A064945 -id:A064945 - cp's OEIS frontend

A386911 Numbers k such that A064945(k) = A064945(k+1).

14, 14229, 194931, 8897930, 278110142

Offset: 1

Michel Marcus, Aug 07 2025

No other terms up to 10^9.

A064945[n_] := Total[#*Range[Length[#], 1, -1]] & [Divisors[n]];q[n_]:=A064945[n]==A064945[n+1];Select[Range[200000],q[#]&] (* James C. McMahon, Aug 07 2025 *)

A064944 a(n) = Sum_{i|n, j|n, j >= i} j.

Original entry on oeis.org

1, 5, 7, 17, 11, 38, 15, 49, 34, 60, 23, 132, 27, 82, 82, 129, 35, 191, 39, 207, 112, 126, 47, 384, 86, 148, 142, 283, 59, 469, 63, 321, 172, 192, 172, 666, 75, 214, 202, 597, 83, 640, 87, 435, 403, 258, 95, 1016, 162, 485, 262, 511, 107, 812, 264, 813, 292, 324

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

			a(6) = max(1,1)+max(1,2)+max(1,3)+max(1,6)+max(2,2)+max(2,3)+max(2,6)+max(3,3)+max(3,6)+max(6,6)=38, or a(6) = dot_product(1,2,3,4)*(1,2,3,6)=1*1+2*2+3*3+4*6=38.

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A060640, A064945, A064946, A064947, A064948, A064949.

Cf. A027750, A135539, A000005, A000203, A064840, A337360, A337297.

a064944 = sum . zipWith (*) [1..] . a027750_row'
-- Reinhard Zumkeller, Jul 14 2015

with(numtheory): seq(add(i*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

A064944[n_] := #.Range[Length[#]] & [Divisors[n]];
Array[A064944, 100] (* Paolo Xausa, Aug 07 2025 *)

a(n) = my(d=divisors(n)); sum(i=1, length(d), i*d[i]); \\ Harry J. Smith, Sep 30 2009

from sympy import divisors
def A064944(n): return sum(a*b for a, b in enumerate(divisors(n),1)) # Chai Wah Wu, Aug 07 2025

a(n) = Sum_{i=1..tau(n)} i*d_i, where {d_i}, i=1..tau(n) is the increasing sequence of divisors of n.
a(n) = Sum_{i=1..A000005(n)} i*A027750(n, i). - Michel Marcus, Jun 10 2015
From Ridouane Oudra, Aug 01 2025: (Start)
a(n) = Sum_{d|n} (n/d)*A135539(n,d).
a(n) = A064946(n) + A000203(n).
a(n) = (A064948(n) + A000203(n))/2.
a(n) = A337360(n) - A064945(n).
a(n) = A064948(n) - A064946(n).
a(n) = A064840(n) - A064947(n). (End)

A064949 a(n) = Sum_{i|n, j|n} min(i,j).

Original entry on oeis.org

1, 5, 6, 15, 8, 32, 10, 37, 23, 42, 14, 100, 16, 52, 52, 83, 20, 125, 22, 132, 64, 72, 26, 252, 45, 82, 76, 162, 32, 286, 34, 177, 88, 102, 88, 397, 40, 112, 100, 336, 44, 352, 46, 222, 208, 132, 50, 572, 75, 239, 124, 252, 56, 416, 120, 414, 136, 162, 62, 916, 64

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

			a(6) = dot_product(7,5,3,1)*(1,2,3,6) = 7*1 + 5*2 + 3*3 + 1*6 = 32.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Harry J. Smith)

Cf. A000005, A000203, A064944, A064945, A135539, A064840.

with(numtheory): seq(add((2*tau(n)-2*i+1)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

Array[Function[{t, d}, Total@ MapIndexed[#1 (2 t - 2 First[#2] + 1) &, d]] @@ {DivisorSigma[0, #], Divisors[#]} &, 61] (* Michael De Vlieger, Oct 25 2021 *)

a(n) = { my(d=divisors(n), t=length(d)); sum(i=1, t, (2*t - 2*i + 1)*d[i]) } \\ Harry J. Smith, Oct 01 2009

A064949(n) = { my(i=0, u=numdiv(n)); sumdiv(n,d,i++; (((2*u)-(2*i))+1)*d); }; \\ Antti Karttunen, Nov 14 2021

a(n) = Sum_{i=1..tau(n)} (2*tau(n)-2*i+1)*d_i, where {d_i}, i=1..tau(n), is increasing sequence of divisors of n.
a(n) = Sum_{i=1..n} A135539(n,i)^2. - Ridouane Oudra, Oct 25 2021
a(n) = A000203(n) * (2*A000005(n)+1) - 2*A064944(n). - Amiram Eldar, Jan 13 2025
From Ridouane Oudra, Aug 13 2025: (Start)
a(n) = A064945(n) + A064947(n).
a(n) = 2*A064947(n) + A000203(n).
a(n) = 2*A064945(n) - A000203(n).
a(n) = 2*A064840(n) - A064948(n). (End)

A064946 a(n) = Sum_{i|n, j|n, j>i} j.

Original entry on oeis.org

0, 2, 3, 10, 5, 26, 7, 34, 21, 42, 11, 104, 13, 58, 58, 98, 17, 152, 19, 165, 80, 90, 23, 324, 55, 106, 102, 227, 29, 397, 31, 258, 124, 138, 124, 575, 37, 154, 146, 507, 41, 544, 43, 351, 325, 186, 47, 892, 105, 392, 190, 413, 53, 692, 192, 693, 212, 234, 59

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

			a(6) = dot_product(0,1,2,3)*(1,2,3,6) = 0*1 + 1*2 + 2*3 + 3*6 = 26.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A064944, A064945, A064947, A064948, A064949.

Cf. A000005, A000203, A064840, A337297, A027750.

with(numtheory): seq(add((i-1)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

A064946[n_] := #.Range[Length[#]] & [Rest[Divisors[n]]];
Array[A064946, 100] (* Paolo Xausa, Aug 14 2025 *)

a(n) = my(d=divisors(n)); sum(i=2, length(d), (i - 1)*d[i]); \\ Harry J. Smith, Oct 01 2009

a(n) = Sum_{i=1..tau(n)} (i-1)*d_i, where {d_i}, i=1..tau(n), is the increasing sequence of the divisors of n.
a(n) = A064944(n) - A000203(n). - Amiram Eldar, Dec 23 2024
From Ridouane Oudra, Aug 06 2025: (Start)
a(n) = A064948(n) - A064944(n).
a(n) = A064840(n) - A064945(n).
a(n) = A337297(n) - A064947(n).
a(n) = (A064948(n) - A000203(n))/2. (End)

A064947 a(n) = Sum_{i|n, j|n, j>i} i.

Original entry on oeis.org

0, 1, 1, 4, 1, 10, 1, 11, 5, 12, 1, 36, 1, 14, 14, 26, 1, 43, 1, 45, 16, 18, 1, 96, 7, 20, 18, 53, 1, 107, 1, 57, 20, 24, 20, 153, 1, 26, 22, 123, 1, 128, 1, 69, 65, 30, 1, 224, 9, 73, 26, 77, 1, 148, 24, 147, 28, 36, 1, 374, 1, 38, 77, 120, 26, 168, 1, 93, 32, 165, 1, 411, 1, 44

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

For given n, iterate a(n); a(a(n)); a(a(a(n))); ... Does this iterative process always lead to a(a(...(a(n))...)) = 1? - Ctibor O. Zizka, Apr 17 2008

No. For example, a(4) = 4, a(14) = 14, and a(99) = 99. - Jason Yuen, Jan 07 2025

			a(6) = dot_product(3,2,1,0)*(1,2,3,6) = 3*1 + 2*2 + 1*3 + 0*6 = 10.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A064840, A337297, A027750.

with(numtheory): seq(add((tau(n)-i)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

Table[t = DivisorSigma[0, n]; Total@ MapIndexed[(t - First[#2])*#1 &, Divisors[n]], {n, 120}] (* Michael De Vlieger, Jan 07 2025 *)

a(n) = my(d=divisors(n), t=length(d)); sum(i=1, t - 1, (t - i)*d[i]); \\ Harry J. Smith, Oct 01 2009

a(n) = Sum_{i=1..tau(n)} (tau(n)-i)*d_i, where {d_i}, i=1..tau(n), is increasing sequence of divisors of n.
From Ridouane Oudra, Aug 07 2025: (Start)
a(n) = A064945(n) - A000203(n).
a(n) = A064840(n) - A064944(n).
a(n) = A064949(n) - A064945(n).
a(n) = A337297(n) - A064946(n).
a(n) = (A064949(n) - A000203(n))/2. (End)

A197410 Product of cumulative sums of divisors of n.

Original entry on oeis.org

1, 3, 4, 21, 6, 216, 8, 315, 52, 432, 12, 80640, 14, 720, 864, 9765, 18, 176904, 20, 232848, 1408, 1512, 24, 149299200, 186, 2016, 2080, 460992, 30, 274827168, 32, 615195, 2880, 3240, 3744, 13333320000, 38, 3960, 3808, 680400000, 42, 702079488, 44, 1270080

Offset: 1

Franklin T. Adams-Watters, Oct 14 2011

When n is prime, a(n) = n + 1.

			a(4) = 21 because the divisors of 4 are 1, 2 and 4, their cumulative sums are 1, 3 and 7, and 1 * 3 * 7 = 21.
a(5) = 6 because the divisors of 5 are 1 and 5, their cumulative sums are 1 and 6, and 1 * 6 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000203, A007955, A064945.

Cf. A027750.

a197410 = product . scanl1 (+) . a027750_row
-- Reinhard Zumkeller, Jan 26 2013

Table[Times@@Table[Plus@@Take[Divisors[n], k], {k, DivisorSigma[0, n]}], {n, 44}] (* Alonso del Arte, Oct 14 2011 *)
Table[Times@@Accumulate[Divisors[n]],{n,50}] (* Harvey P. Dale, Aug 15 2013 *)

a(n)=local(ds,sd);ds=divisors(n);prod(k=1,#ds,sd+=ds[k])

A220849 a(n) = Product_{d|n} Product_{d_x|n , d_x <= d} d_x.

Original entry on oeis.org

1, 2, 3, 16, 5, 432, 7, 1024, 81, 2000, 11, 71663616, 13, 5488, 10125, 1048576, 17, 816293376, 19, 2048000000, 27783, 21296, 23, 219122084616339456, 625, 35152, 59049, 15420489728, 29, 2550916800000000000, 31, 34359738368, 107811, 78608, 214375

Offset: 1

Jaroslav Krizek, Dec 22 2012

nonn

Conjecture: sequence is injective (all terms of this sequence occur only once).

The subsequence of fixed points consists of 1 together with the primes (A008578). - Bernard Schott, Oct 26 2021

			The divisors of 6 are 1, 2, 3, 6. a(n) = 1*(1*2)*(1*2*3)*(1*2*3*6) = 1*2*6*36 = 432.

Seiichi Manyama, Table of n, a(n) for n = 1..2000 (terms 1..100 from Jaroslav Krizek)

Cf. A000292, A006881, A007955, A008578 (fixed points), A064945, A266265.

a[n_] := Module[{d = Divisors[n], nd}, nd = Length[d]; Product[d[[i]]^(nd - i + 1), {i, 1, nd}]]; Array[a, 35] (* Amiram Eldar, Oct 23 2021 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, vecprod(select(x->(x<=d[k]), d))); \\ Michel Marcus, Oct 23 2021

a(p) = p for prime p.
From Bernard Schott, Oct 29 2021: (Start)
For p prime and k >= 0, a(p^k) = p^A000292(k).
For n = p*q, p < q primes (A006881), a(n) = p*n^3. (End)

a(24) corrected by Seiichi Manyama, Oct 23 2021

A246916 Sum of the cumulative sums of all the permutations of divisors of number n.

Original entry on oeis.org

1, 9, 12, 84, 18, 720, 24, 900, 156, 1080, 36, 70560, 42, 1440, 1440, 11160, 54, 98280, 60, 105840, 1920, 2160, 72, 10886400, 372, 2520, 2400, 141120, 90, 13063680, 96, 158760, 2880, 3240, 2880, 165110400, 114, 3600, 3360, 16329600, 126, 17418240, 132, 211680

Offset: 1

Jaroslav Krizek, Sep 12 2014

nonn

For number n there are A130674(n) = tau(n)! = A000005(n)! permutations of divisors of number n and the same number of their cumulative sums. This sequence is sequence of sums of these sums.

Sequences A064945 and A064944 are sequences of minimal and maximal values of cumulative sums of all the permutations of divisors of number n.

			For n = 4; there are tau(4)! = 6 permutations of divisors of number 4: (1, 2, 4); (1, 4, 2); (2, 1, 4); (2, 4, 1); (4, 1, 2); (4, 2, 1). Sum of their cumulative sums = 11 + 13 + 12 + 15 + 16 + 17 = 84.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A001710, A064945, A064944, A130674.

[SumOfDivisors(n)*(Order(AlternatingGroup(NumberOfDivisors(n)+1))): n in [1..100]];

A246916[n_] := DivisorSigma[1, n]*(DivisorSigma[0, n] + 1)!/2;
Array[A246916, 50] (* Paolo Xausa, Aug 08 2025 *)

A001710(n) = if( n<2, n>=0, n!/2);
A246916(n) = (sigma(n) * A001710(numdiv(n) + 1)); \\ Antti Karttunen, Sep 10 2017

a(n) = A130674(n) * (A064945(n) + A064944(n)) / 2 = (tau(n))! * ((Sum_{i=1..tau(n)} (tau(n) - i + 1)*d_i) + (Sum_{i=1..tau(n)} i*d_i)) / 2; where {d_i}, i = 1..tau(n) is the increasing sequence of divisors of n.

a(n) = sigma(n) * A001710(tau(n) + 1) = A000203(n) * A001710(A000005(n)+1).

cp's OEIS Frontend

A386911 Numbers k such that A064945(k) = A064945(k+1).

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A064944 a(n) = Sum_{i|n, j|n, j >= i} j.

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A064949 a(n) = Sum_{i|n, j|n} min(i,j).

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A064946 a(n) = Sum_{i|n, j|n, j>i} j.

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A064947 a(n) = Sum_{i|n, j|n, j>i} i.

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A197410 Product of cumulative sums of divisors of n.

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A220849 a(n) = Product_{d|n} Product_{d_x|n , d_x <= d} d_x.

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