A091954 -id:A091954 - cp's OEIS frontend

A352056 Sum of the 10th powers of the divisor complements of the odd proper divisors of n.

0, 1024, 59049, 1048576, 9765625, 60467200, 282475249, 1073741824, 3486843450, 10000001024, 25937424601, 61918412800, 137858491849, 289254656000, 576660215299, 1099511627776, 2015993900449, 3570527693824, 6131066257801, 10240001048576, 16680163512499

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^10 * Sum_{d|10, d<10, d odd} 1 / d^10 = 10^10 * (1/1^10 + 1/5^10) = 10000001024.

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), A352055 (k=9), this sequence (k=10).

Cf. A000035, A006519, A013669, A321814.

A352056[n_]:=DivisorSum[n,1/#^10&,#A352056,50] (* Paolo Xausa, Aug 10 2023 *)
a[n_] := DivisorSigma[-10, n/2^IntegerExponent[n, 2]] * n^10 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^10 * sigma(n >> valuation(n, 2), -10) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^10 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^10 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 19 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321814(n) * A006519(n)^10 - A000035(n).

Sum_{k=1..n} a(k) = c * n^11 / 11, where c = 2047*zeta(11)/2048 = 1.00000566605... . (End)

A293216 a(n) = Product_{d|n, dA260443(d).

Original entry on oeis.org

1, 2, 2, 6, 2, 36, 2, 30, 12, 108, 2, 2700, 2, 180, 216, 210, 2, 48600, 2, 40500, 360, 1620, 2, 661500, 36, 2700, 1080, 94500, 2, 153090000, 2, 2310, 3240, 3780, 1080, 4465125000, 2, 40500, 5400, 69457500, 2, 34445250000, 2, 21262500, 4082400, 56700, 2, 560290500, 60, 127575000, 7560, 49612500, 2, 1205583750000, 9720, 254677500, 81000, 132300, 2

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A048675, A260443, A293217 (restricted growth sequence transform), A293214 (a variant).

Cf. also A001065, A091954.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
A293216(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A260443(d))); m; };

a(n) = Product_{d|n, dA260443(d).
For all n >= 0, a(2^n) = A002110(n).
For all n >= 1, A007814(a(n)) = A091954(n) and A048675(a(n)) = A001065(n).

A293451 Number of proper divisors of n of the form 4k+1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 3, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 1, 2, 1, 1, 4, 2, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, Oct 19 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001620, A001826, A091954, A121262, A256778, A293513, A293901.

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

PARI
```
A293451(n) = sumdiv(n,d,(d
				
```

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

a(n) = A091954(n) - A293513(n).

a(n) = A001826(n) - A121262(n-1).

G.f.: Sum_{k>=1} x^(8*k-6) / (1 - x^(4*k-3)). - Ilya Gutkovskiy, Apr 14 2021

Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,4) - (2 - gamma)/4 = A256778 - (2 - A001620)/4 = 0.354593... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A325937 Expansion of Sum_{k>=1} (-1)^(k + 1) * x^(2*k) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, -1, 2, 1, 1, -1, 1, 1, 3, -2, 1, 1, 1, -1, 3, 1, 1, -3, 2, 1, 3, -1, 1, 1, 1, -3, 3, 1, 3, -2, 1, 1, 3, -3, 1, 1, 1, -1, 5, 1, 1, -5, 2, 1, 3, -1, 1, 1, 3, -3, 3, 1, 1, -3, 1, 1, 5, -4, 3, 1, 1, -1, 3, 1, 1, -5, 1, 1, 5, -1, 3, 1, 1, -5

Offset: 1

Ilya Gutkovskiy, Sep 09 2019

Number of odd proper divisors of n minus number of even proper divisors of n.

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

Cf. A032741, A048272, A058344, A091954, A275495 (partial sums), A325939.

nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(2 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[-DivisorSum[n, (-1)^# &, # < n &], {n, 1, 80}]

A325937(n) = -sumdiv(n, d, if(d==n,0,((-1)^d))); \\ Antti Karttunen, Sep 20 2019

G.f.: Sum_{k>=2} x^k / (1 + x^k).

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

a(n) = A048272(n) + (-1)^n.

A293513 Number of proper divisors of n of the form 4k+3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 1, 2, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 1, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 2, 2, 0, 1, 0, 0, 2, 1, 2, 2, 0, 0, 2, 0, 0, 2, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 0, 1, 2, 0, 0, 2, 0, 0, 4

Offset: 1

Antti Karttunen, Oct 19 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001842, A091954, A121262, A293451, A293903.

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

PARI
```
A293513(n) = sumdiv(n,d,(d
				
```

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

a(n) = A091954(n) - A293451(n).

a(n) = A001842(n) - A121262(n+1).

G.f.: Sum_{k>=1} x^(8*k-2) / (1 - x^(4*k-1)). - Ilya Gutkovskiy, Apr 14 2021

Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,4) - (2 - gamma)/4 = A256846 - (2 - A001620)/4 = -0.430804... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A096157 Numbers whose numbers of odd and even proper divisors differ at most by 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 18, 19, 20, 22, 23, 26, 28, 29, 30, 31, 34, 37, 38, 41, 42, 43, 44, 46, 47, 50, 52, 53, 54, 58, 59, 61, 62, 66, 67, 68, 70, 71, 73, 74, 76, 78, 79, 82, 83, 86, 89, 90, 92, 94, 97, 98, 101, 102, 103, 106, 107, 109, 110, 113

Offset: 1

Reinhard Zumkeller, Jun 18 2004

nonn

A union of 1, 4, primes, quadrupled primes, and all numbers of the form 4k+2. - Ivan Neretin, Aug 12 2015

			Proper divisors of 42 are {1,2,3,6,7,14,21}: 4=#{1,3,7,21} are odd and 3=#{2,6,14} are even, as abs(4-3)=1, 42 is a term.

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

Cf. A032741, A091954.

filter:= n -> if n::odd then isprime(n) elif n mod 4 = 0 then isprime(n/4) else true fi:
filter(1):= true: filter(4):= true:
select(filter, [$1..1000]); # Robert Israel, Aug 12 2015

okQ[n_] := With[{dd = Most[Divisors[n]]},
     Abs[2 Count[dd, _?OddQ] - Length[dd]] <= 1];
Select[Range[1000], okQ] (* Jean-François Alcover, Dec 16 2021 *)

is(n)=if(bittest(n,0),isprime(n),bittest(n,1)||isprime(n>>2))||n<9 \\ M. F. Hasler, Aug 13 2015

Abs(A032741(a(n)) - 2*A091954(a(n))) <= 1.

A096158 Number of permutations of proper divisors of n such that the sum of adjacent pairs of divisors is prime.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Jun 18 2004

nonn

For n>4: a(n)=2 iff (n/2,2+n/2) is twin prime pair, a(2*A001359(n))=2;

a(A096160(n)) > 0.

			Proper divisors of n=12 are {1,2,3,4,6}:
[2,3,4,1,6]->(2+3,3+4,4+1,1+6)=(5,7,5,7),
[4,3,2,1,6]->(4+3,3+2,2+1,1+6)=(7,5,3,7),
[6,1,2,3,4]->(6+1,1+2,2+3,3+4)=(7,3,5,7) and
[6,1,4,3,2]->(6+1,1+4,4+3,3+2)=(7,5,7,5): therefore a(12)=4.

Cf. A032741, A091954, A000005, A096159.

A368684 Number of partitions of n into 2 parts such that the smaller part divides both n and floor(n/2).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 4, 1, 2, 1, 6, 1, 2, 1, 4, 1, 4, 1, 5, 1, 2, 1, 6, 1, 2, 1, 6, 1, 4, 1, 4, 1, 2, 1, 8, 1, 3, 1, 4, 1, 4, 1, 6, 1, 2, 1, 8, 1, 2, 1, 6, 1, 4, 1, 4, 1, 4, 1, 9, 1, 2, 1, 4, 1, 4, 1, 8, 1, 2, 1, 8, 1, 2, 1, 6, 1, 6

Offset: 1

Wesley Ivan Hurt, Jan 03 2024

Essentially, A000005 interspersed with 1's [prepend 0].

Number of divisors of A057979(n+1) for n >= 2.

Index entries for sequences related to partitions.

Bisections: A060576, A000005.

Cf. A001620, A057979, A091954.

with(numtheory): 0, seq(2*tau(n) - tau(2*n) + (n mod 2), n=2..100); # Ridouane Oudra, Jan 18 2025

Join[{0}, Table[DivisorSigma[0, (n+2+(n-2)*(-1)^n)/4], {n, 2, 100}]]

a(n) = if(n == 1, 0, numdiv((n+2+(n-2)*(-1)^n)/4)); \\ Amiram Eldar, Jan 28 2025

a(n) = A000005(A057979(n+1)) for n >= 2.
a(2n-1) = A060576(n), a(2n) = A000005(n).
a(n) = d(floor((n+1)/2))^((n+1) mod 2), for n >= 2.
a(n) = d( (n+2+(n-2)*(-1)^n)/4 ) for n >= 2.
a(n) = Sum_{k=1..floor(n/2)} c(n/k) * c(floor(n/2)/k), where c(m) = 1 - ceiling(m) + floor(m).
a(n) = A000005(n) - A091954(n), for n > 1. - Ridouane Oudra, Jan 18 2025
Sum_{k=1..n} a(k) ~ (log(n/2) + 2*gamma)*n/2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 28 2025

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cp's OEIS Frontend

A352056 Sum of the 10th powers of the divisor complements of the odd proper divisors of n.

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A293216 a(n) = Product_{d|n, dA260443(d).

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A293451 Number of proper divisors of n of the form 4k+1.

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A325937 Expansion of Sum_{k>=1} (-1)^(k + 1) * x^(2*k) / (1 - x^k).

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A293513 Number of proper divisors of n of the form 4k+3.

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A096157 Numbers whose numbers of odd and even proper divisors differ at most by 1.

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A096158 Number of permutations of proper divisors of n such that the sum of adjacent pairs of divisors is prime.

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A368684 Number of partitions of n into 2 parts such that the smaller part divides both n and floor(n/2).

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