A190998 -id:A190998 - cp's OEIS frontend

A105852 a(n) = sigma(n) mod 9.

1, 3, 4, 7, 6, 3, 8, 6, 4, 0, 3, 1, 5, 6, 6, 4, 0, 3, 2, 6, 5, 0, 6, 6, 4, 6, 4, 2, 3, 0, 5, 0, 3, 0, 3, 1, 2, 6, 2, 0, 6, 6, 8, 3, 6, 0, 3, 7, 3, 3, 0, 8, 0, 3, 0, 3, 8, 0, 6, 6, 8, 6, 5, 1, 3, 0, 5, 0, 6, 0, 0, 6, 2, 6, 7, 5, 6, 6, 8, 6, 4, 0, 3, 8, 0, 6, 3, 0, 0, 0, 4, 6, 2, 0, 3, 0, 8, 0, 3, 1, 3, 0, 5, 3, 3

Offset: 1

Shyam Sunder Gupta, May 05 2005

If gcd(m,n) = 1 then a(m*n) = (a(m) * a(n)) mod 9. - Robert Israel, Sep 14 2014

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

Cf. A000203, A010878, A190998 (digital root of sigma(n)).

seq(numtheory:-sigma(n) mod 9, n=1..1000); # Robert Israel, Sep 14 2014

Mod[DivisorSigma[1, Range[100]], 9] (* Wesley Ivan Hurt, Apr 25 2023 *)

PARI
```
a(n)=sigma(n)%9
```

a(n) = A010878(A000203(n)). - Michel Marcus, Sep 14 2014

Name corrected and keyword base removed by Michel Marcus, Sep 14 2014

A231817 Multiplicative digital root of concatenation of all divisors of n (A037278).

Original entry on oeis.org

1, 2, 3, 8, 5, 8, 7, 8, 4, 0, 1, 6, 3, 0, 5, 0, 7, 0, 9, 0, 8, 8, 6, 8, 0, 4, 6, 0, 8, 0, 3, 0, 4, 6, 0, 0, 2, 8, 8, 0, 4, 0, 2, 0, 0, 6, 6, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 6, 4, 0, 0, 0, 0, 8, 0, 2, 0, 7, 0, 2, 8, 0, 0, 8, 0, 8, 0, 0, 6, 8, 0, 0, 0, 0

Offset: 1

Jaroslav Krizek, Nov 13 2013

Also multiplicative digital root of A190997 (product of digits of all the divisors of n) or A007955 (product of divisors of n).
Conjecture: a(n) = 0 for almost all n.
793 of the first 1000 terms are zeros, and 9147 out of the first 10000 terms are zeros. - Harvey P. Dale, Jul 30 2019

			For n=12: 1*2*3*4*6*1*2=288, 2*8*8=128, 1*2*8=16, 1*6=6; a(12)=6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A037278, A007955, A190998 (associative digital root of digits of all the divisors of n), A031347 (multiplicative digital root of n).

Table[NestWhile[Times@@IntegerDigits[#]&,Times@@Flatten[ IntegerDigits/@ Divisors[ n]], #>9&],{n,90}] (* Harvey P. Dale, Jul 30 2019 *)

A240597 Numbers k such that sigma(k) == k (mod 9).

Original entry on oeis.org

1, 15, 24, 42, 60, 64, 69, 78, 90, 100, 114, 123, 133, 147, 153, 177, 186, 198, 222, 231, 240, 258, 259, 270, 276, 288, 289, 306, 339, 360, 366, 393, 402, 403, 414, 429, 438, 447, 459, 474, 477, 492, 495, 501, 507, 511, 522, 582, 588, 594, 600

Offset: 1

Ivan N. Ianakiev, Sep 13 2014

That is, numbers k that satisfy the following:
A010878(k) = A105852(k) or A010878(k) = A010878(A000203(k)).
A010888(k) = A190998(k) or A010888(k) = A010888(A000203(k)).

			sigma(15) = 24. 24 == 15 (mod 9), therefore 15 is in the sequence.

Ivan N. Ianakiev, Table of n, a(n) for n = 1..10000

Cf. A000203, A190998, A010878, A105852, A010888.

Select[Range[1000],Mod[#,9]==Mod[DivisorSigma[1,#],9]&]

A010888(a(n)) = A010888(A000203(a(n))).

A010888(a(n)) = A190998(a(n)).

A316570 Multiplicative digital root of sigma(n).

Original entry on oeis.org

1, 3, 4, 7, 6, 2, 8, 5, 3, 8, 2, 6, 4, 8, 8, 3, 8, 4, 0, 8, 6, 8, 8, 0, 3, 8, 0, 0, 0, 4, 6, 8, 6, 0, 6, 9, 8, 0, 0, 0, 8, 0, 6, 6, 0, 4, 6, 8, 5, 4, 4, 4, 0, 0, 4, 0, 0, 0, 0, 6, 2, 0, 0, 4, 6, 6, 6, 2, 0, 6, 4, 0, 6, 4, 8, 0, 0, 6, 0, 6, 2, 2, 6, 6, 0, 6, 0

Offset: 1

Jaroslav Krizek, Jul 07 2018

Multiplicative digital root of A000203(n).

			For n = 12: sigma(12) = 28; a(n) = 6 because 2 * 8 = 16; 1 * 6 = 6.

Cf. A000203, A031347.

Cf. A190998 (additive digital root of sigma(n)).

m:= n-> `if`(n<10, n, m(mul(d, d=convert(n, base, 10)))):
a:= n-> m(numtheory[sigma](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 21 2018

a031347(n) = while(n>9, n=prod(i=1, #n=digits(n), n[i])); n;
a(n) = a031347(sigma(n)); \\ Michel Marcus, Jul 07 2018

a(n) = A031347(A000203(n)).

cp's OEIS Frontend

A105852 a(n) = sigma(n) mod 9.

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A231817 Multiplicative digital root of concatenation of all divisors of n (A037278).

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A240597 Numbers k such that sigma(k) == k (mod 9).

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Mathematica

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A316570 Multiplicative digital root of sigma(n).

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Author

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Examples

Crossrefs

Programs

Maple

PARI

Formula