A046055 -id:A046055 - cp's OEIS frontend

A046054 Incrementally largest number of nonisomorphic finite Abelian groups as a function of order.

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 303, 385, 405, 490, 505, 528, 539, 627, 675, 693, 707, 792, 880, 891, 945, 1002, 1155, 1232, 1255, 1485, 1617, 1925, 1936, 2079, 2450, 2541, 2695, 3135, 3267, 3430, 3465, 3960, 4235

Offset: 1

Eric W. Weisstein

nonn

Records in A000688. - Artur Jasinski, Mar 14 2008

Amiram Eldar, Table of n, a(n) for n = 1..1111 (terms 1..216 from Charlie Neder)
Eric Weisstein's World of Mathematics, Abelian Group
Index entries for sequences related to groups

Cf. A000688, A046055, A046056.

a = {}; b = 0; f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; k = Array[f, 1000000]; Do[If[k[[m]] > b, b = k[[m]]; AppendTo[a, b]], {m, 1, Length[k]}]; a (* using a procedure from Robert G. Wilson v; Artur Jasinski, Mar 14 2008 *)

a(n) = A000688(A046055(n)). - Amiram Eldar, Aug 20 2019

More terms from David Wasserman, Feb 06 2002

A146541 Binomial transform of A010688.

Original entry on oeis.org

1, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Philippe Deléham, Oct 31 2008

Hankel transform is := 1,-48,0,0,0,0,0,0,0,...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000007, A011782, A000079, A003945, A046055, A146523, A082505, A135092.

Essentially the same as A132479, A077552, A020707.

Join[{1},2^Range[3,40]] (* Harvey P. Dale, Feb 28 2016 *)

Vec((1+6*x)/(1-2*x) + O(x^50)) \\ Colin Barker, Mar 17 2016

a(0)=1, a(n) = 2^(n+2) for n>0.
a(n) = Sum_{k, 0..n} A109466(n,k)*A146534(k).
a(n) = A132479(n), n>1. - R. J. Mathar, Nov 02 2008
G.f.: (1+6*x) / (1-2*x). - Colin Barker, Mar 17 2016

Corrected and extended by Harvey P. Dale, Feb 28 2016

A131127 Table read by rows: 2*A007318(n,m) - A167374(n,m).

Original entry on oeis.org

1, 3, 1, 2, 5, 1, 2, 6, 7, 1, 2, 8, 12, 9, 1, 2, 10, 20, 20, 11, 1, 2, 12, 30, 40, 30, 13, 1, 2, 14, 42, 70, 70, 42, 15, 1, 2, 16, 56, 112, 140, 112, 56, 17, 1, 2, 18, 72, 168, 252, 252, 168, 72, 19, 1, 2, 20, 90, 240, 420, 504, 420, 240, 90, 21, 1, 2, 22, 110, 330, 660, 924, 924, 660, 330, 110, 23, 1

Offset: 0

Gary W. Adamson, Jun 16 2007

Row sums = A000079(n+1), n>0.
Warning: row sums are not A046055! - N. J. A. Sloane, Jul 08 2009
Row sums = A151821(n+1), n>=0. - Alois P. Heinz, Jul 13 2009
A167374 is a modified version of the pair operator A097806 with (1,1,1,...) in the main diagonal and (-1,-1,-1,...) in the subdiagonal.

			First few rows of the triangle:
  1;
  3,  1;
  2,  5,  1;
  2,  6,  7,  1;
  2,  8, 12,  9,  1;
  2, 10, 20, 20, 11,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A007318, A097806, A151821, A167374.

T:= (n, m)-> 2*binomial(n, m) -(-1)^(n+m)*`if`(n=m or n=m+1, 1, 0): seq(seq(T(n,m), m=0..n), n=0..12); # Alois P. Heinz, Jul 13 2009

T[n_, m_] := 2*Binomial[n, m] - (-1)^(n+m)*If[n == m || n == m+1, 1, 0];
Table[Table[T[n, m], {m, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 19 2016, translated from Maple *)

Edited by N. J. A. Sloane and R. J. Mathar, Jul 09 2009
Corrected and extended by Alois P. Heinz, Jul 13 2009
Definition simplified by Georg Fischer, Jun 07 2023

A384784 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

Original entry on oeis.org

1, 16, 64, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

The least term that is not a power of 2 is a(47) = 2^35 * 3^10.
Indices of records in A188585.
All the terms are powerful numbers since A188585(1) = 1 and A188585(n) = 0 if n is a nonpowerful number.
The corresponding record values are 1, 2, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..713
Amiram Eldar, Table of n, a(n), A188585(a(n)) for n = 1..713
Index entries for sequences related to powerful numbers.

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A025475, A046055, A188585, A384783, A384786 (cubefull analog).

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A384786 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

Original entry on oeis.org

1, 64, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

The least term that is not a power of 2 is a(85) = 2^61 * 3^18.
Indices of records of the multiplicative function f(n) with f(p^e) = A008483(e).
All the terms are cubefull numbers since f(1) = 1 and f(n) = 0 if n is a noncubefull number.
The corresponding record values are 1, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..567
Amiram Eldar, Table of n, a(n), and record values for n = 1..567
Index entries for sequences related to powerful numbers.

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A008483, A046055, A246549, A384784 (powerful analog), A384785.

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1] - PartitionsP[e-2] + PartitionsP[e-3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A384791 Numbers with a record number of ways in which they can be expressed as b^3 * c^4 * d^5, with b, c and d >= 1.

Original entry on oeis.org

1, 256, 4096, 32768, 262144, 1048576, 8388608, 16777216, 134217728, 268435456, 1073741824, 4294967296, 8589934592, 34359738368, 68719476736, 110075314176, 549755813888, 557256278016, 1761205026816, 4458050224128, 7044820107264, 8916100448256, 56358560858112, 71328803586048

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

The least term that is not a power of 2 is a(16) = 2^24 * 3^8.	
Indices of records of the multiplicative function f(n) with f(p^e) = A008680(e).
All the terms are cubefull numbers since f(1) = 1 and f(n) = 0 if n is a noncubefull number.
The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, ... (see the link for more values).
Every exponent must be the index of the first occurrence of A008680(e) in A008680. So possible exponents of prime factors of terms are 0, 8, 12, 15, 18, 20, ... - David A. Corneth, Jun 30 2025

			256 in the sequence as 256 = 1^3 * 4^4 * 1^5 = 2^3 * 1^4 * 2^5 so there are two ways to write 256 as b^3 * c^4 * d^5, with b, c and d >= 1 and no smaller positive integer can be written in at least two ways like that. - _David A. Corneth_, Jun 30 2025

David A. Corneth, Table of n, a(n) for n = 1..810 (first 216 terms from Amiram Eldar, terms <= 10^120)
David A. Corneth, Table of n, a(n), exponents of prime factors and record values for n = 1..810
Index entries for sequences related to powerful numbers.

Subsequence of A025487, A036966 and A181800.

Cf. A008680, A046055, A384789, A384790 (powerful analog).

f[p_, e_] := Floor[(1+(-1)^e)*(-1)^Floor[e/2]/8 + (e^2 + 12*e + 90)/120]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A384790 Numbers with a record number of ways in which they can be expressed as b^2 * c^3, with b and c >= 1.

Original entry on oeis.org

1, 64, 4096, 46656, 2985984, 191102976, 2176782336, 12230590464, 46656000000, 2985984000000, 34012224000000, 191102976000000, 2176782336000000, 139314069504000000, 351298031616000000, 4001504141376000000, 22483074023424000000, 256096265048064000000, 16390160963076096000000

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

Indices of records of A370256.
All the terms are powerful numbers since A370256(1) = 1 and A370256(n) = 0 if n is a nonpowerful number.
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..61.
Amiram Eldar, n, a(n), A370256(a(n)) for n = 1..61.
Index entries for sequences related to powerful numbers.

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A002182 (sixth root), A046055, A057523, A370256, A384791 (cubefull analog).

f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A074231 Numbers n such that Kronecker(8,n) = mu(gcd(8,n)).

Original entry on oeis.org

1, 4, 7, 8, 9, 12, 15, 16, 17, 20, 23, 24, 25, 28, 31, 32, 33, 36, 39, 40, 41, 44, 47, 48, 49, 52, 55, 56, 57, 60, 63, 64, 65, 68, 71, 72, 73, 76, 79, 80, 81, 84, 87, 88, 89, 92, 95, 96, 97, 100, 103, 104, 105, 108, 111, 112, 113, 116, 119, 120, 121, 124, 127, 128, 129

Offset: 1

Jon Perry, Sep 17 2002

nonn

A Chebyshev transform of (1+2x)/(1-2x) (A046055) given by G(x)->(1/(1+x^2))G(x/(1+x^2)). - Paul Barry, Oct 27 2004

Essentially the same as A047538.

for (x=1,200, for (y=1,200,if (kronecker(x,y)==moebius(gcd(x,y)),write("km.txt",x,";",y," : ",kronecker(x,y)))))

[lucas_number1(n+2, 0, 1)+2*n for n in range(1, 66)] # Zerinvary Lajos, Mar 09 2009

From Paul Barry, Oct 27 2004: (Start)
G.f.: (1+x)^2/((1+x^2)*(1-2x+x^2));
e.g.f.: exp(x)(2+2x) - cos(x);
a(n) = 2n + 2 - cos(Pi*n/2);
a(n) = Sum_{k=0..n} (0^k + 4^k)*cos(Pi*(n-k)/2);
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k(2*2^(n-2k)-0^(n-2k));
a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) - a(n-4). (End)

A124846 Triangle read by rows: T(n,k) = (2 - (-1)^k)*binomial(n,k) (0 <= k <= n).

Original entry on oeis.org

1, 1, 3, 1, 6, 1, 1, 9, 3, 3, 1, 12, 6, 12, 1, 1, 15, 10, 30, 5, 3, 1, 18, 15, 60, 15, 18, 1, 1, 21, 21, 105, 35, 63, 7, 3, 1, 24, 28, 168, 70, 168, 28, 24, 1, 1, 27, 36, 252, 126, 378, 84, 108, 9, 3, 1, 30, 45, 360, 210, 756, 210, 360, 45, 30, 1, 1, 33, 55, 495, 330, 1386, 462, 990

Offset: 0

Gary W. Adamson, Nov 10 2006

			First few rows of the triangle:
  1;
  1,  3;
  1,  6,  1;
  1,  9,  3,  3;
  1, 12,  6, 12,  1;
  1, 15, 10, 30,  5,  3;
  ...
A046055(4) = 16 = sum of row 4 terms (1 + 9 + 3 + 3).

Cf. A010684, A046055, A124845.

T:=(n,k)->(2-(-1)^k)*binomial(n,k): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Edited by N. J. A. Sloane, Nov 24 2006

A131131 4A007318 - 3A097806.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 4, 12, 9, 1, 4, 16, 24, 13, 1, 4, 20, 40, 40, 17, 1, 4, 24, 60, 80, 60, 21, 1, 4, 28, 84, 140, 140, 84, 25, 1, 4, 32, 112, 224, 280, 224, 112, 29, 1

Offset: 0

Gary W. Adamson, Jun 16 2007

Row sums = A131130, (1, 2, 10, 26, 52, 98, 190, ...), the binomial transform of (1, 1, 7, 1, 7, 1, ...). Generally, triangles generated from N*A007318 - (N-1)*A097806 have row sums that are binomial transforms of (1, 1, (N-1), 1, (N-1), 1, ...). A095121 = (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...) and = row sums of A131108.

Triangle T(n,k), 0 <= k <= n,read by rows given by [1,3,-4,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

			First few rows of the triangle:
  1;
  1,  1;
  4,  5,  1;
  4, 12,  9,  1;
  4, 16, 24, 13,  1
  4, 20, 40, 40, 17,  1;
  ...

Cf. A131130, A131129, A131128, A131127, A046055, A131108, A095121, A097806.

4*A007318 - 3*A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

G.f.: (1-x*y+3*x^2+3*x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015

cp's OEIS Frontend

A046054 Incrementally largest number of nonisomorphic finite Abelian groups as a function of order.

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A146541 Binomial transform of A010688.

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A131127 Table read by rows: 2*A007318(n,m) - A167374(n,m).

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A384784 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

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A384786 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

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A384791 Numbers with a record number of ways in which they can be expressed as b^3 * c^4 * d^5, with b, c and d >= 1.

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Mathematica

A384790 Numbers with a record number of ways in which they can be expressed as b^2 * c^3, with b and c >= 1.

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Mathematica

A074231 Numbers n such that Kronecker(8,n) = mu(gcd(8,n)).

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A131131 4A007318 - 3A097806.