A245211 -id:A245211 - cp's OEIS frontend

144, 192, 216, 240, 288, 336, 360, 384, 432, 480, 504, 540, 576, 600, 648, 672, 720, 768, 792, 840, 864, 900, 936, 960, 1008, 1056, 1080, 1152, 1200, 1248, 1260, 1296, 1320, 1344, 1440, 1512, 1536, 1560, 1584, 1620, 1632, 1680, 1728, 1800, 1824, 1848, 1872, 1920, 1944, 1980, 2016, 2040, 2100, 2112, 2160, 2240

Offset: 1

Jaroslav Krizek, Jul 23 2014

nonn

If d are divisors of k then values of sequence A245212(k) are by bending moments in point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length k with bracket in point 0 by the schema: A245212(k) = (k * tau(k)) - Sum_{(d

Numbers k such that A038040(k) = k * tau(k) < A245211(k) = Sum_{(d

From Amiram Eldar, Jul 19 2024: (Start)

Numbers whose divisors have a mean abundancy index that is larger than 2.

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 24, 243, 2571, 25583, 254794, 2551559, 25514104, 255112225, ... . Apparently, the asymptotic density of this sequence exists and equals 0.02551... .

The least odd term in this sequence is a(276918705) = 10854718875. (End)

			Number 144 is in sequence because 144 * tau(144) = 2160  < Sum_{(d<144) | 144} (d * tau(d)) = 2226.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000005, A038040, A245211, A245212, A245213.

[n: n in [1..100000] | (2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])) lt 0]

f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2500], s[#] > 2 &]  (* Amiram Eldar, Jul 19 2024 *)

isok(n) = (n*numdiv(n) - sumdiv(n, d, (dMichel Marcus, Aug 06 2014

is(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i,1]; e=f[i,2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 2;} \\ Amiram Eldar, Jul 19 2024

A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 12, 5, 10, 1, 30, 1, 12, 11, 32, 1, 36, 1, 38, 13, 16, 1, 92, 7, 18, 19, 46, 1, 74, 1, 80, 17, 22, 15, 140, 1, 24, 19, 116, 1, 90, 1, 62, 51, 28, 1, 256, 9, 62, 23, 70, 1, 136, 19, 140, 25, 34, 1, 286, 1, 36, 61, 192, 21, 122, 1, 86, 29, 114

Offset: 1

Paolo P. Lava, Feb 19 2015

nonn

a(n) = 1 if n is prime.

			The aliquot parts of 8 are 1, 2, 4 and their sum is 7.
Now, let us calculate the aliquot parts of 1, 2 and 4:
1 => 0;  2 => 1;  4 => 1, 2.  Their sum is 0 + 1 + 1 + 2 = 4.
Let us calculate the aliquot parts of 1, 1, 2:
1 => 0;  1 = > 0; 2 => 1. Their sum is 1.
We have left 1: 1 => 0.
Finally, 7 + 4 + 1 = 12. Therefore a(8) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)
Jon Maiga, Computer-generated formulas for A255242, Sequence Machine.

Cf. A001047, A001065, A001787, A006516, A016127, A016130, A016135, A255243, A330575.

Sequences that appear in the convolution formulas: A000203, A001065, A051953, A062790, A067824, A074206, A157658, A174725, A174726, A191161, A211779, A245211, A323910, A253249.

with(numtheory): P:=proc(q) local a,b,c,k,n,t,v;
for n from 1 to q do b:=0; a:=sort([op(divisors(n))]); t:=nops(a)-1;
while add(a[k],k=1..t)>0 do b:=b+add(a[k],k=1..t); v:=[];
for k from 2 to t do c:=sort([op(divisors(a[k]))]); v:=[op(v),op(c[1..nops(c)-1])]; od;
a:=v; t:=nops(a); od; print(b); od; end: P(10^3);

f[s_] := Flatten[Most[Divisors[#]] & /@ s]; a[n_] := Total@Flatten[FixedPointList[ f, {n}]] - n; Array[a, 100] (* Amiram Eldar, Apr 06 2019 *)

ali(n) = setminus(divisors(n), Set(n));
a(n) = my(list = List(), v = [n]); while (#v, my(w = []); for (i=1, #v, my(s=ali(v[i])); for (j=1, #s, w = concat(w, s[j]); listput(list, s[j]));); v = w;); vecsum(Vec(list)); \\ Michel Marcus, Jul 15 2023

a(1) = 0.
a(2^k) = k*2^(k-1) = A001787(k), for k>=1.
a(n^k) = (n^k-2^k)/(n-2), for n odd prime and k>=1.
In particular:
a(3^k)  = A001047(k-1);
a(5^k)  = A016127(k-1);
a(7^k)  = A016130(k-1);
a(11^k) = A016135(k-1).
From Antti Karttunen, Nov 22 2024: (Start)
a(n) = A330575(n) - n.
Also, following formulas were conjectured by Sequence Machine:
a(n) = (A191161(n)-n)/2.
a(n) = Sum_{d|n} A001065(d)*A074206(n/d). [Compare to David A. Corneth's Apr 13 2020 formula for A330575]
a(n) = Sum_{d|n} A051953(d)*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A174726(n/d).
a(n) = Sum_{d|n} A062790(d)*A253249(n/d).
a(n) = Sum_{d|n} A157658(d)*A191161(n/d).
a(n) = Sum_{d|n} A174725(d)*A211779(n/d).
a(n) = Sum_{d|n} A245211(d)*A323910(n/d).
(End)

A245212 a(n) = n * tau(n) - Sum_{(d

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 13, 15, 20, 25, 21, 25, 25, 37, 43, 31, 33, 46, 37, 53, 63, 61, 45, 41, 64, 73, 74, 81, 57, 95, 61, 63, 103, 97, 115, 70, 73, 109, 123, 101, 81, 147, 85, 137, 166, 133, 93, 57, 132, 170, 163, 165, 105, 154, 187, 161, 183, 169, 117, 131, 121

Offset: 1

Jaroslav Krizek, Jul 23 2014

sign

If d are divisors of n then values of sequence a(n) are the bending moments at point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: a(n) = (n * tau(n)) - Sum_{(d

If a(n) = 0 then n must be > 10^7.

Conjecture: a(n) = sigma(n) iff n is a power of 2 (A000079).

Number n = 72 is the smallest number n such that a(n) < n (see A245213).

Number n = 144 is the smallest number n such that a(n) < 0 (see A245214).

			For n = 6 with divisors [1, 2, 3, 6] we have: a(6) = 6 * tau(6) - (3 * tau(3) + 2 * tau(2) + 1 * tau(1)) = 6*4 - (3*2+2*2+1*1) = 13.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000005, A100484, A245211, A245213, A245214.

[(2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])): n in [1..1000]];

a(n) = sumdiv(n, d, (-1)^(dJens Kruse Andersen, Aug 13 2014

a(n) = A038040(n) - A245211(n).

a(n) = 2 * A038040(n) - A060640(n) = 2 * (n * tau(n)) - Sum_{d | n} (d * tau(d)).

A245213 Numbers n such that A245212(n) < n.

Original entry on oeis.org

72, 96, 120, 144, 180, 192, 216, 240, 288, 336, 360, 384, 432, 480, 504, 528, 540, 576, 600, 624, 648, 672, 720, 756, 768, 792, 840, 864, 900, 936, 960, 972, 1008, 1056, 1080, 1120, 1152, 1176, 1200, 1224, 1248, 1260, 1280, 1296, 1320, 1344, 1368, 1440, 1512

Offset: 1

Jaroslav Krizek, Jul 23 2014

nonn

Numbers n such that A245212(n) = (n * tau(n)) - Sum_((d

If d are divisors of n then values of sequence A245212(n) are the bending moments at point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: A245212(n) = (n * tau(n)) - Sum_((d

			Number 72 is in sequence because A245212(72) = 62 < 72.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000005, A100484, A245211, A245212, A245214.

[n: n in [1..100000] | (2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])) lt n]

Showing 1-4 of 4 results.

cp's OEIS Frontend

A245214 Numbers k such that A245212(k) < 0.

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A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.

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A245212 a(n) = n * tau(n) - Sum_{(d

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A245213 Numbers n such that A245212(n) < n.

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