A219730 -id:A219730 - cp's OEIS frontend

A219729 Sum_{x <= n} largest divisor of x that is <= sqrt(x).

1, 2, 3, 5, 6, 8, 9, 11, 14, 16, 17, 20, 21, 23, 26, 30, 31, 34, 35, 39, 42, 44, 45, 49, 54, 56, 59, 63, 64, 69, 70, 74, 77, 79, 84, 90, 91, 93, 96, 101, 102, 108, 109, 113, 118, 120, 121, 127, 134, 139, 142, 146, 147, 153, 158, 165, 168, 170, 171, 177, 178

Offset: 1

Michel Marcus, Nov 26 2012

nonn

G. Tenenbaum proved that limit(log(a(n)/n^(3/2)))/log(log(n)) is -b with b = 1-(1+loglog 2)/log 2 = 0.08607... (same constant as in A027424 comment) (théorème 1).

Ivan Neretin, Table of n, a(n) for n = 1..10000
Steven Finch, Multiples and divisors, January 27, 2004. [Cached copy, with permission of the author]
Gérald Tenenbaum, Sur deux fonctions de diviseurs, J. London Math. Soc. (1976) s2-14 (3): 521-526; alternative copy.

Cf. A027424, A033676, A219730.

t = Table[d = Divisors[n]; d[[Ceiling[Length[d]/2]]], {n, 100}]; Accumulate[t] (* T. D. Noe, Nov 26 2012 *)

A373083 a(1) = 1; for n > 1, a(n) = a(n-1) + n if n is prime, else a(n) = a(n-1) + largest divisor of n < n.

Original entry on oeis.org

1, 3, 6, 8, 13, 16, 23, 27, 30, 35, 46, 52, 65, 72, 77, 85, 102, 111, 130, 140, 147, 158, 181, 193, 198, 211, 220, 234, 263, 278, 309, 325, 336, 353, 360, 378, 415, 434, 447, 467, 508, 529, 572, 594, 609, 632, 679, 703, 710, 735, 752, 778, 831, 858, 869, 897

Offset: 1

James C. McMahon, Jun 17 2024

Differs from A219730 at a(12).

			From _Michael De Vlieger_, Jun 23 2024: (Start)
Let lpf = A020639(n).
a(2) = 3 since 2 is prime, therefore a(1) + 2 = 3.
a(3) = 6 since 3 is prime, therefore a(2) + 3 = 6.
a(4) = 8 since 4 is not prime, therefore a(3) + 4/lpf(4) = 6 + 2 = 8.
a(5) = 13 since 5 is prime, therefore a(4) + 5 = 13.
a(6) = 16 since 6 is not prime, hence a(5) + 6/lpf(6) = 13 + 3 = 16, etc. (End)

James C. McMahon, Table of n, a(n) for n = 1..1000

Cf. A117818, A219730.

a[1]=1;a[n_]:=If[PrimeQ[n],a[n-1]+n,a[n-1]+Divisors[n][[-2]]];Table[a[n],{n,56}]

alist(N) = my(r, d); vector(N, n, r+=if(2<#d=divisors(n), d[#d-1], d[#d])); \\ Ruud H.G. van Tol, Jul 11 2024

a(n) = a(n-1) + A117818(n) for n > 1. - Michael De Vlieger, Jun 23 2024

A161416 Partial sums of A056737.

Original entry on oeis.org

0, 1, 3, 3, 7, 8, 14, 16, 16, 19, 29, 30, 42, 47, 49, 49, 65, 68, 86, 87, 91, 100, 122, 124, 124, 135, 141, 144, 172, 173, 203, 207, 215, 230, 232, 232, 268, 285, 295, 298, 338, 339, 381, 388, 392, 413, 459, 461, 461, 466, 480, 489, 541, 544, 550, 551, 567, 594

Offset: 1

Omar E. Pol, Jun 21 2009

nonn

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Divisors and pi(x)

Cf. A056737, A018253, A160811, A160812, A161205, A161344, A161345, A161424, A006446, A161828, A161835, A219729, A219730.

a(n) = A219730(n) - A219729(n). - Tamas Sandor Nagy, Jan 20 2024

Extended beyond a(16) by R. J. Mathar, Aug 01 2009

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A219729 Sum_{x <= n} largest divisor of x that is <= sqrt(x).

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A373083 a(1) = 1; for n > 1, a(n) = a(n-1) + n if n is prime, else a(n) = a(n-1) + largest divisor of n < n.

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A161416 Partial sums of A056737.

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