A330575 -id:A330575 - cp's OEIS frontend

A333953 Recursively superabundant numbers: numbers m such that A330575(m)/m > A330575(k)/k for all k < m.

1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 240, 288, 360, 480, 576, 720, 1152, 1440, 2160, 2880, 4320, 5760, 8640, 11520, 17280, 25920, 30240, 34560, 51840, 60480, 69120, 103680, 120960, 172800, 181440, 207360, 241920, 345600, 362880, 414720, 483840, 725760

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

Fink (2019) defined this sequence. He asked whether 720 is the largest term that is also superabundant number (A004394).
He noted that up to 10^6 all the recursively superabundant numbers are also recursively highly composite numbers (A333952), except for 181440 (the next term which is not recursively highly composite is 2177280). He asked whether there are a finite number of numbers that are both recursively highly composite and recursively superabundant (in analogy to A166981).
From David A. Corneth, Apr 13 2020: (Start)
The 76 terms in the b-file are products of primorials (Cf. A025487) and 7-smooth numbers  (Cf. A002473). All terms are in A025487.
Proof: As A330575(n) = Sum_{d|n} A074206(d) * n/d we have A330575(n) / n = Sum_{d|n} A074206(d)/d which is maximal for some prime signature when n is a product of primorials.
Assuming terms below 10^17 are 13-smooth gives the 213 11-smooth numbers in the Corneth a-file. (End)

David A. Corneth, Table of n, a(n) for n = 1..213 (first 76 terms from Amiram Eldar)
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. See section 5.

Cf. A002473, A004394, A166981, A025487, A330575, A333952.

s[1] = 1; s[n_] := s[n] = n + DivisorSum[n, s[#] &, # < n &]; seq={}; rm = 0; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A378217 Dirichlet inverse of A330575.

Original entry on oeis.org

1, -3, -4, 1, -6, 10, -8, 1, 2, 16, -12, -2, -14, 22, 22, 1, -18, -2, -20, -4, 30, 34, -24, -2, 4, 40, 2, -6, -30, -52, -32, 1, 46, 52, 46, -2, -38, 58, 54, -4, -42, -72, -44, -10, -8, 70, -48, -2, 6, -4, 70, -12, -54, -2, 70, -6, 78, 88, -60, 8, -62, 94, -12, 1, 82, -112, -68, -16, 94, -116, -72, -2, -74, 112, -8, -18, 94

Offset: 1

Antti Karttunen, Nov 22 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A330575.

memoA330575 = Map();
A330575(n) = if(1==n,1,my(v); if(mapisdefined(memoA330575,n,&v), v, v = n + sumdiv(n,d,if(dA330575(d),0)); mapput(memoA330575,n,v); (v)));
memoA378217 = Map();
A378217(n) = if(1==n,1,my(v); if(mapisdefined(memoA378217,n,&v), v, v = -sumdiv(n,d,if(dA330575(n/d)*A378217(d),0)); mapput(memoA378217,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA330575(n/d) * a(d).

A333962 a(n) = A330575(A025487(n)).

Original entry on oeis.org

1, 3, 8, 14, 20, 42, 48, 116, 104, 112, 176, 304, 346, 256, 524, 768, 1044, 576, 1472, 1584, 1888, 1056, 2088, 2960, 1280, 3968, 5208, 4544, 3858, 6216, 8032, 2816, 10368, 15960, 10752, 12612, 17712, 10532, 21088, 6144, 22416, 26432, 19128, 24096, 46512, 25088, 38400

Offset: 1

David A. Corneth, Apr 15 2020

nonn

These numbers provide candidate record values of A330575(k)/k as k must be a product of primorials for A330575(k)/k to be a record.

Amiram Eldar, Table of n, a(n) for n = 1..750

Cf. A025487, A330575, A333953.

A333954 Numbers k such that A330575(k) = A330575(k+1).

Original entry on oeis.org

14, 16101, 72926, 97101, 2872701, 7610324

Offset: 1

Amiram Eldar, Apr 11 2020

a(7) > 6*10*8.

			14 is a term since A330575(14) = A330575(15) = 26.

Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019.

Cf. A330575.

Similar sequences: A002961, A064115, A064125, A293183, A306985.

s[1] = 1; s[n_] := s[n] = n + DivisorSum[n, s[#] &, # < n &]; seq = {}; s1 = s[1]; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq

A323910 Dirichlet inverse of the deficiency of n, A033879.

Original entry on oeis.org

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323910, Sequence Machine.

Cf. A033879, A323911, A323912, A359549 (parity of terms).

Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).
From Antti Karttunen, Nov 14 2024: (Start)
Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
(End)

A191161 Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

Original entry on oeis.org

1, 4, 5, 12, 7, 22, 9, 32, 19, 30, 13, 72, 15, 38, 37, 80, 19, 90, 21, 96, 47, 54, 25, 208, 39, 62, 65, 120, 31, 178, 33, 192, 67, 78, 65, 316, 39, 86, 77, 272, 43, 222, 45, 168, 147, 102, 49, 560, 67, 174, 97, 192, 55

Offset: 1

Alonso del Arte, May 26 2011

In wanting to ensure the definition was not arbitrary, I initially thought that 1s had to stop the recursion. But as T. D. Noe showed me, this doesn't have to be the case: the 1s can be included in the recursion.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A191161, Sequence Machine.

Cf. A000203, A191150, A202687, A255242, A378211 (Dirichlet inverse).

Sequences that appear in the convolution formulas: A000010, A000203, A007429, A038040, A060640, A067824, A074206, A174725, A253249, A323910, A323912, A330575.

hsTD[n_] := hsTD[n] = Module[{d = Divisors[n]}, Total[d] + Total[hsTD /@ Most[d]]]; Table[hsTD[n], {n, 100}] (* From T. D. Noe *)

a(n)=sumdiv(n,d,if(dCharles R Greathouse IV, Dec 20 2011

a(n) = sigma(n) + sum_{d | n, d < n} a(d). - Charles R Greathouse IV, Dec 20 2011
From Antti Karttunen, Nov 22 2024: (Start)
Following formulas were conjectured by Sequence Machine:
For n > 1, a(n) = A191150(n) + A074206(n).
a(n) = A330575(n) + A255242(n) = 2*A255242(n) + n = 2*A330575(n) - n.
a(n) = Sum_{d|n} A330575(d).
a(n) = Sum_{d|n} d*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A074206(n/d).
a(n) = Sum_{d|n} A007429(d)*A174725(n/d).
a(n) = Sum_{d|n} A000010(d)*A253249(n/d).
a(n) = Sum_{d|n} A038040(d)*A323912(n/d).
a(n) = Sum_{d|n} A060640(d)*A323910(n/d).
(End)

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)

A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

Original entry on oeis.org

1, 0, -1, 0, -3, 2, -5, 0, -2, 2, -9, 4, -11, 2, 5, 0, -15, 2, -17, 4, 7, 2, -21, 8, -6, 2, -4, 4, -27, -2, -29, 0, 11, 2, 17, 8, -35, 2, 13, 8, -39, -6, -41, 4, 8, 2, -45, 16, -10, -2, 17, 4, -51, 0, 29, 8, 19, 2, -57, 4, -59, 2, 12, 0, 35, -14, -65, 4, 23, -10, -69, 24, -71, 2, 4, 4, 47, -18, -77, 16, -8, 2, -81, -4, 47, 2, 29, 8, -87, 4

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323912, Sequence Machine.
Wikipedia, Dirichlet convolution.

Cf. A083254, A323913.

Sequences that appear in the convolution formulas: A002033, A023900, A046692, A055615, A067824, A074206, A101035, A130054, A174725, A191161, A253249, A323910 (Möbius transform), A328722, A330575.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA083254(n) = (2*eulerphi(n)-n);
v323912 = DirInverse(vector(up_to,n,A083254(n)));
A323912(n) = v323912[n];

A083254(n) = (2*eulerphi(n)-n);
memoA323912 = Map();
A323912(n) = if(1==n,1,my(v); if(mapisdefined(memoA323912,n,&v), v, v = -sumdiv(n,d,if(dA083254(n/d)*A323912(d),0)); mapput(memoA323912,n,v); (v))); \\ Antti Karttunen, Nov 22 2024

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083254(n/d) * a(d).
From Antti Karttunen, Nov 22 2024: (Start)
Following convolution formulas were conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly. The first one is certainly true, because A083254 is Möbius transform of A033879:
a(n) = Sum_{d|n} A323910(d).
a(n) = Sum_{d|n} A023900(d)*A074206(n/d) = Sum_{d|n} A002033(d-1)*A023900(n/d).
a(n) = Sum_{d|n} A055615(d)*A067824(n/d)
a(n) = Sum_{d|n} A046692(d)*A253249(n/d)
a(n) = Sum_{d|n} A130054(d)*A174725(n/d)
a(n) = Sum_{d|n} A101035(d)*A330575(n/d)
a(n) = Sum_{d|n} A191161(d)*A328722(n/d)
(End)

A345139 a(1) = 1; a(n) = a(n-1) + Sum_{d|n, d < n} a(d).

Original entry on oeis.org

1, 2, 3, 6, 7, 13, 14, 23, 27, 37, 38, 63, 64, 81, 92, 124, 125, 171, 172, 225, 243, 284, 285, 396, 404, 471, 502, 606, 607, 762, 763, 919, 961, 1089, 1111, 1397, 1398, 1573, 1641, 1942, 1943, 2300, 2301, 2632, 2762, 3050, 3051, 3682, 3697, 4148, 4277, 4821, 4822, 5541, 5587

Offset: 1

Ilya Gutkovskiy, Jun 09 2021

nonn

Cf. A074206, A084978, A330575.

a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[If[d < n, a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
nmax = 55; A[] = 0; Do[A[x] = (1/(1 - x)) (x + Sum[A[x^k], {k, 2, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x + A(x^2) + A(x^3) + A(x^4) + ...).

A345140 a(1) = 1; a(n+1) = n + Sum_{d|n} a(d).

Original entry on oeis.org

1, 2, 5, 9, 16, 22, 36, 44, 64, 79, 108, 120, 171, 185, 238, 275, 347, 365, 477, 497, 624, 687, 820, 844, 1071, 1113, 1313, 1410, 1671, 1701, 2094, 2126, 2489, 2636, 3020, 3108, 3732, 3770, 4288, 4504, 5192, 5234, 6151, 6195, 7046, 7415, 8284, 8332, 9702, 9788, 11007, 11411, 12759, 12813, 14639

Offset: 1

Ilya Gutkovskiy, Jun 09 2021

nonn

Cf. A003238, A068336, A330575.

a[1] = 1; a[n_] := a[n] = n - 1 + Sum[a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 55}]
nmax = 55; A[] = 0; Do[A[x] = x (1 + x/(1 - x)^2 + Sum[A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * (1 + x / (1 - x)^2 + A(x) + A(x^2) + A(x^3) + ...).

cp's OEIS Frontend

A333953 Recursively superabundant numbers: numbers m such that A330575(m)/m > A330575(k)/k for all k < m.

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A378217 Dirichlet inverse of A330575.

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A333962 a(n) = A330575(A025487(n)).

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A333954 Numbers k such that A330575(k) = A330575(k+1).

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A323910 Dirichlet inverse of the deficiency of n, A033879.

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A191161 Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

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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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A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

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