A139711 -id:A139711 - cp's OEIS frontend

A063655 Smallest semiperimeter of integral rectangle with area n.

2, 3, 4, 4, 6, 5, 8, 6, 6, 7, 12, 7, 14, 9, 8, 8, 18, 9, 20, 9, 10, 13, 24, 10, 10, 15, 12, 11, 30, 11, 32, 12, 14, 19, 12, 12, 38, 21, 16, 13, 42, 13, 44, 15, 14, 25, 48, 14, 14, 15, 20, 17, 54, 15, 16, 15, 22, 31, 60, 16, 62, 33, 16, 16, 18, 17, 68, 21, 26

Offset: 1

Floor van Lamoen, Jul 24 2001

nonn

Similar to A027709, which is minimal perimeter of polyomino of n cells, or equivalently, minimal perimeter of rectangle of area at least n and with integer sides. Present sequence is minimal semiperimeter of rectangle with area exactly n and with integer sides. - Winston C. Yang (winston(AT)cs.wisc.edu), Feb 03 2002
Semiperimeter b+d, d >= b, of squarest (smallest d-b) integral rectangle with area bd = n. That is, b = largest divisor of n <= sqrt(n), d = smallest divisor of n >= sqrt(n). a(n) = n+1 iff n is noncomposite (1 or prime). - Daniel Forgues, Nov 22 2009
From Juhani Heino, Feb 05 2019: (Start)
Basis for any thickness "frames" around the minimal area. Perimeter can be thought as the 0-thick frame, it is obviously 2a(n). Thickness 1 is achieved by laying unit tiles around the area, there are 2(a(n)+2) of them. Thickness 2 comes from the second such layer, now there are 4(a(n)+4) and so on. They all depend only on a(n), so they share this structure:
Every n > 1 is included. (For different thicknesses, every integer that can be derived from these with the respective formula. So, the perimeter has every even n > 2.)
For each square n > 1, a(n) = a(n-1).
a(1), a(2) and a(6) are the only unique values - the others appear multiple times.
(End)
Gives a discrete Uncertainty Principle. A complex function on an abelian group of order n and its Discrete Fourier Transform must have at least a(n) nonzero entries between them. This bound is achieved by the indicator function on a subgroup of size closest to sqrt(n). - Oscar Cunningham, Oct 10 2021
Also two times the median divisor of n, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). The version for mean instead of median is A057020/A057021. Other doubled medians of multisets are: A360005 (prime indices), A360457 (distinct prime indices), A360458 (distinct prime factors), A360459 (prime factors), A360460 (prime multiplicities), A360555 (0-prepended differences). - Gus Wiseman, Mar 18 2023

			Since 15 = 1*15 = 3*5 and the 3*5 rectangle gives smallest semiperimeter 8, we have a(15)=8.

T. D. Noe, Table of n, a(n) for n = 1..10000
Roy Meshulam, An uncertainty inequality for finite abelian groups, arXiv:math/0312407 [math.CO], 2003.
Roy Meshulam, An uncertainty inequality for finite abelian groups, European Journal of Combinatorics, 27 (2006) 63-67.

Cf. A027709, A033676, A033677, A092510, A162348.
Positions of odd terms are A139710.
Positions of even terms are A139711.
A000005 counts divisors, listed by A027750.
A000975 counts subsets with integer median.
Cf. A003601, A026424, A057020/A057021, A325347, A359893, A359901.

A063655 := proc(n)
    local i,j;
    for i from floor(sqrt(n)) to 1 by -1 do
        j := floor(n/i) ;
        if i*j = n then
            return i+j;
        end if;
    end do:
end proc:
seq(A063655(n), n=1..80); # Winston C. Yang, Feb 03 2002

Table[d = Divisors[n]; len = Length[d]; If[OddQ[len], 2*Sqrt[n], d[[len/2]] + d[[1 + len/2]]], {n, 100}] (* T. D. Noe, Mar 06 2012 *)
Table[2*Median[Divisors[n]],{n,100}] (* Gus Wiseman, Mar 18 2023 *)

A063655(n) = { my(c=1); fordiv(n,d,if((d*d)>=n,if((d*d)==n,return(2*d),return(c+d))); c=d); (0); }; \\ Antti Karttunen, Oct 20 2017

from sympy import divisors
def A063655(n):
    d = divisors(n)
    l = len(d)
    return d[(l-1)//2] + d[l//2] # Chai Wah Wu, Jun 14 2019

a(n) = A033676(n) + A033677(n).
a(n) = A162348(2n-1) + A162348(2n). - Daniel Forgues, Sep 29 2014
a(n) = Min_{d|n} (n/d + d). - Ridouane Oudra, Mar 17 2024

Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Dean Hickerson, Jul 26 2001

A360556 Numbers > 1 whose first differences of 0-prepended prime indices have integer median.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 26, 27, 28, 29, 30, 31, 32, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 83, 84, 86, 87, 89

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so 1617 is not in the sequence.

For mean instead of median we have A340610.
Positions of even terms in A360555.
The complement is A360557 (without 1).
These partitions are counted by A360688.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A112798 lists prime indices, length A001222, sum A056239.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A000975, A026424, A078175, A316413, A360009, A360558, A360669, A360681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],IntegerQ[Median[Differences[Prepend[prix[#],0]]]]&]

A360550 Numbers > 1 whose distinct prime indices have integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 73, 78, 79, 80, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 94, 97, 100

Offset: 1

Gus Wiseman, Feb 14 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Distinct prime indices are listed by A304038.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is in the sequence.
The prime indices of 330 are {1,2,3,5},  with distinct parts {1,2,3,5}, with median 5/2, so 330 is not in the sequence.

For mean instead of median we have A326621.
Positions of even terms in A360457.
The complement (without 1) is A360551.
Partitions with these Heinz numbers are counted by A360686.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices, length A001221, sum A066328.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A316413, A360006, A360009, A360248, A360453.

Select[Range[2,100],IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]

A139710 Numbers k such that the sum of (the largest divisor of k that is <= sqrt(k)) and (the smallest divisor of k that is >= sqrt(k)) is odd.

Original entry on oeis.org

2, 6, 10, 12, 14, 18, 20, 22, 26, 28, 30, 34, 38, 40, 42, 44, 46, 50, 52, 54, 56, 58, 62, 66, 68, 70, 72, 74, 76, 78, 82, 84, 86, 88, 90, 92, 94, 98, 102, 104, 106, 108, 110, 114, 116, 118, 122, 124, 126, 130, 132, 134, 136, 138, 142, 146, 148, 150, 152, 154, 156, 158

Offset: 1

Leroy Quet, Apr 30 2008

nonn

All terms of this sequence are even.

A139711 contains all positive integers not in this sequence and vice versa.

			The divisors of 12 are 1,2,3,4,6,12. The middle 2 divisors are 3 and 4. The sum of these is 7, which is odd. So 12 is included in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000035, A033676, A033677, A063655, A139711.

A033676 := proc(n) local d ; for d from floor(sqrt(n)) to 1 by -1 do if n mod d = 0 then RETURN(d) ; fi ; od: end: A033677 := proc(n) n/A033676(n) ; end: isA139710 := proc(n) RETURN ( ( A033676(n)+A033677(n) ) mod 2 = 1 ) ; end: for n from 1 to 300 do if isA139710(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, May 11 2008

centralDivisors:=#[[({Floor[#],Ceiling[#]}&[(1+#)/2&[Length[#]]])]]&[Divisors[#]]&;
Select[Range[500],OddQ[Total[#]]&[centralDivisors[#]]&](* Peter J. C. Moses, May 31 2019 *)
Select[Range[158],!IntegerQ[Median[Divisors[#]]] &] (* Stefano Spezia, Mar 14 2023 *)

b(n) = {local(d); d=divisors(n); d[(length(d)+1)\2] + d[length(d)\2+1]};
for(n=1, 180, if(b(n)%2==1, print1(n, ", ")) ) \\ G. C. Greubel, May 31 2019

{k: A000035(A033676(k) + A033677(k)) = 1}. - R. J. Mathar, May 11 2008

More terms from R. J. Mathar, May 11 2008

A360551 Numbers > 1 whose distinct prime indices have non-integer median.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192, 196

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A325700 in having 330 and lacking 462.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Distinct prime indices are listed by A304038.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is not in the sequence.
The prime indices of 462 are {1,2,4,5}, with distinct parts {1,2,4,5}, with median 3, so 462 is not in the sequence.

For mean instead of median we have the complement of A326621.
Positions of odd terms in A360457.
The complement (without 1) is A360550, counted by A360686.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices, length A001221, sum A066328.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A316413, A348551, A360006, A360009, A360248, A360453.

Select[Range[2,100],!IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]

A360554 Numbers > 1 whose unordered prime signature has non-integer median.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 200, 207, 208, 212, 236, 242, 244, 245, 261, 268, 272, 275, 279, 284, 288, 292, 304, 316, 320, 325, 332, 333

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A187039 in having 2520 and lacking 1 and 12600.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The unordered prime signature of 2520 is {3,2,1,1}, with median 3/2, so 2520 is in the sequence.
The unordered prime signature of 12600 is {3,2,2,1}, with median 2, so 12600 is not in the sequence.

A subset of A030231.
For mean instead of median we have A070011.
Positions of odd terms in A360460.
The complement is A360553 (without 1), counted by A360687.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A304038, A316413, A326621, A348551, A360006, A360009, A360248, A360453.

Select[Range[2,100],!IntegerQ[Median[Last/@FactorInteger[#]]]&]

A360557 Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.

Original entry on oeis.org

4, 10, 15, 22, 24, 25, 33, 34, 36, 40, 46, 51, 54, 55, 56, 62, 69, 77, 82, 85, 88, 93, 94, 100, 104, 115, 118, 119, 121, 123, 134, 135, 136, 141, 146, 152, 155, 161, 166, 177, 184, 187, 194, 196, 201, 205, 206, 217, 218, 219, 220, 221, 225, 232, 235, 240, 248

Offset: 1

Gus Wiseman, Feb 17 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so 1617 is in the sequence.

For mean instead of median complement we have A340610, counted by A168659.
For mean instead of median we have A360668, counted by A200727.
Positions of odd terms in A360555.
The complement is A360556 (without 1), counted by A360688.
These partitions are counted by A360691.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551, complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A287352 lists 0-prepended first differences of prime indices.
A325347 counts partitions with integer median, complement A307683.
A355536 lists first differences of prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A000975, A026424, A078175, A316413, A360009, A360558, A360669, A360681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],!IntegerQ[Median[Differences[Prepend[prix[#],0]]]]&]

A360553 Numbers > 1 whose unordered prime signature has integer median.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A067340 in having 60.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The unordered prime signature of 60 is {1,1,2}, with median 1, so 60 is in the sequence.
The unordered prime signature of 1260 is {1,1,2,2}, with median 3/2, so 1260 is not in the sequence.

For mean instead of median we have A067340, complement A070011.
Positions of even terms in A360460.
The complement is A360554 (without 1).
These partitions are counted by A360687.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A112798 lists prime indices, length A001222, sum A056239.
A124010 lists prime signature.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360454 = numbers whose prime indices and signature have the same median.
Cf. A000975, A026424, A078174, A078175, A316413, A326621, A360453.

Select[Range[2,100],IntegerQ[Median[Last/@FactorInteger[#]]]&]

A360552 Numbers > 1 whose distinct prime factors have integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime factors of 900 are {2,2,3,3,5,5}, with distinct parts {2,3,5}, with median 3, so 900 is in the sequence.

For mean instead of median we have A078174, complement of A176587.
The complement is A100367 (without 1).
Positions of even terms in A360458.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A323171/A323172 = mean of distinct prime factors, indices A326619/A326620.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A056239, A078175, A175352, A316413, A326621, A360009.

Select[Range[2,100],IntegerQ[Median[First/@FactorInteger[#]]]&]

A361565 a(n) is the numerator of the median of divisors of n.

Original entry on oeis.org

1, 3, 2, 2, 3, 5, 4, 3, 3, 7, 6, 7, 7, 9, 4, 4, 9, 9, 10, 9, 5, 13, 12, 5, 5, 15, 6, 11, 15, 11, 16, 6, 7, 19, 6, 6, 19, 21, 8, 13, 21, 13, 22, 15, 7, 25, 24, 7, 7, 15, 10, 17, 27, 15, 8, 15, 11, 31, 30, 8, 31, 33, 8, 8, 9, 17, 34, 21, 13, 17, 36, 17, 37, 39, 10

Offset: 1

Stefano Spezia, Mar 15 2023

			a(9) = 3 since the divisors of 9 are 1, 3, 9, and their median is 3.
a(12) = 7 since the divisors of 12 are 1, 2, 3, 4, 6, 12, and their median is 7/2.

Winston de Greef, Table of n, a(n) for n = 1..10000
Index entries for sequences related to divisors of numbers

Cf. A027750, A033676, A033677, A071090, A139710, A139711, A361566 (denominator).

a[n_]:=Numerator[Median[Divisors[n]]]; Array[a,75]

a(n) = my(d=divisors(n), m=#d+1); numerator((d[m\2] + d[m-m\2])/2); \\ Michel Marcus, Mar 16 2023

a(n) = numerator((A033676(n) + A033677(n))/2).
a(p) = (1 + p)/2 if p is an odd prime.
For p a prime, a(p^k) = (p^((k-1)/2) + p^((k+1)/2))/2 if k is odd, a(p^k) = p^(k/2) if k is even.

cp's OEIS Frontend

A063655 Smallest semiperimeter of integral rectangle with area n.

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A360556 Numbers > 1 whose first differences of 0-prepended prime indices have integer median.

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A360550 Numbers > 1 whose distinct prime indices have integer median.

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A139710 Numbers k such that the sum of (the largest divisor of k that is <= sqrt(k)) and (the smallest divisor of k that is >= sqrt(k)) is odd.

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A360551 Numbers > 1 whose distinct prime indices have non-integer median.

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A360554 Numbers > 1 whose unordered prime signature has non-integer median.

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A360557 Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.

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A360553 Numbers > 1 whose unordered prime signature has integer median.

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