A183002 -id:A183002 - cp's OEIS frontend

A183003 a(n) = A183002(n)/2.

0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 7, 8, 9, 11, 11, 13, 13, 15, 16, 17, 17, 20, 21, 22, 23, 25, 25, 28, 28, 30, 31, 32, 33, 37, 37, 38, 39, 42, 42, 45, 45, 47, 49, 50, 50, 54, 55, 57, 58, 60, 60, 63, 64, 67, 68, 69, 69, 74, 74, 75, 77, 80, 81, 84, 84, 86, 87, 90, 90, 95, 95, 96, 98, 100, 101, 104, 104, 108, 110, 111, 111, 116, 117, 118, 119, 122, 122, 127, 128, 130, 131, 132, 133, 138, 138, 140, 142, 146

Offset: 1

Omar E. Pol, Jan 27 2011

For n >= 2, a(n) is the number of partitions of n-1 into 3 parts such that the largest part is greater than or equal to the product of the other two. For example, a(9) = 4 since the partitions for 8 would be 1+1+6 = 1+2+5 = 1+3+4 = 2+2+4, but not 2+3+3 since 2*3 > 3. - Wesley Ivan Hurt, Jan 03 2022

Conjecture: partial sums of A072670. - Sean A. Irvine, Jul 14 2022

Cf. A000005, A001620, A072670, A183002.

Accumulate[Table[d = DivisorSigma[0, n]; If[OddQ[d], d - 1, d - 2], {n, 100}]]/2

a(n) = sum(k=1, n, numdiv(k) - 2 + numdiv(k)%2)/2; \\ Michel Marcus, Jan 04 2022

a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k..floor((n-k-1)/2)} sign(floor((n-i-k-1)/(i*k))). - Wesley Ivan Hurt, Jan 03 2022
a(n) = (1/2) * Sum_{k=1..n} (tau(k)-2 + (tau(k) mod 2)), tau = A000005. - Alois P. Heinz, Jan 04 2022
a(n) ~ n * (log(n) + 2*gamma - 3) / 2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 19 2024

A161840 Number of noncentral divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 2, 2, 2, 0, 4, 0, 2, 2, 4, 0, 4, 0, 4, 2, 2, 0, 6, 2, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 8, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 2, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 10, 0, 2, 4, 6, 2, 6, 0, 4, 2, 6, 0, 10, 0, 2, 4, 4, 2, 6, 0, 8, 4, 2, 0, 10, 2, 2, 2, 6, 0, 10, 2, 4, 2, 2, 2, 10, 0, 4, 4, 8

Offset: 1

Omar E. Pol, Jun 21 2009

Noncentral divisors in the following sense: if we sort the divisors of n in natural order, there is one "central", median divisor if the number of divisors tau(n) = A000005(n) is odd, and there are two "central" divisors if tau(n) is even. a(n) is the number of divisors not counting the median or two central divisors.

			The divisors of 4 are 1, 2, 4 so the noncentral divisors of 4 are 1, 4 because its central divisor is 2.
The divisors of 12 are 1, 2, 3, 4, 6, 12 so the noncentral divisors of 12 are 1, 2, 6, 12 because its central divisors  are 3, 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A001620, A049240, A010052, A161841, A169695, A183002, A183003, A200213, A323643.

A000005 := proc(n) numtheory[tau](n) ; end: A010052 := proc(n) if issqr(n) then 1; else 0 ; fi; end: A161840 := proc(n) A000005(n)+A010052(n)-2 ; end: seq(A161840(n),n=1..100) ; # R. J. Mathar, Jul 04 2009

If[EvenQ[#],#-2,#-1]&/@DivisorSigma[0,Range[100]] (* Harvey P. Dale, Sep 22 2024 *)

A161840(n) = numdiv(n)+issquare(n)-2; \\ Antti Karttunen, Jul 07 2017

(define (A161840 n) (+ (A000005 n) (A010052 n) -2)) ;; Antti Karttunen, Jul 07 2017

a(n) = tau(n)-2 + (tau(n) mod 2), tau = A000005.
a(n) = A000005(n) - A049240(n) - 1.
a(n) = A000005(n) + A010052(n) - 2.
a(n) = A000005(n) - A169695(n).
For n >= 2, a(n) = A200213(n) + 2*A010052(n). - Antti Karttunen, Jul 07 2017
a(n) = 2*A072670(n-1). - Omar E. Pol, Jul 08 2017
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024

More terms from R. J. Mathar, Jul 04 2009

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A183003 a(n) = A183002(n)/2.

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A161840 Number of noncentral divisors of n.

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