A296086 -id:A296086 - cp's OEIS frontend

A324886 a(n) = A276086(A108951(n)).

2, 3, 5, 9, 7, 25, 11, 15, 35, 49, 13, 625, 17, 121, 117649, 225, 19, 1225, 23, 2401, 1771561, 169, 29, 875, 717409, 289, 55, 14641, 31, 184877, 37, 21, 4826809, 361, 36226650889, 1500625, 41, 529, 24137569, 77, 43, 143, 47, 28561, 1127357, 841, 53, 1715, 902613283, 514675673281, 47045881, 83521, 59, 3025, 8254129, 214358881, 148035889, 961, 61

Offset: 1

Antti Karttunen, Mar 30 2019

nonn

Cf. A034386, A108951, A117366, A276086, A324887, A324888, A324896, A329047, A342920, A344592, A346106, A346107.
Permutation of A324576.
Cf. also A346105.

With[{b = MixedRadix[Reverse@ Prime@ Range@ 120]}, Array[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]], b] &, 58]] (* Michael De Vlieger, Nov 18 2019 *)
A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
(* b is A108951 *)
b[n_] := b[n] = Module[{pe = FactorInteger[n], p, e}, If[Length[pe] > 1, Times @@ b /@ Power @@@ pe, {{p, e}} = pe; Times @@ (Prime[Range[ PrimePi[p]]]^e)]]; b[1] = 1;
a[n_] := A276086[b[n]];
Array[a, 100] (* Jean-François Alcover, Dec 01 2021, after _Antti Karttunen in A296086 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324886(n) = A276086(A108951(n));

a(n) = A276086(A108951(n)).
a(n) = A117366(n) * A324896(n).
A001222(a(n)) = A324888(n).
A020639(a(n)) = A117366(n).
A032742(a(n)) = A324896(n).
a(A000040(n)) = A000040(1+n).
From Antti Karttunen, Jul 09 2021: (Start)
For n > 1, a(n) = A003961(A329044(n)).
a(n) = A346091(n) * A344592(n).
a(n) = A346106(n) /  A346107(n).
A003415(a(n)) = A329047(n).
A003557(a(n)) = A344592(n).
A342001(a(n)) = A342920(n) = A329047(n) / A344592(n).
(End)

A322073 Number of binary self-dual codes that share the most common weight distribution for binary self-dual codes of length 2n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 10, 16, 53, 553, 14536

Offset: 1

Nathan J. Russell, Nov 25 2018

The weight distribution list [w_0,w_1,...,w_30] = [1,0,0,6,0,53,0,339,0,1782,5284,0,8919,...,1] where w_i is the number of codewords of weight i, represents the most common weight distribution for the binary self-dual codes of length 2*15 = 30. There are a(15) = 16 binary self-dual codes that have this weight enumerator. Ellipses are used to shorten the list since the list is symmetrical (i.e., w_n = w_{30-n}).

The indexing of the list is 2n since there are no binary self-dual codes of odd length.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 7,252-282,338-393.

Cf. A296086, A321969.

A322074 Maximum number of codewords a binary self-dual code of length 4n can have with Hamming weight 2n (half of length).

Original entry on oeis.org

2, 14, 32, 198, 512, 2972, 8192, 45638, 131072

Offset: 1

Nathan J. Russell, Nov 25 2018

All binary self-dual codes of length 2*18 = 36 have a(9) = 131072 or fewer codewords with Hamming weight 18. In fact, there is only one binary self-dual code of length 36 that has 131072 codewords with Hamming weight 18.
There is at least one binary self-dual code of length 4n having a(n) codewords of weight 2n. However, the code may not be unique. There are two binary self-dual codes of length 4*4=16 having a(4)=198 codewords with Hamming weight 2*4=8.
All binary self-dual codes must be even length and all codewords must have an even Hamming weight. Only codewords with a length that is a multiple of 4 can have codewords with a Hamming weight equal to half the length of the code.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, 2003, Page 7, 252-330, 338-393.

Cf. A322073, A321969, A296086, A001405, A000984.

cp's OEIS Frontend

A324886 a(n) = A276086(A108951(n)).

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A322073 Number of binary self-dual codes that share the most common weight distribution for binary self-dual codes of length 2n.

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A322074 Maximum number of codewords a binary self-dual code of length 4n can have with Hamming weight 2n (half of length).

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