A354100 -id:A354100 - cp's OEIS frontend

A329963 Numbers k such that sigma(k) is not divisible by 3.

1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 144, 148, 151, 156, 157, 163, 171, 172, 175, 181, 183, 189, 192, 193, 199, 201, 208, 211, 217, 219, 223, 225, 228, 229

Offset: 1

John L. Drost, Nov 25 2019

nonn

A number k is in the sequence iff in its prime factorization, all primes p == 1 (mod 3) occur to such a power p^e that e != 2 (mod 3), and all primes == 2 (mod 3) occur to even powers. (3 can occur to any power.) This sequence is similar but not identical to many others; in particular, 343 is in this sequence, but not in A034022. (And here we don't have 196, although it is in A034022). - First sentence corrected and additional notes added by Antti Karttunen, Jul 03 2024, see also Robert Israel's Nov 09 2016 comment in A087943.

The asymptotic density of this sequence is 0 (Dressler, 1975). - Amiram Eldar, Jul 23 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 15 ff. [Note: the "if and only if" condition given in the beginning of Theorem 7.1 is for A003136, not for this sequence. - _Antti Karttunen_, Jul 04 2024]
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Complement of A087943. Positions of zeros in A354100, nonzeros in A074941.
Cf. A000203, A353815 (characteristic function).
Setwise difference A003136 \ A088535.
Subsequences: A002476, A068228, A351537, A374135.
Cf. also A088232.
Not the same as A034022.

[k:k in [1..200]| DivisorSigma(1,k) mod 3 ne 0]; // Marius A. Burtea, Jan 02 2020

select(t -> numtheory:-sigma(t) mod 3 <> 0, [$1..200]); # Robert Israel, Jan 01 2020

Select[Range[200], !Divisible[DivisorSigma[1, #], 3] &] (* Amiram Eldar, Nov 25 2019 *)

isok(k) = (sigma(k) % 3) != 0; \\ Michel Marcus, Nov 26 2019

isA329963 = A353815; \\ Antti Karttunen, Jul 03 2024

More terms from Joshua Oliver, Nov 26 2019

Data section further extended up to a(71), to better differentiate from nearby sequences - Antti Karttunen, Jul 04 2024

A354096 a(n) = A354092(sigma(A354091(n))).

Original entry on oeis.org

1, 3, 1, 31, 3, 3, 1, 39, 13, 9, 9, 31, 7, 3, 3, 295, 3, 39, 2, 93, 1, 27, 6, 39, 133, 21, 2, 31, 21, 9, 1, 1953, 9, 9, 3, 403, 19, 6, 7, 117, 3, 3, 5, 279, 39, 18, 27, 295, 57, 399, 3, 217, 6, 6, 27, 39, 2, 63, 9, 93, 31, 3, 13, 19531, 21, 27, 11, 93, 6, 9, 21, 507, 37, 57, 133, 62, 9, 21, 2, 885, 25, 9, 18, 31, 9

Offset: 1

Antti Karttunen, May 17 2022

Cf. A000203, A003627, A007949, A010872, A329963, A354091, A354092, A354093, A354095.

Cf. also A326042, A354088 for variants.

A354091(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), for(i=1+primepi(f[k,1]),oo,if(2==(prime(i)%3), f[k,1]=prime(i); break)))); factorback(f); };
A354092(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), if(2==f[k,1], f[k,1]--, forstep(i=primepi(f[k,1])-1,0,-1,if(2==(prime(i)%3), f[k,1]=prime(i); break))))); factorback(f); };
A354096(n) = A354092(sigma(A354091(n)));

Multiplicative with a(p^e) = A354092((q^(e+1)-1)/(q-1)), where q = A003627(1+n) if p = A003627(n), otherwise q = p.
a(n) = A354092(A354093(n)) = A354095(A354091(n)) = A354092(A000203(A354091(n))).
For all n >= 1, A010872(a(n)) = A010872(A354095(n)).
For all k in A329963, A007949(a(k)) = A007949(sigma(k)) = A354100(k) = 0.

A354099 The 3-adic valuation of Euler totient function phi.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 2, 0, 2, 1, 1, 1, 0, 3, 0, 0, 1, 0, 1, 0, 0, 0, 1, 2, 0, 1, 0, 2, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1

Offset: 1

Antti Karttunen, May 17 2022

nonn

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

Cf. A000010, A007949, A074942.
Cf. A088232 (positions of zeros), A066498 (of terms > 0).
Cf. also A354100.

a[n_] := IntegerExponent[EulerPhi[n], 3]; Array[a, 100] (* Amiram Eldar, May 17 2022 *)

PARI
```
A354099(n) = valuation(eulerphi(n),3);
```

A354099(n) = { my(f=factor(n)); sum(k=1,#f~,valuation((f[k,1]-1)*(f[k,1]^(f[k,2]-1)), 3)); }; \\ Demonstrates the additivity.

a(n) = A007949(A000010(n)).

Additive with a(p^e) = A007949((p-1)*p^(e-1)).

A379473 a(n) is the highest power of 3 dividing the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 1, 3, 3, 1, 3, 1, 9, 3, 1, 1, 3, 3, 1, 9, 3, 1, 3, 1, 9, 3, 3, 1, 3, 1, 1, 3, 9, 1, 9, 3, 27, 3, 1, 1, 3, 1, 9, 3, 3, 1, 3, 3, 9, 3, 1, 3, 3, 9, 1, 27, 3, 9, 3, 1, 9, 3, 3, 1, 3, 1, 1, 3, 9, 1, 9, 3, 9, 9, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 9, 3, 1, 27, 3, 3, 9, 9, 9, 1, 3, 1, 9, 3, 9, 1, 9, 3, 1, 3, 27, 1, 3, 3

Offset: 1

Antti Karttunen, Dec 27 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n).

Cf. A000203, A000244, A038500, A354100, A379484 [= a(A379481(n))].
Cf. A329963 (positions of 1's), A087943 (of terms > 1).
Cf. also A082903.

a[n_] := 3^IntegerExponent[DivisorSigma[1, n], 3]; Array[a, 100] (* Amiram Eldar, Dec 27 2024 *)

PARI
```
A379473(n) = (3^valuation(sigma(n),3));
```

Multiplicative with a(p^e) = A038500((p^(e+1)-1)/(p-1)).
a(n) = A038500(A000203(n)).
a(n) = A000244(A354100(n)).

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A329963 Numbers k such that sigma(k) is not divisible by 3.

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A354096 a(n) = A354092(sigma(A354091(n))).

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A354099 The 3-adic valuation of Euler totient function phi.

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A379473 a(n) is the highest power of 3 dividing the sum of divisors of n.

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