The On-Line Encyclopedia of Integer Sequences (OEIS)

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A327566

Partial sums of the infinitary divisors sum function: a(n) = Sum_{k=1..n} isigma(k), where isigma is A049417.
(history; published version)

#10 by Giovanni Resta at Mon Sep 30 04:04:57 EDT 2019

STATUS

proposed

approved

#9 by Amiram Eldar at Mon Sep 30 04:01:46 EDT 2019

STATUS

editing

proposed

#8 by Amiram Eldar at Mon Sep 30 03:45:19 EDT 2019

LINKS

Amiram Eldar, <a href="/A327566/b327566.txt">Table of n, a(n) for n = 1..10000</a>

STATUS

approved

editing

#7 by Susanna Cuyler at Tue Sep 17 08:26:11 EDT 2019

STATUS

proposed

approved

#6 by Amiram Eldar at Tue Sep 17 05:57:41 EDT 2019

STATUS

editing

proposed

#5 by Amiram Eldar at Tue Sep 17 05:45:51 EDT 2019

FORMULA

a(n) ~ c * n^2, where c = 0.730718... (A327569A327574).

CROSSREFS

Cf. A049417 (isigma), A327569A327574.

#4 by Amiram Eldar at Tue Sep 17 05:35:09 EDT 2019

MATHEMATICA

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], _?(# == 1 &)])); isigma[1] = 1; isigma[n_] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); Accumulate[Array[isigma, 52]]

#3 by Amiram Eldar at Tue Sep 17 05:33:09 EDT 2019

NAME

allocated for Amiram EldarPartial sums of the infinitary divisors sum function: a(n) = Sum_{k=1..n} isigma(k), where isigma is A049417.

DATA

1, 4, 8, 13, 19, 31, 39, 54, 64, 82, 94, 114, 128, 152, 176, 193, 211, 241, 261, 291, 323, 359, 383, 443, 469, 511, 551, 591, 621, 693, 725, 776, 824, 878, 926, 976, 1014, 1074, 1130, 1220, 1262, 1358, 1402, 1462, 1522, 1594, 1642, 1710, 1760, 1838, 1910, 1980

OFFSET

1,2

COMMENTS

Differs from A307159 at n >= 16.

REFERENCES

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.

LINKS

Graeme L. Cohen and Peter Hagis, Jr., <a href="http://dx.doi.org/10.1155/S0161171293000456">Arithmetic functions associated with infinitary divisors of an integer</a>, International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 2 (1993), pp. 373-383.

FORMULA

a(n) ~ c * n^2, where c = 0.730718... (A327569).

MATHEMATICA

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], _?(# == 1 &)])); isigma[1] = 1; isigma[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); Accumulate[Array[isigma, 52]]

CROSSREFS

Cf. A049417 (isigma), A327569.

Cf. A024916 (all divisors), A064609 (unitary), A307042 (exponential), A307159 (bi-unitary).

KEYWORD

allocated

nonn

AUTHOR

Amiram Eldar, Sep 17 2019

STATUS

approved

editing

#2 by Amiram Eldar at Tue Sep 17 05:28:26 EDT 2019

KEYWORD

allocating

allocated

#1 by Amiram Eldar at Tue Sep 17 05:28:26 EDT 2019

NAME

allocated for Amiram Eldar

KEYWORD

allocating

STATUS

approved