A091954 -id:A091954 - cp's OEIS frontend

A352038 Sum of the 10th powers of the odd proper divisors of n.

0, 1, 1, 1, 1, 59050, 1, 1, 59050, 9765626, 1, 59050, 1, 282475250, 9824675, 1, 1, 3486843451, 1, 9765626, 282534299, 25937424602, 1, 59050, 9765626, 137858491850, 3486843451, 282475250, 1, 576660215300, 1, 1, 25937483651, 2015993900450, 292240875, 3486843451

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 9765626; a(10) = Sum_{d|10, d<10, d odd} d^10 = 1^10 + 5^10 = 9765626.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors.

Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), this sequence (k=10).

Cf. A000035, A013669, A321814.

f[2, e_] := 1; f[p_, e_] := (p^(10*e+10) - 1)/(p^10 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^10, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

from math import prod
from sympy import factorint
def A352038(n): return prod((p**(10*(e+1))-1)//(p**10-1) for p, e in factorint(n).items() if p > 2) - (n**10 if n % 2 else 0) # Chai Wah Wu, Mar 01 2022

a(n) = Sum_{d|n, d

G.f.: Sum_{k>=1} (2*k-1)^10 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A321814(n) - n^10*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^11, where c = (zeta(11)-1)/22 = 0.0000224631... . (End)

A352047 Sum of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 2, 3, 4, 5, 8, 7, 8, 12, 12, 11, 16, 13, 16, 23, 16, 17, 26, 19, 24, 31, 24, 23, 32, 30, 28, 39, 32, 29, 48, 31, 32, 47, 36, 47, 52, 37, 40, 55, 48, 41, 64, 43, 48, 77, 48, 47, 64, 56, 62, 71, 56, 53, 80, 71, 64, 79, 60, 59, 96, 61, 64, 103, 64, 83, 96, 67, 72, 95, 96, 71, 104

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 12; a(10) = 10 * Sum_{d|10, d<10, d odd} 1 / d = 10 * (1/1 + 1/5) = 12.

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

Cf. A000035, A000593, A002131, A006519, A111003, A300415.

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), this sequence (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

A352047 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if type(d,'odd') and d < n then
            a := a+n/d ;
        end if;
    end do:
    a ;
end proc:
seq(A352047(n),n=1..30) ; # R. J. Mathar, Mar 09 2022

Table[n DivisorSum[n, 1/# &, # < n && OddQ[#] &], {n, 72}] (* Michael De Vlieger, Mar 02 2022 *)
a[n_] := DivisorSigma[-1, n / 2^IntegerExponent[n, 2]] * n - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n*sumdiv(n, d, if ((d%2) && (dMichel Marcus, Mar 02 2022

a(n) = n * sigma(n >> valuation(n, 2), -1) - n % 2; \\ Amiram Eldar, Oct 13 2023

from math import prod
from sympy import factorint
def A352047(n): return prod(p**e if p == 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) - n % 2 # Chai Wah Wu, Mar 02 2022

a(n) = n * Sum_{d|n, d

a(2n+1) = A000593(2n+1) - 1. - Chai Wah Wu, Mar 01 2022

G.f.: Sum_{k>=2} k * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

exp( 2*Sum_{n>=1} a(n)*x^n/n ) is the g.f. of A300415. - Paul D. Hanna, May 15 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A000593(n) * A006519(n) - A000035(n).

Sum_{k=1..n} a(k) = c * n^2 / 2, where c = Pi^2/8 = 1.23370055... (A111003). (End)

A352048 Sum of the squares of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 4, 9, 16, 25, 40, 49, 64, 90, 104, 121, 160, 169, 200, 259, 256, 289, 364, 361, 416, 499, 488, 529, 640, 650, 680, 819, 800, 841, 1040, 961, 1024, 1219, 1160, 1299, 1456, 1369, 1448, 1699, 1664, 1681, 2000, 1849, 1952, 2365, 2120, 2209, 2560, 2450, 2604, 2899, 2720

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^2 * Sum_{d|10, d<10, d odd} 1 / d^2 = 10^2 * (1/1^2 + 1/5^2) = 104.

Robert Israel, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), this sequence (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A006519, A050999, A076577, A233091.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^2, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..60]); # Robert Israel, Apr 03 2023

a[n_] := n^2 DivisorSum[n, If[# < n && OddQ[#], 1/#^2, 0]&];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, May 11 2023 *)
a[n_] := DivisorSigma[-2, n/2^IntegerExponent[n, 2]] * n^2 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^2*sumdiv(n, d, if ((dMichel Marcus, May 11 2023

a(n) = n^2 * sigma(n >> valuation(n, 2), -2) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^2 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^2 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A050999(n) * A006519(n)^2 - A000035(n).

Sum_{k=1..n} a(k) = c * n^3 / 3, where c = 7*zeta(3)/8 = 1.0517997... (A233091). (End)

A352049 Sum of the cubes of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 8, 27, 64, 125, 224, 343, 512, 756, 1008, 1331, 1792, 2197, 2752, 3527, 4096, 4913, 6056, 6859, 8064, 9631, 10656, 12167, 14336, 15750, 17584, 20439, 22016, 24389, 28224, 29791, 32768, 37295, 39312, 43343, 48448, 50653, 54880, 61543, 64512, 68921, 77056, 79507

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^3 * Sum_{d|10, d<10, d odd} 1 / d^3 = 10^3 * (1/1^3 + 1/5^3) = 1008.

Robert Israel, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), this sequence (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A006519, A051000, A300707.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^3, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..50]); # Robert Israel, Apr 03 2023

A352049[n_]:=DivisorSum[n,1/#^3&,#A352049,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-3, n/2^IntegerExponent[n, 2]] * n^3 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^3 * sigma(n >> valuation(n, 2), -3) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^3 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^3 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A051000(n) * A006519(n)^3 - A000035(n).

Sum_{k=1..n} a(k) = c * n^4 / 4, where c = 15*zeta(4)/16 = 1.01467803... (A300707). (End)

A352050 Sum of the 4th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 16, 81, 256, 625, 1312, 2401, 4096, 6642, 10016, 14641, 20992, 28561, 38432, 51331, 65536, 83521, 106288, 130321, 160256, 196963, 234272, 279841, 335872, 391250, 456992, 538083, 614912, 707281, 821312, 923521, 1048576, 1200643, 1336352, 1503651, 1700608, 1874161

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^4 * Sum_{d|10, d<10, d odd} 1 / d^4 = 10^4 * (1/1^4 + 1/5^4) = 10016.

Robert Israel, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), this sequence (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013663, A051001.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^4, d = select(`<`,numtheory:-divisors(m),n))
end proc:map(f, [$1..40]); # Robert Israel, Apr 03 2023

A352050[n_]:=DivisorSum[n,1/#^4&,#A352050,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-4, n/2^IntegerExponent[n, 2]] * n^4 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^4 * sigma(n >> valuation(n, 2), -4) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^4 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^4 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A051001(n) * A006519(n)^4 - A000035(n).

Sum_{k=1..n} a(k) = c * n^5 / 5, where c = 31*zeta(5)/32 = 1.00452376... . (End)

A352051 Sum of the 5th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 32, 243, 1024, 3125, 7808, 16807, 32768, 59292, 100032, 161051, 249856, 371293, 537856, 762743, 1048576, 1419857, 1897376, 2476099, 3201024, 4101151, 5153664, 6436343, 7995392, 9768750, 11881408, 14408199, 17211392, 20511149, 24407808, 28629151, 33554432, 39296687

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^5 * Sum_{d|10, d<10, d odd} 1 / d^5 = 10^5 * (1/1^5 + 1/5^5) = 100032.

Robert Israel, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), this sequence (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013664, A051002.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^5, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..40]); # Robert Israel, Apr 03 2023

A352051[n_]:=DivisorSum[n,1/#^5&,#A352051,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-5, n/2^IntegerExponent[n, 2]] * n^5 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^5 * sigma(n >> valuation(n, 2), -5) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^5 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^5 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A051002(n) * A006519(n)^5 - A000035(n).

Sum_{k=1..n} a(k) = c * n^6 / 6, where c = 63*zeta(6)/64 = 1.00144707... . (End)

A352052 Sum of the 6th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 64, 729, 4096, 15625, 46720, 117649, 262144, 532170, 1000064, 1771561, 2990080, 4826809, 7529600, 11406979, 16777216, 24137569, 34058944, 47045881, 64004096, 85884499, 113379968, 148035889, 191365120, 244156250, 308915840, 387952659, 481894400, 594823321

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^6 * Sum_{d|10, d<10, d odd} 1 / d^6 = 10^6 * (1/1^6 + 1/5^6) = 1000064.

Robert Israel, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), this sequence (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013665, A321810.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^6, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..30]); # Robert Israel, Apr 03 2023

Table[n^6*DivisorSum[n, 1/#^6 &, And[# < n, OddQ[#]] &], {n, 29}] (* Michael De Vlieger, Apr 04 2023 *)
a[n_] := DivisorSigma[-6, n/2^IntegerExponent[n, 2]] * n^6 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^6*sumdiv(n, d, if ((dMichel Marcus, Apr 04 2023

a(n) = n^6 * sigma(n >> valuation(n, 2), -6) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^6 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^6 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321810(n) * A006519(n)^6 - A000035(n).

Sum_{k=1..n} a(k) = c * n^7 / 7, where c = 127*zeta(7)/128 = 1.000471548... . (End)

A352053 Sum of the 7th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 128, 2187, 16384, 78125, 280064, 823543, 2097152, 4785156, 10000128, 19487171, 35848192, 62748517, 105413632, 170939687, 268435456, 410338673, 612500096, 893871739, 1280016384, 1801914271, 2494358016, 3404825447, 4588568576, 6103593750, 8031810304, 10465138359

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^7 * Sum_{d|10, d<10, d odd} 1/d^7 = 10^7 * (1/1^7 + 1/5^7) = 10000128.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), this sequence (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013666, A321811.

A352053[n_]:=DivisorSum[n,1/#^7&,#A352053,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-7, n/2^IntegerExponent[n, 2]] * n^7 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^7 * sigma(n >> valuation(n, 2), -7) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^7 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^7 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321811(n) * A006519(n)^7 - A000035(n).

Sum_{k=1..n} a(k) = c * n^8 / 8, where c = 255*zeta(8)/256 = 1.000155179... . (End)

A352054 Sum of the 8th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 256, 6561, 65536, 390625, 1679872, 5764801, 16777216, 43053282, 100000256, 214358881, 430047232, 815730721, 1475789312, 2563287811, 4294967296, 6975757441, 11021640448, 16983563041, 25600065536, 37828630723, 54875873792, 78310985281, 110092091392, 152588281250

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^8 * Sum_{d|10, d<10, d odd} 1 / d^8 = 10^8 * (1/1^8 + 1/5^8) = 100000256.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), this sequence (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013667, A321812.

A352054[n_]:=DivisorSum[n,1/#^8&,#A352054,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-8, n/2^IntegerExponent[n, 2]] * n^8 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^8 * sigma(n >> valuation(n, 2), -8) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^8 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^8 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 19 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321812(n) * A006519(n)^8 - A000035(n).

Sum_{k=1..n} a(k) = c * n^9 / 9, where c = 511*zeta(9)/512 = 1.0000513451... . (End)

A352055 Sum of the 9th powers of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 512, 19683, 262144, 1953125, 10078208, 40353607, 134217728, 387440172, 1000000512, 2357947691, 5160042496, 10604499373, 20661047296, 38445332183, 68719476736, 118587876497, 198369368576, 322687697779, 512000262144, 794320419871, 1207269218304, 1801152661463, 2641941757952

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^9 * Sum_{d|10, d<10, d odd} 1 / d^9 = 10^9 * (1/1^9 + 1/5^9) = 1000000512.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), this sequence (k=9), A352056 (k=10).

Cf. A000035, A006519, A013668, A321813.

A352055[n_]:=DivisorSum[n,1/#^9&,#A352055,50] (* Paolo Xausa, Aug 10 2023 *)
a[n_] := DivisorSigma[-9, n/2^IntegerExponent[n, 2]] * n^9 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^9 * sigma(n >> valuation(n, 2), -9) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^9 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^9 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 19 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321813(n) * A006519(n)^9 - A000035(n).

Sum_{k=1..n} a(k) = c * n^10 / 10, where c = 1023*zeta(10)/1024 = 1.0000170413... . (End)

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A352038 Sum of the 10th powers of the odd proper divisors of n.

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A352047 Sum of the divisor complements of the odd proper divisors of n.

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A352048 Sum of the squares of the divisor complements of the odd proper divisors of n.

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A352049 Sum of the cubes of the divisor complements of the odd proper divisors of n.

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A352050 Sum of the 4th powers of the divisor complements of the odd proper divisors of n.

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A352051 Sum of the 5th powers of the divisor complements of the odd proper divisors of n.

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A352052 Sum of the 6th powers of the divisor complements of the odd proper divisors of n.

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