A046951 -id:A046951 - cp's OEIS frontend

A351602 a(n) = n^4 * Sum_{d^2|n} 1 / d^4.

1, 16, 81, 272, 625, 1296, 2401, 4352, 6642, 10000, 14641, 22032, 28561, 38416, 50625, 69888, 83521, 106272, 130321, 170000, 194481, 234256, 279841, 352512, 391250, 456976, 538002, 653072, 707281, 810000, 923521, 1118208, 1185921, 1336336, 1500625, 1806624, 1874161, 2085136

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), this sequence (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013664.

f[p_, e_] := p^4*(p^(4*e) - p^(4*Floor[(e - 1)/2]))/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^4*sumdiv(n, d, if (issquare(d), 1/d^2)); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^4*(p^(4*e) - p^(4*floor((e-1)/2)))/(p^4 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(6)/5 = Pi^6/4725 = 0.203468... . - Amiram Eldar, Nov 13 2022

A351603 a(n) = n^5 * Sum_{d^2|n} 1 / d^5.

Original entry on oeis.org

1, 32, 243, 1056, 3125, 7776, 16807, 33792, 59292, 100000, 161051, 256608, 371293, 537824, 759375, 1082368, 1419857, 1897344, 2476099, 3300000, 4084101, 5153632, 6436343, 8211456, 9768750, 11881376, 14407956, 17748192, 20511149, 24300000, 28629151, 34635776, 39135393

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), this sequence (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013665.

f[p_, e_] := p^5*(p^(5*e) - p^(5*Floor[(e - 1)/2]))/(p^5 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^5*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^5))); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^5*(p^(5*e) - p^(5*floor((e-1)/2)))/(p^5 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^6, where c = zeta(7)/6 = 0.168058... . - Amiram Eldar, Nov 13 2022

A351604 a(n) = n^6 * Sum_{d^2|n} 1 / d^6.

Original entry on oeis.org

1, 64, 729, 4160, 15625, 46656, 117649, 266240, 532170, 1000000, 1771561, 3032640, 4826809, 7529536, 11390625, 17043456, 24137569, 34058880, 47045881, 65000000, 85766121, 113379904, 148035889, 194088960, 244156250, 308915776, 387951930, 489419840, 594823321, 729000000

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), this sequence (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013666.

f[p_, e_] := p^6*(p^(6*e) - p^(6*Floor[(e - 1)/2]))/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^6*sumdiv(n, d, if (issquare(d), 1/d^3)); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^6*(p^(6*e) - p^(6*floor((e-1)/2)))/(p^6 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(8)/7 = Pi^8/66150 = 0.143439... . - Amiram Eldar, Nov 13 2022

A351605 a(n) = n^7 * Sum_{d^2|n} 1 / d^7.

Original entry on oeis.org

1, 128, 2187, 16512, 78125, 279936, 823543, 2113536, 4785156, 10000000, 19487171, 36111744, 62748517, 105413504, 170859375, 270548992, 410338673, 612499968, 893871739, 1290000000, 1801088541, 2494357888, 3404825447, 4622303232, 6103593750, 8031810176, 10465136172

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), this sequence (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013667.

f[p_, e_] := p^7*(p^(7*e) - p^(7*Floor[(e - 1)/2]))/(p^7 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^7*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^7))); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^7*(p^(7*e) - p^(7*floor((e-1)/2)))/(p^7 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^8, where c = zeta(9)/8 = 0.125251... . - Amiram Eldar, Nov 13 2022

A351606 a(n) = n^8 * Sum_{d^2|n} 1 / d^8.

Original entry on oeis.org

1, 256, 6561, 65792, 390625, 1679616, 5764801, 16842752, 43053282, 100000000, 214358881, 431661312, 815730721, 1475789056, 2562890625, 4311810048, 6975757441, 11021640192, 16983563041, 25700000000, 37822859361, 54875873536, 78310985281, 110505295872, 152588281250

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), this sequence (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013668.

f[p_, e_] := p^8*(p^(8*e) - p^(8*Floor[(e - 1)/2]))/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^8*sumdiv(n, d, if (issquare(d), 1/d^4)); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^8*(p^(8*e) - p^(8*floor((e-1)/2)))/(p^8 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(10)/9 = Pi^10/841995 = 0.1112216... . - Amiram Eldar, Nov 13 2022

A351607 a(n) = n^9 * Sum_{d^2|n} 1 / d^9.

Original entry on oeis.org

1, 512, 19683, 262656, 1953125, 10077696, 40353607, 134479872, 387440172, 1000000000, 2357947691, 5169858048, 10604499373, 20661046784, 38443359375, 68853956608, 118587876497, 198369368064, 322687697779, 513000000000, 794280046581, 1207269217792, 1801152661463

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), this sequence (k=9), A351608 (k=10).

Cf. A013669.

f[p_, e_] := p^9*(p^(9*e) - p^(9*Floor[(e - 1)/2]))/(p^9 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^9*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^9))); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^9*(p^(9*e) - p^(9*floor((e-1)/2)))/(p^9 - 1). - Sebastian Karlsson, Mar 03 2022

Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(11)/10 = 0.100049... . - Amiram Eldar, Nov 13 2022

A351608 a(n) = n^10 * Sum_{d^2|n} 1 / d^10.

Original entry on oeis.org

1, 1024, 59049, 1049600, 9765625, 60466176, 282475249, 1074790400, 3486843450, 10000000000, 25937424601, 61977830400, 137858491849, 289254654976, 576650390625, 1100586418176, 2015993900449, 3570527692800, 6131066257801, 10250000000000, 16679880978201

Offset: 1

Wesley Ivan Hurt, Feb 14 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), this sequence (k=10).

Cf. A013670.

f[p_, e_] := p^10*(p^(10*e) - p^(10*Floor[(e - 1)/2]))/(p^10 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 13 2022 *)

a(n) = n^10*sumdiv(n, d, if (issquare(d), 1/d^5)); \\ Michel Marcus, Feb 15 2022

Multiplicative with a(p^e) = p^10*(p^(10*e) - p^(10*floor((e-1)/2)))/(p^10 - 1). - Sebastian Karlsson, Mar 03 2022

Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(12)/11 = 691*Pi^12/7023641625 = 0.090931... . - Amiram Eldar, Nov 13 2022

A056626 Number of non-unitary square divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Labos Elemer, Aug 08 2000

nonn

			n = p^u prime power has u+1 square divisors of which 2 (i.e., 1 and n) are unitary but u-1 are not unitary, so a(p^u) = u - 1. E.g., n = 4^4 = 256, has 5 square divisors {1, 4, 16, 64, 256} of which {4, 16, 64} are not unitary, so a(256)=3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000188, A008833, A034444, A046951, A055229, A056624, A162641.

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, ! CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n])- Times @@ f2 @@@ fct; Array[a, 100] (* Amiram Eldar, Sep 26 2022 *)

a(n) = {my(f = factor(n), r=0, m = 0); prod(i=1,#f~,f[i,2]>>1 + 1) - 2^(omega(f) - omega(core(f)))} \\ David A. Corneth, Jul 28 2017

a(n) = sumdiv(n, d, if(gcd(d, n/d)!=1, issquare(d))); \\ Michel Marcus, Jul 29 2017

from math import prod
from sympy import factorint
def A056626(n):
    f = factorint(n).values()
    return prod((e>>1)+1 for e in f)-(1<Chai Wah Wu, Aug 04 2024

a(n) = A046951(n) - 2^r(n), where r(n) is the number of distinct prime factors of the largest unitary square divisor of n. [Corrected by Amiram Eldar, Aug 03 2024]
a(n) = A046951(n) - 2^(A162641(n)). -  David A. Corneth, Jul 28 2017
From Amiram Eldar, Sep 26 2022: (Start)
a(n) = A046951(n) - A056624(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)*(1 - 1/zeta(3)) = 0.27650128922802056073... . (End)

a(32) and a(96) corrected by Michael De Vlieger, Jul 29 2017

A140773 Consider the products of all pairs of (not necessarily distinct) positive divisors of n. a(n) is the number of these products that divide n. a(n) also is the number of the products that are divisible by n.

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 2, 6, 4, 5, 2, 10, 2, 5, 5, 9, 2, 10, 2, 10, 5, 5, 2, 16, 4, 5, 6, 10, 2, 14, 2, 12, 5, 5, 5, 20, 2, 5, 5, 16, 2, 14, 2, 10, 10, 5, 2, 24, 4, 10, 5, 10, 2, 16, 5, 16, 5, 5, 2, 28, 2, 5, 10, 16, 5, 14, 2, 10, 5, 14, 2, 32, 2, 5, 10, 10, 5, 14, 2, 24, 9, 5, 2, 28, 5, 5, 5, 16, 2, 28, 5

Offset: 1

Leroy Quet, May 29 2008

nonn

Number of 3D grids of n congruent boxes with two different edge lengths, in a box, modulo rotation (cf. A034836 for cubes instead of boxes and A007425 for boxes with three different edge lengths; cf. A000005 for the 2D case). - Manfred Boergens, Feb 25 2021

Number of distinct faces obtainable by arranging n unit cubes into a cuboid. - Chris W. Milson, Mar 14 2021

			The divisors of 20 are 1,2,4,5,10,20. There are 10 pairs of divisors whose product divides 20: 1*1=1, 1*2=2, 1*4=4, 1*5=5, 1*10=10, 1*20=20, 2*2=4, 2*5=10, 2*10=20, 4*5 = 20. Likewise, there are 10 products that are divisible by 20: 4*5=20, 2*10=20, 4*10=40, 10*10=100, 1*20=20, 2*20=40, 4*20=80, 5*20=100, 10*20=200, 20*20=400. So a(20) = 10.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Chris W. Milson, Constructing Cuboids
Chris W. Milson, A faster algorithm for a(n)

Cf. A140774.
Cf. A000005, A034836, A038548, A007425, A046951.
Cf. A369255 (parity), A369256 (positions of odd terms), A378213 (Dirichlet inverse).

(* first do *) Needs["Combinatorica`"] (* then *) f[n_] := Count[ n/Times @@@ Union[Sort /@ Tuples[Divisors@ n, 2]], Integer]; Array[f, 91] (* _Robert G. Wilson v, May 31 2008 *)
d=Divisors[n]; r=Length[d]; Sum[Ceiling[Length[Divisors[d[[j]]]]/2],{j,r}] (* Manfred Boergens, Feb 25 2021 *)

\\ Two implementations, after the two different interpretations given by the author of the sequence:
A140773v1(n) = { my(ds = divisors(n),s=0); for(i=1,#ds,for(j=i,#ds,if(!(n%(ds[i]*ds[j])),s=s+1))); s; }
A140773v2(n) = { my(ds = divisors(n),s=0); for(i=1,#ds,for(j=i,#ds,if(!((ds[i]*ds[j])%n),s=s+1))); s; }
\\ Antti Karttunen, May 19 2017

Python
```
# See C. W. Milson link.
```

a(n) = Sum_{m|n} A038548(m) = Sum_{m|n} ceiling(d(m)/2), where d(m) = number of divisors of m (A000005). - Manfred Boergens, Feb 25 2021
a(n) = Sum_{d|n} A135539(d,n/d). - Ridouane Oudra, Jul 10 2021
a(n) = (A007425(n) + A046951(n))/2. - Ridouane Oudra, Apr 10 2024
G.f.: Sum_{k>=1} Sum_{d|k} x^(k^2/d)/(1 - x^k). - Miles Wilson, Jun 12 2025

Corrected and extended by Robert G. Wilson v, May 31 2008

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 20, 6, 8, 2, 48, 2, 8, 8, 84, 2, 48, 2, 48, 8, 8, 2, 320, 6, 8, 20, 48, 2, 128, 2, 264, 8, 8, 8, 864, 2, 8, 8, 320, 2, 128, 2, 48, 48, 8, 2, 2688, 6, 48, 8, 48, 2, 320, 8, 320, 8, 8, 2, 3072, 2, 8, 48, 1560, 8, 128, 2, 48, 8, 128, 2, 11520, 2, 8, 48, 48, 8, 128, 2, 2688, 84, 8, 2, 3072, 8, 8, 8, 320

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

			For n = 24, its divisors larger than one are: 2, 3, 4, 6, 8, 12, 24. Only 2 has valuation > 1, namely A286561(24,2) = 3 (as 2^3 divides 24), while the other six have valuation 1. Thus a(24) = prime(1)^6 * prime(3) = 64*5 = 320.
For n = 64, its divisors larger than one are: 2, 4, 8, 16, 32, 64. We see that 2^6 = 4^3 = 8^2 = 64, while valuation of the last three 16, 32 and 64 is 1. Thus a(64) = prime(1)^3 * prime(2) * prime(3) * prime(6) = 2^3 * 3 * 5 * 13 = 1560.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A286561.

Cf. also A293515, A293524, A294876, A293442.

A293514(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(valuation(n,d)))); m; };

a(n) = Product_{d|n, d>1} A000040(A286561(n,d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A056595(n) [See A046951.]
1+A056239(a(n)) = A169594(n).
A064989(a(n)) = A293515(n).

cp's OEIS Frontend

A351602 a(n) = n^4 * Sum_{d^2|n} 1 / d^4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351603 a(n) = n^5 * Sum_{d^2|n} 1 / d^5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351604 a(n) = n^6 * Sum_{d^2|n} 1 / d^6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351605 a(n) = n^7 * Sum_{d^2|n} 1 / d^7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351606 a(n) = n^8 * Sum_{d^2|n} 1 / d^8.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351607 a(n) = n^9 * Sum_{d^2|n} 1 / d^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A351608 a(n) = n^10 * Sum_{d^2|n} 1 / d^10.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A056626 Number of non-unitary square divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica