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A297575 Numbers whose sum of divisors is divisible by 10. 1
19, 24, 27, 29, 38, 40, 54, 56, 57, 58, 59, 76, 79, 87, 88, 89, 95, 104, 108, 109, 114, 116, 118, 120, 133, 135, 136, 139, 145, 149, 152, 158, 168, 171, 174, 177, 178, 179, 184, 189, 190, 199, 203, 209, 216, 218, 228, 229, 232, 236, 237, 239, 247, 248 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Of the first 10^2, 10^6, and 10^10 positive integers, 17%, 40.8%, and 48.4%, respectively, are in the sequence (See Robert G. Wilson's table below). So the conjecture here is that if this trend continues then as the numbers approach infinity, the percentage of those numbers whose sum of divisors is divisible by 10 approaches 100%.

From David A. Corneth, Jan 01 2018: (Start)

If a(n) and m > 0 are coprime then a(n) * m is in the sequence.

A030433 is a subsequence.

Let p(n, d) be a prime ending in base-10 digit d and m != n gives p(m, d) != p(n, d). For p(n, 2) and p(n, 5) we must have n = 1 (one unique value). Then we could describe families of terms as factorizations in terms of these primes. 24 would give the family p(1, 2)^3 * p(1, 3).

For d coprime to 10 we could replace p(n, d) with p(n, 10 - d) and give another family of terms. Constructing terms could then be done by selecting primes for p(n, 1), p(n, 3), p(n, 7) and p(n, 9) from A030430, A030431, A030432 and A030433 respectively. (End)

From Robert G. Wilson v, Jan 03 2017: (Start)

The following array has for n powers of 10 and k is the number of integers <= 10^n which are == k (mod 10).

Array begins:

========================================================================================

n\k|         0     1          2     3          4     5          6     7          8     9

---|------------------------------------------------------------------------------------

_1 |         0     1          1     2          1     1          1     1          2     0

_2 |        17     6         16     4         21     2         12     4         17     1

_3 |       274    17        178    16        177     7        145     7        173     6

_4 |      3352    47       1690    44       1727    23       1498    27       1563    29

_5 |     37709   130      16012   144      16069    77      14649    87      15022   101

_6 |    408270   406     151936   424     152079   249     141527   293     144481   335

_7 |   4327266  1255    1451931  1341    1452017   787    1368110   923    1395278  1092

_8 |  45278675  3876   13963299  4206   13963783  2494   13268616  2998   13508556  3497

_9 | 469680154 11975  134976927 13185  134999718  7903  129084019  9684  131205200 11235

10 |4842279472 37545 1310133910 41349 1310381140 25017 1259051325 31136 1277983443 35663

  ... (End)

From Robert Israel, Jan 07 2018: (Start)

A number with prime factorization Product_j (p_j)^(e_j) is in the sequence if and only if

1) e_j is odd for some odd p_j, and

2) For some j, either p_j == 1 (mod 5) and e_j == 4 (mod 5), or p_j == 2 or 3 (mod 5) and e_j == 3 (mod 4), or p_j == 4 (mod 5) and e_j is odd.

By the strong form of Dirichlet's theorem, the product of 1-1/p for primes == 4 (mod 5) is 0, which implies that the asymptotic density of the sequence is 1.

(End)

REFERENCES

I. Niven, H. S. Zuckermann and H. L. Montgomery, An Introduction to the Theory of Numbers, John Wiley and Sons, 1991, pages 4-20.

LINKS

Zoltan Galantai, Table of n, a(n) for n = 1..1591

Zoltan Galantai, List of numbers up to 3 million whose sum of divisors is divisible by 10

EXAMPLE

19 is in the sequence, since sigma(19) = 20, which is divisible by 10.

20 is not in the sequence as sigma(20) = 42, which isn't divisible by 10.

MAPLE

select(n -> numtheory:-sigma(n) mod 10 = 0, [$1..1000]); # Robert Israel, Jan 02 2018

MATHEMATICA

Select[Range@250, Mod[DivisorSigma[1, #], 10] == 0 &] (* Robert G. Wilson v, Jan 03 2018 *)

PROG

(PARI) is(n) = sigma(n) % 10 == 0 \\ David A. Corneth, Jan 01 2018

CROSSREFS

Cf. A000203, A008592, A030430, A030431, A030432, A030433, A291422, A292217.

Sequence in context: A284495 A160077 A076353 * A093020 A105504 A151900

Adjacent sequences:  A297572 A297573 A297574 * A297576 A297577 A297578

KEYWORD

nonn

AUTHOR

Zoltan Galantai, Jan 01 2018

STATUS

approved

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Last modified November 16 17:25 EST 2018. Contains 317275 sequences. (Running on oeis4.)