A320226 -id:A320226 - cp's OEIS frontend

A002541 a(n) = Sum_{k=1..n-1} floor((n-k)/k).

0, 1, 2, 4, 5, 8, 9, 12, 14, 17, 18, 23, 24, 27, 30, 34, 35, 40, 41, 46, 49, 52, 53, 60, 62, 65, 68, 73, 74, 81, 82, 87, 90, 93, 96, 104, 105, 108, 111, 118, 119, 126, 127, 132, 137, 140, 141, 150, 152, 157, 160, 165, 166, 173, 176, 183, 186, 189, 190, 201, 202, 205

Offset: 1

N. J. A. Sloane

Number of pairs (a, b) with 1 <= a < b <= n, a | b.
The sequence shows how many digit "skips" there have been from 2 to n, a skip being either a prime factor or product thereof. Every time you have a place where you have X skips and the next skip value is X+1, you will have a prime number since a prime number will only add exactly one more skip to get to it. a(n) = Sum_{x=2..n} floor(n/x) - Sum_{x=2..n-1} floor( (n-1)/x) = 1 when prime. - Marius-Paul Dumitrean (marius(AT)neldor.com), Feb 19 2007
A027749(a(n)+1) = n; A027749(a(n)+2) = A020639(n+1). - Reinhard Zumkeller, Nov 22 2003
Number of partitions of n into exactly 2 types of part, where one part is 1, e.g., n=7 gives 1111111, 111112, 11122, 1222, 11113, 133, 1114, 115 and 16, so a(7)=9. - Jon Perry, May 26 2004
The sequence of partial sums of A032741. Idea of proof: floor((n-k)/k) - floor((n-k-1)/k) only increases by 1 when k | n. - George Beck, Feb 12 2012
Also the number of integer partitions of n whose non-1 parts are all equal and with at least one non-1 part. - Gus Wiseman, Oct 07 2018

			From _Gus Wiseman_, Oct 07 2018: (Start)
The integer partitions whose non-1 parts are all equal and with at least one non-1 part:
  (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)
       (21)  (22)   (41)    (33)     (61)      (44)       (81)
             (31)   (221)   (51)     (331)     (71)       (333)
             (211)  (311)   (222)    (511)     (611)      (441)
                    (2111)  (411)    (2221)    (2222)     (711)
                            (2211)   (4111)    (3311)     (6111)
                            (3111)   (22111)   (5111)     (22221)
                            (21111)  (31111)   (22211)    (33111)
                                     (211111)  (41111)    (51111)
                                               (221111)   (222111)
                                               (311111)   (411111)
                                               (2111111)  (2211111)
                                                          (3111111)
                                                          (21111111)
(End)

J. P. Gram, Undersoegelser angaaende maengden af primtal under en given graense, Det Kongelige Danskevidenskabernes Selskabs Skrifter, series 6, vol. 2 (1884), 183-288; see Tab. VII: Vaerdier af Funktionen psi(n) og andre numeriske Funktioner, pp. 281-288, especially p. 281.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Antidiagonal sums of array A003988. Antidiagonal sums of A004199.
Cf. A000005, A006218, A020639, A027749, A032741.
Cf. A003238, A070776, A126656, A320224, A320225, A320226.

a002541 n = sum $ zipWith div [n - 1, n - 2 ..] [1 .. n - 1]
-- Reinhard Zumkeller, Jul 05 2013

a:= proc(n) option remember; `if`(n<2, 0,
      numtheory[tau](n)-1+a(n-1))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 12 2021

Table[Sum[Floor[(n-k)/k],{k,n-1}],{n,100}] (* Harvey P. Dale, May 02 2011 *)

a(n)=sum(k=1,n-1, n\k-1) \\ Charles R Greathouse IV, Feb 07 2017

first(n)=my(v=vector(n),s); for(k=1,n, v[k]=-k+s+=numdiv(k)); v \\ Charles R Greathouse IV, Feb 07 2017

from math import isqrt
def A002541(n): return (sum(n//k for k in range(1,isqrt(n)+1))<<1)-isqrt(n)**2-n # Chai Wah Wu, Oct 20 2023

a(n) = -n + Sum_{k=1..n} tau(k). - Vladeta Jovovic, Oct 17 2002
G.f.: 1/(1-x) * Sum_{k>=2} x^k/(1-x^k). - Benoit Cloitre, Apr 23 2003
a(n) = Sum_{i=2..n} floor(n/i). - Jon Perry, Feb 02 2004
a(n) = (Sum_{i=2..n} ceiling((n+1)/i)) - n + 1. - Jon Perry, May 26 2004 [corrected by Jason Yuen, Jul 31 2024]
a(n) = A006218(n) - n. Proof: floor((n-k)/k)+1 = floor(n/k). Then Sum_{k=1..n-1} floor((n-k)/k)+(n-1)+1 = Sum_{k=1..n-1} floor(n/k) + floor(n/n) = Sum_{k=1..n} floor(n/k); i.e., a(n) + n = A006218(n). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(n) = A161886(n) - (2n-1). - Eric Desbiaux, Jul 10 2013
a(n+1) = Sum_{k=1..n} A004199(n-k+1,k). - L. Edson Jeffery, Aug 31 2014
a(n) = -Sum_{i=1..n} floor((n-2i+1)/(n-i+1)). - Wesley Ivan Hurt, May 08 2016
a(n) = Sum_{i=1..floor(n/2)} floor((n-i)/i). - Wesley Ivan Hurt, Nov 16 2017
a(n) = Sum_{k=1..n-1} (A000005(n-k) - 1). - Gus Wiseman, Oct 07 2018
a(n) ~ n * (log(n) + 2*EulerGamma - 2). - Rok Cestnik, Dec 19 2020

More terms from David W. Wilson

A320230 Matula-Goebel numbers of rooted trees in which the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 48, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 64, 67, 68, 71, 72, 74, 76, 79, 80, 81, 82, 83, 86, 88, 89, 92, 96, 97

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

A number is in the sequence iff it belongs to A070776 and its prime indices already belong to the sequence. A prime index of n is a number m such that prime(m) divides n.

Cf. A002541, A070776, A214577, A294336, A317710, A320222, A320226.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
smakQ[n_]:=And[SameQ@@DeleteCases[primeMS[n],1],And@@smakQ/@DeleteCases[primeMS[n],1]];Select[Range[100],smakQ[#]&]

is(n) = while((n>>=valuation(n,2)) > 1, isprimepower(n,&n) || return(0); n=primepi(n)); 1; \\ Kevin Ryde, Apr 04 2021

A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

First differs from A331871 in lacking 1589.
Lone-child-avoiding means there are no unary branchings.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A002541, A070776, A167865, A214577, A317710, A320222, A320226.
The same-tree version is A291441.
Not requiring lone-child-avoidance gives A320230.
The enumeration of these trees by vertices is A320268.
The semi-lone-child-avoiding version is A331936.
If the non-leaf branches are all different instead of equal we get A331965.
The fully-achiral case is A331967.
Achiral rooted trees are counted by A003238.
MG-numbers of lone-child-avoiding rooted trees are A291636.
Cf. A061775, A196050, A276625, A280996, A306202, A331912, A331916, A331966.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
hmakQ[n_]:=And[!PrimeQ[n],SameQ@@DeleteCases[primeMS[n],1],And@@hmakQ/@primeMS[n]];Select[Range[1000],hmakQ[#]&]

Updated with corrected terminology by Gus Wiseman, Feb 06 2020

A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 44, 70, 119, 189, 314, 506, 830, 1336, 2186, 3522, 5737, 9266, 15047, 24313, 39444, 63759, 103322, 167098, 270616, 437714, 708676, 1146390, 1855582, 3002017, 4858429, 7860454, 12720310, 20580764, 33303260, 53884144, 87190964

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

This is a weaker condition than achirality (cf. A167865).

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(3) = 1 through a(8) = 9 rooted trees:
  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)
               (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))
                        (oo(oo))  (oo(ooo))   (oo(oooo))
                                  (ooo(oo))   (ooo(ooo))
                                  ((oo)(oo))  (oooo(oo))
                                  (o(o(oo)))  (o(o(ooo)))
                                              (o(oo)(oo))
                                              (o(oo(oo)))
                                              (oo(o(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001678, A002541, A003238, A010766, A070776, A014668, A126656, A167865, A317099, A317100, A317712, A320222, A320226, A320269.

saum[n_]:=Sum[If[DeleteCases[ptn,1]=={},1,saum[DeleteCases[ptn,1][[1]]]],{ptn,Select[IntegerPartitions[n-1],And[Length[#]!=1,SameQ@@DeleteCases[#,1]]&]}];
Array[saum,20]

seq(n)={my(v=vector(n)); v[1]=1; for(n=3, n, v[n] = 1 + sum(k=2, n-2, (n-1)\k*v[k])); v} \\ Andrew Howroyd, Oct 26 2018

a(1) = 1; a(2) = 0; a(n > 2) = 1 + Sum_{k = 2..n-2} floor((n-1)/k) * a(k).

A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 2, 5, 3, 4, 3, 6, 5, 6, 3, 7, 6, 7, 6, 9, 6, 7, 6, 11, 9, 10, 7, 10, 9, 10, 9, 14, 11, 12, 9, 13, 12, 13, 10, 15, 14, 15, 14, 17, 12, 13, 12, 19, 17, 18, 15, 18, 17, 18, 15, 20, 17, 18, 17, 22, 21, 22, 17, 23, 20, 21, 20, 23, 20, 21, 20, 27, 26, 27, 22, 25, 22, 23, 22

Offset: 1

Ilya Gutkovskiy, May 25 2020

nonn

a(n) = number of terms among {ceiling(n/k)}, 1 <= k <= n, that are odd.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002541, A006218, A048272, A059851, A075997, A120885, A320226, A325939, A332682, A333194.

b:= n-> add((-1)^d, d=numtheory[divisors](n)):
a:= proc(n) option remember; `if`(n>0, 1+b(n-1)+a(n-1), 0) end:
seq(a(n), n=1..80);  # Alois P. Heinz, May 25 2020

Table[Sum[Mod[Ceiling[n/k], 2], {k, 1, n}], {n, 1, 80}]
Table[n - Sum[DivisorSum[k, (-1)^(# + 1) &], {k, 1, n - 1}], {n, 1, 80}]
nmax = 80; CoefficientList[Series[x/(1 - x) (1 + Sum[x^(2 k)/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, ceil(n/k) % 2); \\ Michel Marcus, May 25 2020

from math import isqrt
def A330926(n): return n+(s:=isqrt(n-1))**2-((t:=isqrt(m:=n-1>>1))**2<<1)-(sum((n-1)//k for k in range(1,s+1))-(sum(m//k for k in range(1,t+1))<<1)<<1) # Chai Wah Wu, Oct 23 2023

G.f.: (x/(1 - x)) * (1 + Sum_{k>=1} x^(2*k) / (1 + x^k)).
a(n) = n - Sum_{k=1..n-1} A048272(k).
a(n) = A075997(n-1) + 1.

cp's OEIS Frontend

A002541 a(n) = Sum_{k=1..n-1} floor((n-k)/k).

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A320230 Matula-Goebel numbers of rooted trees in which the non-leaf branches directly under any given node are all equal.

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A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).

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A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.

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A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

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