A183003 -id:A183003 - cp's OEIS frontend

A161840 Number of noncentral divisors of n.

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

A183002 a(n) is the total number of noncentral divisors in all positive integers <= n.

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 4, 6, 8, 10, 10, 14, 14, 16, 18, 22, 22, 26, 26, 30, 32, 34, 34, 40, 42, 44, 46, 50, 50, 56, 56, 60, 62, 64, 66, 74, 74, 76, 78, 84, 84, 90, 90, 94, 98, 100, 100, 108, 110, 114, 116, 120, 120, 126, 128, 134, 136, 138, 138, 148, 148, 150, 154

Offset: 1

Omar E. Pol, Jan 27 2011

The original name was: Partial sums of A161840.

Cf. A000005, A001620, A161840, A183003.

with(numtheory):
b:= n-> (x-> x-2+(x mod 2))(tau(n)):
a:= proc(n) option remember; b(n) +`if`(n=1, 0, a(n-1)) end:
seq(a(n), n=1..100);

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

lista(nmax) = {my(s = 0, d); for(n = 1, nmax, d = numdiv(n); s += (d + d%2 - 2); print1(s, ", ")); } \\ Amiram Eldar, Jan 19 2024

a(n) = Sum_{k=1..n} (tau(k)-2 + (tau(k) mod 2)), tau = A000005.

a(n) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 19 2024

New name from Omar E. Pol, Jan 04 2022

A350497 Sum of the largest parts in all the partitions of n into 3 parts whose largest part is greater than or equal to the product of the other two.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 7, 12, 19, 27, 32, 48, 55, 68, 84, 109, 120, 149, 162, 196, 223, 249, 266, 323, 359, 392, 430, 484, 509, 586, 614, 678, 728, 775, 831, 952, 989, 1044, 1106, 1219, 1261, 1379, 1424, 1520, 1627, 1698, 1748, 1919, 2009, 2124, 2213, 2332, 2392, 2552, 2655, 2827

Offset: 0

Wesley Ivan Hurt, Jan 03 2022

nonn

			a(7) = 12 since we have 7 = 1+1+5 = 1+2+4 = 1+3+3, and the sum of the largest parts in each partition is 5+4+3 = 12. The partition 2+2+3 is not included since 2*2 > 3.

David A. Corneth, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A072670, A183003, A350534.

Table[Sum[Sum[(n - i - k) Sign[Floor[(n - i - k)/(i*k)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
Table[Total[Select[IntegerPartitions[n,{3}],#[[1]]>=Times@@Rest[#]&][[All,1]]],{n,0,60}] (* Harvey P. Dale, Aug 22 2022 *)

first(n) = my(res=vector(n, i, [0, 0])); for(i = 1, n\2, for(j = i, n\i, c = i + j + i * j; if(c <= n, res[c][1]++; res[c][2] += i*j))); forstep(i = n, 1, -1, for(j = i + 1, n, res[j][2] += ((j-i) * res[i][1] + res[i][2]))); concat(0, vector(#res, i, res[i][2])) \\ David A. Corneth, Jan 07 2022

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((n-i-k)/(i*k))) * (n-i-k).

a(0) = 0 prepended by David A. Corneth, Jan 09 2022

A350534 Sum of the largest parts of the partitions of n into 3 parts whose largest part is equal to the product of the other two.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 4, 4, 0, 11, 0, 6, 8, 16, 0, 18, 0, 21, 12, 10, 0, 40, 16, 12, 16, 31, 0, 52, 0, 36, 20, 16, 24, 88, 0, 18, 24, 74, 0, 76, 0, 51, 60, 22, 0, 121, 36, 60, 32, 61, 0, 100, 40, 108, 36, 28, 0, 198, 0, 30, 88, 125, 48, 124, 0, 81, 44, 140, 0, 243, 0, 36, 104

Offset: 0

Wesley Ivan Hurt, Jan 04 2022

nonn

			a(13) = 6 since we have 13 = 1+6+6, whose largest part is 6. Partitions not counted: 1+1+11, 1+2+10, 1+3+9, 1+4+8, 1+5+7, 2+2+9, 2+3+8, 2+4+7, 2+5+6, 3+3+7, 3+4+6, 3+5+5, 4+4+5.

David A. Corneth, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A072670, A183003, A350497.

Table[Sum[Sum[(n - i - k) KroneckerDelta[(n - i - k), (i*k)], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 0, 100}]

first(n) = {my(res = vector(n)); for(i = 1, n \ 2, for(j = i, n\i, c = i + j + i*j; if(c <= n, res[c] += i*j))); concat(0, res)} \\ David A. Corneth, Jan 07 2022

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} [n-i-k = i*k] * (n-i-k), where [ ] is the Iverson bracket.

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A161840 Number of noncentral divisors of n.

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A183002 a(n) is the total number of noncentral divisors in all positive integers <= n.

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A350497 Sum of the largest parts in all the partitions of n into 3 parts whose largest part is greater than or equal to the product of the other two.

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A350534 Sum of the largest parts of the partitions of n into 3 parts whose largest part is equal to the product of the other two.

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Programs

Mathematica

PARI

Formula