A344872 -id:A344872 - cp's OEIS frontend

A369659 Non-multiples of 3 whose arithmetic derivative, or equally, the sum of prime factors (with multiplicity) is a multiple of 3.

1, 8, 14, 20, 26, 35, 38, 44, 50, 62, 64, 65, 68, 74, 77, 86, 92, 95, 110, 112, 116, 119, 122, 125, 134, 143, 146, 155, 158, 160, 161, 164, 170, 185, 188, 194, 196, 203, 206, 208, 209, 212, 215, 218, 221, 230, 236, 242, 254, 275, 278, 280, 284, 287, 290, 299, 302, 304, 305, 314, 323, 326, 329, 332, 335, 341, 343

Offset: 1

Antti Karttunen, Feb 10 2024

nonn

This is a subsequence of A373475, containing all its terms that are not multiples of 3. (See comments in A373475 for a proof). The first difference from A373475 is at n=4186, where A373475(4186) = 19683 = 3^9, the value which is missing from this sequence. - Antti Karttunen, Jun 07 2024
From Antti Karttunen, Jun 11 2024: (Start)
A multiplicative semigroup: if m and n are in the sequence, then so is m*n.
Numbers that are not multiples of 3, and the multiplicities of prime factors of the forms 3m+1 (A002476) and 3m-1 (A003627) are equal modulo 3.
Like A373597, which is a subsequence, also this sequence can be viewed as a kind of k=3 variant of A046337.
A289142, numbers whose sum of prime factors (with multiplicity, A001414) is a multiple of 3, is generated (as a multiplicative semigroup) by the union of this sequence with {3}.
A327863, numbers whose arithmetic derivative is a multiple of 3, is generated by this sequence and A008591.
A373478, numbers that are in the intersection of A289142 and A327863, is generated by the union of this sequence with {9, 27}.
A373475, numbers that are in the intersection of A289142 and A369644 (positions of multiples of 3 in A083345), is generated by the union of this sequence with {19683}, where 19683 = 3^9.
(End)
The integers in the multiplicative subgroup of positive rationals generated by semiprimes of the form 3m+2 (A344872) and cubes of primes except 27. - Peter Munn, Jun 19 2024

			280 = 2*2*2*5*7 is included as it is not a multiple of 3, and one of its prime factors (7) is of the form 3m+1 and four are of the form 3m-1, and because 4 == 1 (mod 3). Also, A001414(280) = 18, and A003415(280) = 516, both of which are multiples of 3. - _Antti Karttunen_, Jun 12 2024

Antti Karttunen, Table of n, a(n) for n = 1..33911
Index entries for sequences which agree for a long time but are different

Cf. A001414, A002476, A003415, A003627, A083345, A369658 (characteristic function).
Intersection of A001651 and A327863.
Intersection of A001651 and A373475.
Setwise difference A373475 \ A373476.
Subsequence of A369644, which is a subsequence of A327863, and also of the following sequences: A289142, A373475, A373478.
Includes A030078 \ {27}, A344872 and A373597 as subsequences.
Cf. also A046337, A360110, A369969 for cases k=2, 4, 5 of "Nonmultiples of k whose arithmetic derivative is a multiple of k".
Cf. also A374044.

PARI
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\\ See A369658.
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Name amended with an alternative definition by Antti Karttunen, Jun 11 2024

A344703 Numbers k for which sigma(k^2) and psi(k^2) share a factor, where sigma is the sum of divisors, A000203, and psi is the Dedekind psi function, A001615.

Original entry on oeis.org

14, 21, 26, 28, 35, 38, 39, 42, 52, 56, 57, 62, 63, 65, 70, 74, 76, 77, 78, 82, 84, 86, 93, 95, 98, 99, 104, 105, 111, 112, 114, 117, 119, 122, 124, 126, 129, 130, 133, 134, 140, 143, 146, 148, 152, 154, 155, 156, 158, 161, 166, 168, 171, 172, 175, 182, 183, 185, 186, 189, 190, 194, 195, 198, 201, 203, 206, 208, 209

Offset: 1

Antti Karttunen and Peter Munn, May 27 2021

nonn

Numbers k for which A344695(k^2) > 1.
It can be shown that sigma(m) and psi(m) share a factor if m is nonsquare. (See A344695 for more detail.) So here we consider only square numbers, m = k^2.
For prime p, sigma(p^2) and psi(p^2) are coprime, since sigma(p^2) = p^2 + p + 1 = psi(p^2) + 1. So all terms are composite. We can say more, since for prime p and positive integer e, psi(p^(2*e)) = p^(2*e-1) * (p+1), whereas sigma(p^(2*e)) can be shown to be congruent to 1 modulo p and to 1 modulo p+1, so shares no factors with p^(2*e-1) * (p+1). So all terms are divisible by more than one prime.
If k is in the sequence, m*k is also present for any positive integer m coprime to k.

			Sigma (A000203) and the Dedekind psi function (A001615) are both multiplicative, so we gain insight by showing the values of these functions using their multiplicative properties:-
sigma(14^2) = sigma(2^2) * sigma(7^2) = 7 * 57 = 7 * (3*19).
psi(14^2) = psi(2^2) * psi(7^2) = 6 * 56 = (2*3) * (2^3*7).
So sigma(14^2) and psi(14^2) share factors 3 and 7, so 14 is in the sequence.
Looking in particular at the shared factor 3, we see it is present in sigma(7^2) and psi(2^2). For prime p, sigma(p^2) and psi(p^2) equate to polynomials in p, so we deduce 3 divides sigma(p^2) for p congruent to 7 modulo 3, and divides psi(p^2) for p congruent to 2 modulo 3. From this we see all products of a prime of the form 3m+1 and a prime of the form 3m+2 are in the sequence; for instance (3*4+1) * (3*1+2) = 13 * 5 = 65.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A000290, A001615, A344695.

Subsequences: A344872.

filter:= proc(k) local n,F, sig, psi, t;
   n:= k^2;
   F:= map(t -> [t[1],2*t[2]], ifactors(k)[2]);
   sig:= mul((t[1]^(1+t[2])-1)/(t[1]-1),t=F);
   psi:= n*mul(1+1/t[1],t=F);
   igcd(sig,psi) > 1
end proc:
select(filter, [$1..300]); # Robert Israel, Jan 06 2024

filter[k_] := Module[{n, F, sig, psi},
   n = k^2;
   F = {#[[1]], 2 #[[2]]}& /@ FactorInteger[k];
   sig = Product[(t[[1]]^(1 + t[[2]]) - 1)/(t[[1]] - 1), {t, F}];
   psi = n*Product[1 + 1/t[[1]], {t, F}];
   GCD[sig, psi] > 1];
Select[Range[300], filter] // Quiet (* Jean-François Alcover, May 23 2024, after Robert Israel *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344695(n) = gcd(sigma(n), A001615(n));
isA344703(n) = (A344695(n^2)>1);

cp's OEIS Frontend

A369659 Non-multiples of 3 whose arithmetic derivative, or equally, the sum of prime factors (with multiplicity) is a multiple of 3.

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