Cf. also A106737, A290077 (tau and phi similarly permuted).

nn = 76; a[0] = 1; Do[Set[a[n], Prime[1 + DigitCount[n, 2, 0]]*a[n - 2^Floor@ Log2@ n]], {n, nn}]; Array[DivisorSigma[1, a[#]] &, nn, 0] (* Michael De Vlieger, Aug 03 2022 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324054(n) = sigma(A005940(1+n));

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };

from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime
def A324054(n): return prod(((p:=prime(len(a)+1))**(b+1)-1)//(p-1) for a, b in Counter(accumulate(bin(n)[2:].split('1')[:0:-1])).items()) # Chai Wah Wu, Mar 10 2023

a(n) = A000203(A005940(1+n)).

a(n) = A324056(n) * A038712(1+n).

A324111 Numbers n for which A324108(n) = A324054(n-1) and which are neither prime powers nor of the form 2^i * p^j, where p is an odd prime, with either exponent i or j > 0.

Original entry on oeis.org

1, 87, 174, 348, 696, 1392, 2091, 2784, 4182, 5568, 8364, 11136, 16683, 16728, 22272, 33215, 33366, 33456, 44544, 66430, 66732, 66912, 89088, 132860, 133464, 133824, 178176, 265720, 266928, 267179, 267648, 356352, 531440, 533856, 534358, 535296, 712704, 1062880, 1066877, 1067712, 1068716, 1070592, 1319235, 1425408

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Setwise difference of A324109 and A070776.
Setwise difference of A070537 and A324110.
If an odd number n > 1 is present, then all 2^k * n are present also. Odd terms > 1 are given in A324112.

			87 is a term, as 87 = 3*29, A324054(3-1) = 4, A324054(29-1) = 156, and A324108(87) = 4*156 = 624 = A324054(87-1).

Antti Karttunen, Table of n, a(n) for n = 1..173

Cf. A070537, A070776, A324109, A324110, A324112.

A000265(n) = (n/2^valuation(n, 2));
A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324111(n) = ((1!=omega(n))&&(1!=omega(A000265(n)))&&(A324054(n-1)==A324108(n)));
for(n=1,2^20,if(isA324111(n), print1(n,", ")))

cp's OEIS Frontend

A324054 a(n) = A000203(A005940(1+n)).

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A324111 Numbers n for which A324108(n) = A324054(n-1) and which are neither prime powers nor of the form 2^i * p^j, where p is an odd prime, with either exponent i or j > 0.

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