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A192718 Elements of A192628 which are congruent to 7 (mod 8) (equivalently, 7 (mod 16)). 1
7, 55, 71, 87, 103, 119, 183, 263, 279, 343, 375, 391, 439, 455, 519, 551, 567, 583, 615, 631, 647, 695, 711, 727, 759, 775, 791, 823, 855, 871, 887, 903, 951, 967, 1015, 1047, 1079, 1095, 1111, 1127, 1159, 1175, 1191, 1223, 1239, 1271, 1303, 1319, 1367 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is the subsequence/subset of A192628 which contains elements congruent to 7 modulo 8.  Equivalently, these elements are also congruent to 7 modulo 16.

By partitioning A192628 into congruence classes k modulo 8, it turns out that it contains only elements congruent to 0, 1, 3, and 7 modulo 8.  Further, the congruence classes 0, 1, and 3 modulo 8 are vanishing--having a density asymptotic to 0.

However, the 7 modulo 8 congruence classes appears to have nonzero density, conjectured 1/32.  A current upper bound on its density (thus the entire density of A192628) is 1/16.

REFERENCES

J. Cooper and A. Riasanovsky, On the reciprocal of the binary generating function for the sum-of-divisors, Journal of Integer Sequences (accepted).

J. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.

LINKS

Table of n, a(n) for n=1..49.

J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8

PROG

(Sage)

prec = 2^12

R = PowerSeriesRing(GF(2), 'q', default_prec = prec)

q = R.gen()

sigma = lambda x : 1 if x == 0 else sum(Integer(x).divisors())

SigmaSeries = sum([sigma(m)*q^m for m in range(prec)])

SigmaBarSeries = 1/SigmaSeries

SigmaBarList = SigmaBarSeries.exponents()

SigmaBar7Mod8 = [m for m in SigmaBarList if mod(m, 8) == 7]

print(SigmaBar7Mod8)

CROSSREFS

Sequence in context: A198149 A203878 A043077 * A014637 A062212 A272864

Adjacent sequences:  A192715 A192716 A192717 * A192719 A192720 A192721

KEYWORD

nonn

AUTHOR

Alexander Riasanovsky, Dec 31 2012

STATUS

approved

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Last modified May 25 18:25 EDT 2022. Contains 354071 sequences. (Running on oeis4.)