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A060831 Number of sums less than or equal to n of sequences of consecutive positive integers (including sequences of length 1). 18
0, 1, 2, 4, 5, 7, 9, 11, 12, 15, 17, 19, 21, 23, 25, 29, 30, 32, 35, 37, 39, 43, 45, 47, 49, 52, 54, 58, 60, 62, 66, 68, 69, 73, 75, 79, 82, 84, 86, 90, 92, 94, 98, 100, 102, 108, 110, 112, 114, 117, 120, 124, 126, 128, 132, 136, 138, 142, 144, 146, 150, 152, 154, 160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of odd divisors present in [1,..,n]. E.g., for a(7), we consider the odd divisors of 1,2,3,4,5,6,7, which gives 1,1,2,1,2,2,2=11. - Jon Perry, Mar 22 2004

Starting with 1 = row sums of triangle A168508. - Gary W. Adamson, Nov 27 2009

The subsequence of primes in this sequence begins, through a(100): 2, 5, 7, 11, 17, 19, 23, 29, 37, 43, 47, 73, 79, 173, 181, 223, 227, 229, 233, 263. - Jonathan Vos Post, Feb 13 2010

Apart from the initial zero, a(n) is also the total number of subparts of the symmetric representations of sigma of all positive integers <= n. Hence, a(n) is also the total number of subparts in the terraces of the step pyramid with n levels described in A245092. For more information see A279387 and A237593. - Omar E. Pol, Dec 17 2016

a(n) is also the total number of partitions of all positive integers <= n into an odd number of equal parts. - Omar E. Pol, May 14 2017

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = n + floor(n/3) + floor(n/5) + floor(n/7) + floor(n/9) + ...

a(n) = a(n-1) + A001227(n).

a(n) = A006218(n) - A006218(floor(n/2)).

a(n) = Sum_{i=1..n} A001227(i). [Trivial inversion of the first differences formula. - Jonathan Vos Post, Feb 13 2010]

a(n) = Sum_{i=1..ceiling(n/2)} A000005(n-i+1). - Wesley Ivan Hurt, Sep 30 2013

G.f.: (1/(1 - x))*Sum_{k>=1} x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 23 2016

a(n) ~ n*(log(2*n) + 2*gamma - 1) / 2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 30 2019

EXAMPLE

a(7) = 11 since 1, 2, 3, 4, 5, 6, 7, 1+2, 2+3, 3+4, 1+2+3 are all 7 or less.

MAPLE

A060831 := proc(n)

    add(numtheory[tau](n-i+1), i=1..ceil(n/2)) ;

end proc:

seq(A060831(n), n=0..100) ; # Wesley Ivan Hurt, Oct 02 2013

MATHEMATICA

f[n_] := Sum[ -(-1^k)Floor[n/(2k - 1)], {k, n}]; Table[ f[n], {n, 0, 65}] (* Robert G. Wilson v, Jun 16 2006 *)

PROG

(PARI) a(n)=local(c); c=0; for(i=1, n, c+=sumdiv(i, X, X%2)); c for(i=1, 100, print1(", "a(i)))

(PARI) for (n=0, 1000, s=n; d=3; while (n>=d, s+=n\d; d+=2); write("b060831.txt", n, " ", s); ) \\ Harry J. Smith, Jul 12 2009

(PARI) a(n)=my(n2=n\2); sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2-sum(k=1, sqrtint(n2), n2\k)*2+sqrtint(n2)^2 \\ Charles R Greathouse IV, Jun 18 2015

CROSSREFS

Cf. A006218, A168508, A279387.

Sequence in context: A325101 A301728 A005152 * A073727 A075692 A112235

Adjacent sequences:  A060828 A060829 A060830 * A060832 A060833 A060834

KEYWORD

nonn

AUTHOR

Henry Bottomley, May 01 2001

STATUS

approved

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Last modified October 18 05:45 EDT 2019. Contains 328146 sequences. (Running on oeis4.)