A136404 -id:A136404 - cp's OEIS frontend

A176471 Duplicate of A136404.

1, 4, 16, 36, 144, 576, 900, 3600, 14400, 32400, 44100, 129600, 176400, 705600, 1587600, 2822400, 6350400, 21344400, 57153600, 85377600, 192099600, 341510400, 768398400, 3073593600, 6915585600, 12294374400, 14428814400, 32464832400, 57715257600, 129859329600

Offset: 1

Ethan Ward (etkewa(AT)gmail.com), Apr 18 2010

dead

Same as A136404. - Georg Fischer, Oct 12 2018

Previous name was: Highly composite square numbers for which the number of divisors increases to a record.

Cf. A126098 (square root of these terms).

Cf. A136404. - Georg Fischer, Oct 12 2018

With[{sqs=Range[10000]^2},Union[Flatten[Table[Select[sqs, DivisorSigma[ 0,#]>n&,1],{n,500}]]]] (* Harvey P. Dale, Apr 28 2012 *)
t = {1}; sig = {1}; n = 1; Do[While[n++; d = DivisorSigma[0, n^2]; d <= sig[[-1]]]; AppendTo[sig, d]; AppendTo[t, n^2], {30}]; t (* T. D. Noe, Apr 30 2012 *)

Corrected and extended by Harvey P. Dale, Apr 28 2012

A166721 Squares for which no smaller square has the same number of divisors.

Original entry on oeis.org

1, 4, 16, 36, 64, 144, 576, 900, 1024, 1296, 3600, 4096, 5184, 9216, 14400, 32400, 36864, 44100, 46656, 65536, 82944, 129600, 176400, 230400, 262144, 331776, 589824, 705600, 746496, 810000, 921600, 1166400, 1587600, 2073600, 2359296, 2822400, 2985984, 3240000

Offset: 1

Alexander Isaev (i2357(AT)mail.ru), Oct 20 2009

From Jon E. Schoenfield, Mar 03 2018: (Start)
Numbers k^2 such there is no positive m < k such that A000005(m^2) = A000005(k^2).
Square terms in A007416. (End)

			The positive squares begin 1, 4, 9, 16, 25, 36, 49, 64, ..., and their corresponding numbers of divisors are 1, 3, 3, 5, 3, 9, 3, 7, ...; thus, a(1)=1, a(2)=4, 9 is not a term (it has the same number of divisors as does 4; the same is true of 25, 49, etc.), a(3)=16, a(4)=36, a(5)=64, ... - _Jon E. Schoenfield_, Mar 03 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..300 from Alois P. Heinz)

Cf. A000005, A005179, A007416, A025487, A048691, A136404, A166722.

 Sort[Module[{nn=2000,tbl},tbl=Table[{n^2,DivisorSigma[0,n^2]},{n,nn}];Table[ SelectFirst[ tbl,#[[2]]==k&],{k,nn}]][[All,1]]/."NotFound"->Nothing] (* Harvey P. Dale, Jun 06 2022 *)

lista(nn) = {v = []; for (n=1, nn, d = numdiv(n^2); if (! vecsearch(v, d), print1(n^2, ", "); v = Set(concat(v, d))););} \\ Michel Marcus, Mar 04 2018

Proper definition and substantial editing by Jon E. Schoenfield, Mar 03 2018

A166722 a(n) is the number of divisors of A166721(n).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 21, 27, 11, 25, 45, 13, 35, 33, 63, 75, 39, 81, 49, 17, 55, 105, 135, 99, 19, 65, 51, 189, 77, 125, 117, 147, 225, 165, 57, 243, 91, 175, 23, 85, 315, 195, 297, 153, 231, 95, 405, 245, 69, 375, 351, 119, 275, 441, 171, 121, 273, 567, 495, 255, 525

Offset: 1

Alexander Isaev (i2357(AT)mail.ru), Oct 20 2009

This is a permutation of the odd numbers A005408. - Alois P. Heinz, Mar 04 2018

			a(8) = A000005(A166721(8)) = A000005(900) = A000005(2^2 * 3^2 * 5^2) = (2+1)*(2+1)*(2+1) = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..300 from Alois P. Heinz)

Cf. A000005, A005408, A048691, A136404, A152674, A166721.

a(n) = A000005(A166721(n)).

Proper definition (and removal of obscure Comments entries) by Jon E. Schoenfield, Mar 03 2018

A250029 Maximum number of binary strings with symmetrically partitioned n 1's and n 0's, counted up to isomorphism.

Original entry on oeis.org

1, 1, 1, 4, 9, 16, 36, 144, 400, 900, 3600, 11025, 28224, 78400, 254016, 705600, 2286144, 6350400, 25401600, 85377600, 250905600, 768398400, 3073593600, 10600761600, 32464832400, 129859329600, 456536705625

Offset: 0

Andrei Cretu, Nov 11 2014

nonn

The number of binary strings, counted up to isomorphism, that can be constructed by taking an equal number (n) of 0's and 1's and partitioning both the 0's and the 1's into m runs using the same partition, can be written as:
dualseq[partition[n]]=m!^2/(Prod_j(m_j!))^2
where m_j is the multiplicity of runs of length j.
The numbers satisfy the relations Sum_j(m_j)=m, Sum_j(j*m_j)=n.
The strings obtained in this manner are a subset of those in A247651.
Both the finest and coarsest partitions of n minimize dualseq[partition[n]]. In this sense, dualseq[partition[n]] is another relative measure of the complexity of the partition and the associated binary strings.
a[n] is the number of strings, counted up to isomorphism, that can be generated based on the partition that maximizes dualseq[partition[n]].

			n=0 gives the empty string.
n=1 and the only possible partition generate 01 (and the isomorphic 10).
For n=2, both possible partitions generate, up to isomorphism, 1 string, 0011 (1100), and respectively 0101 (1010).
For n=3, the optimal partition is {1,2}, generating, up to isomorphism, 4 strings: 001011 (110100), 001101 (110010), 010011 (101100) and 011001 (100110).
For n=4, the optimal partition is {1,1,2}, generating, up to isomorphism, 9 strings: 00101011 (11010100), 00101101 (11010010), 00110101 (11001010), 01001011 (10110100), 01001101 (10110010), 01010011 (10101100), 01011001 (10100110), 01100101 (10011010) and 01101001 (10010110).

Cf. A247651, A046952, A092266, A136404, A176471, A065886.

dualseq[p_]:=Factorial[Length[p]]^2/Apply[Times,Map[Factorial[Count[p,#1]]&,Range[Max[Length[p]]]]]^2
a[n_]:=Max[Map[dualseq,IntegerPartitions[n]]]
Table[a[n],{n,0,25}] (* after A130670 *)

a[n]=Max[m!^2/(Prod_j(m_j!))^2] where Sum_j(m_j)=m, Sum_j(j*m_j)=n, over all partitions of n.

A364726 Admirable numbers with more divisors than any smaller admirable number.

Original entry on oeis.org

12, 24, 84, 120, 672, 24384, 43065, 78975, 81081, 261261, 523776, 9124731, 13398021, 69087249, 91963648, 459818240, 39142675143, 51001180160

Offset: 1

Amiram Eldar, Aug 05 2023

The corresponding numbers of divisors are 6, 8, 12, 16, 24, 28, 32, 36, 40, 48, 80, 90, 96, 120, 144, 288, 360, 480, ... .
If there are infinitely many even perfect numbers (A000396), then this sequence is infinite, because if p is a Mersenne prime exponent (A000043) and q is an odd prime that does not divide 2^p-1, then 2^(p-1)*(2^p-1)*q is an admirable number with 4*p divisors (see A165772).
a(19) > 10^11.

Cf. A000005, A000043, A000396, A109745, A111592, A165772.

Similar sequences: A002182, A136404, A335008, A335317, A348198, A359963, A359964.

admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2];
seq[kmax_] := Module[{s = {}, dm = 0, d1}, Do[d1 = DivisorSigma[0, k]; If[d1 > dm && admQ[k], dm = d1; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6]

isadm(n) = {my(ab=sigma(n)-2*n); ab>0 && ab%2 == 0 && ab/2 < n && n%(ab/2) == 0;}
lista(kmax) = {my(dm = 0, d1); for(k = 1, kmax, d1 = numdiv(k); if(d1 > dm && isadm(k), dm = d1; print1(k,", ")));}

cp's OEIS Frontend

A176471 Duplicate of A136404.

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A166721 Squares for which no smaller square has the same number of divisors.

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A166722 a(n) is the number of divisors of A166721(n).

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A250029 Maximum number of binary strings with symmetrically partitioned n 1's and n 0's, counted up to isomorphism.

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Mathematica

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A364726 Admirable numbers with more divisors than any smaller admirable number.

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