A351035 -id:A351035 - cp's OEIS frontend

A347385 Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).

1, 1, 4, 1, 6, 4, 8, 1, 12, 6, 12, 4, 14, 8, 24, 1, 18, 12, 20, 6, 32, 12, 24, 4, 30, 14, 36, 8, 30, 24, 32, 1, 48, 18, 48, 12, 38, 20, 56, 6, 42, 32, 44, 12, 72, 24, 48, 4, 56, 30, 72, 14, 54, 36, 72, 8, 80, 30, 60, 24, 62, 32, 96, 1, 84, 48, 68, 18, 96, 48, 72, 12, 74, 38, 120, 20, 96, 56, 80, 6, 108, 42, 84, 32, 108

Offset: 1

Antti Karttunen, Aug 31 2021

Coincides with A000593 on A122132.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000265, A001615, A122132, A185199, A206787, A336651, A347386, A347387, A351035.

Cf. also A000593, A053575.

f[p_, e_] := If[p == 2, 1, (p + 1)*p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2021 *)

A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));

Multiplicative with a(2^e) = 1, a(p^e) = (p+1)*p^(e-1) for all odd primes p.
a(n) = A001615(A000265(n)).
a(n) = A206787(n) * A336651(n). - Antti Karttunen, Feb 11 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). - Amiram Eldar, Nov 19 2022
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s))*(4^s-2^(s+1))/(4^s-1). - Amiram Eldar, Jan 04 2023

A351040 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336158(i) = A336158(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 17, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 25, 4, 27, 15, 28, 8, 29, 16, 30, 1, 31, 17, 32, 9, 33, 17, 34, 5, 35, 18, 36, 10, 33, 19, 37, 3, 38, 20, 39, 11, 40, 21, 41, 6, 42

Offset: 1

Antti Karttunen, Jan 31 2022

Restricted growth sequence transform of the ordered triplet [A336158(n), A206787(n), A336651(n)].
For all i, j >= 1:
  A003602(i) = A003602(j) => a(i) = a(j),
  a(i) = a(j) => A336390(i) = A336390(j) => A336391(i) = A336391(j),
  a(i) = a(j) => A347374(i) = A347374(j),
  a(i) = a(j) => A351036(i) = A351036(j) => A113415(i) = A113415(j),
  a(i) = a(j) => A351461(i) = A351461(j).
From Antti Karttunen, Nov 23 2023: (Start)
Conjectured to be equal to the lexicographically earliest infinite sequence such that b(i) = b(j) => A000593(i) = A000593(j), A336158(i) = A336158(j) and A336467(i) = A336467(j), for all i, j >= 1 (this was the original definition). In any case it holds that a(i) = a(j) => b(i) = b(j) for all i, j >= 1. See comment in A351461.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65539

Cf. A000265, A000593, A336158, A336467.
Cf. also A003602, A336390, A336391, A351037, A366891.
Differs from A347374 for the first time at n=103, where a(103) = 48, while A347374(103) = 30.
Differs from A351035 for the first time at n=175, where a(175) = 80, while A351035(175) = 78.
Differs from A351036 for the first time at n=637, where a(637) = 272, while A351036(637) = 261.

up_to = 65539;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A336158(n) = A046523(A000265(n));
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d));
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351040(n) = [A336158(n), A206787(n), A336651(n)];
v351040 = rgs_transform(vector(up_to, n, Aux351040(n)));
A351040(n) = v351040[n];

Original definition moved to the comment section and replaced with a definition that is at least as encompassing, and conjectured to be equal to the original one. - Antti Karttunen, Nov 23 2023

A351036 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

Offset: 1

Antti Karttunen, Jan 30 2022

Restricted growth sequence transform of the ordered pair [A000593(n), A336158(n)], where A000593(n) is the sum of odd divisors of n, and A336158(n) is the least representative of the prime signature of the odd part of n.
For all i, j:
  A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),
  a(i) = a(j) => A113415(i) = A113415(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A000593, A003602, A046523, A113415, A336158.
Cf. also A351037.
Differs from A347374 for the first time at n=103, where a(103) = 48, while A347374(103) = 30.
Differs from A351035 for the first time at n=175, where a(175) = 80, while A351035(175) = 78.
Differs from A351040 for the first time at n=637, where a(637) = 261, while A351040(637) = 272.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
A000593(n) = sigma(A000265(n));
Aux351036(n) = [A000593(n), A336158(n)];
v351036 = rgs_transform(vector(up_to, n, Aux351036(n)));
A351036(n) = v351036[n];

cp's OEIS Frontend

A347385 Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).

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A351040 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336158(i) = A336158(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

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A351036 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j) and A336158(i) = A336158(j), for all i, j >= 1.

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