A059796 -id:A059796 - cp's OEIS frontend

A331591 a(n) is the number of distinct prime factors of A225546(n), or equally, number of distinct prime factors of A293442(n).

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

a(n) is the number of terms in the unique factorization of n into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between this factorization, canonical (prime power) factorization and A225546.
The result depends only on the prime signature of n.
a(n) is the number of distinct bit-positions where there is a 1-bit in the binary representation of an exponent in the prime factorization of n. - Antti Karttunen, Feb 05 2020
The first 3 is a(128) = a(2^1 * 2^2 * 2^4) = 3 and in general each m occurs first at position 2^(2^m-1) = A058891(m+1). - Peter Munn, Mar 07 2022

			From _Peter Munn_, Jan 28 2020: (Start)
The factorization of 6 into powers of squarefree numbers with distinct exponents that are powers of 2 is 6 = 6^(2^0) = 6^1, which has 1 term. So a(6) = 1.
Similarly, 40 = 10^(2^0) * 2^(2^1) = 10^1 * 2^2 = 10 * 4, which has 2 terms. So a(40) = 2.
Similarly, 320 = 5^(2^0) * 2^(2^1) * 2^(2^2) = 5^1 * 2^2 * 2^4 = 5 * 4 * 16, which has 3 terms. So a(320) = 3.
10^100 (a googol) factorizes in this way as 10^4 * 10^32 * 10^64. So a(10^100) = 3.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A000120, A005117, A048675, A225546, A267116, A293442, A293447, A299090, A329332, A337533, A337534.
Sequences with related definitions: A001221, A331309, A331592, A331593, A331740.
Positions of records: A058891.
Positions of 1's: A340682.
Sequences used to express relationships between the terms: A007913, A008833, A059796, A331590.

Array[PrimeNu@ If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] (* Michael De Vlieger, Jan 24 2020 *)
f[e_] := Position[Reverse[IntegerDigits[e, 2]], 1] // Flatten; a[n_] := CountDistinct[Flatten[f /@ FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Dec 23 2023 *)

A331591(n) = if(1==n,0,my(f=factor(n),u=#binary(vecmax(f[, 2])),xs=vector(u),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),xs[i]++)); m<<=1); #select(x -> (x>0),xs));

A331591(n) = if(1==n, 0, hammingweight(fold(bitor, factor(n)[, 2]))); \\ Antti Karttunen, Feb 05 2020

A331591(n) = if(n==1, 0, (core(n)>1) + A331591(core(n,1)[2])) \\ Peter Munn, Mar 08 2022

a(n) = A001221(A293442(n)) = A001221(A225546(n)).
From Peter Munn, Jan 28 2020: (Start)
a(n) = A000120(A267116(n)).
a(n) = a(A007913(n)) + a(A008833(n)).
For m >= 2, a(A005117(m)) = 1.
a(n^2) = a(n).
(End)
a(n) <= A331740(n) <= A048675(n) <= A293447(n). - Antti Karttunen, Feb 05 2020
From Peter Munn, Mar 07 2022: (Start)
a(n) <= A299090(n).
a(A337533(n)) = A299090(A337533(n)).
a(A337534(n)) < A299090(A337534(n)).
max(a(n), a(k)) <= a(A059796(n, k)) = a(A331590(n, k)) <= a(n) + a(k).
(End)

A003877 Degrees of irreducible representations of symmetric group S_13.

Original entry on oeis.org

1, 1, 12, 12, 65, 65, 66, 66, 208, 208, 220, 220, 429, 429, 429, 429, 429, 429, 495, 495, 572, 572, 792, 792, 924, 936, 936, 1287, 1287, 1365, 1365, 1430, 1430, 2574, 2574, 2574, 2574, 2860, 2860, 3003, 3003, 3432, 3432, 3432, 3432, 3432, 3432, 3575, 3575, 3640, 3640, 4004, 4004, 4212, 4212, 4290, 4290, 5005, 5005, 5148, 5148, 5720, 5720, 6006, 6006, 6435, 6435, 6864, 6864, 7371, 7371, 7800, 7800, 8580, 8580, 8580, 9009, 9009, 9360, 9360, 10296, 10296, 11440, 11440, 11583, 11583, 12012, 12012, 12012, 12012, 12870, 12870, 15015, 15015, 16016, 17160, 17160, 20592, 20592, 21450, 21450

Offset: 1

N. J. A. Sloane

All 101 terms of this finite sequence are shown.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Cf. A003869 - A003876, A059796.

Row n=13 of A060240.

A003877 := List(Irr(CharacterTable("S13")), chi->chi[1]);; Sort(A003877); # Eric M. Schmidt, Jul 18 2012

Magma
```
// See A003875 for Magma code.
```

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i < 1, 0, Flatten@ Table[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]];
T[n_] := g[n, n, {}];
Sort[T[13]] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

More terms from Emeric Deutsch, May 13 2004

cp's OEIS Frontend

A331591 a(n) is the number of distinct prime factors of A225546(n), or equally, number of distinct prime factors of A293442(n).

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A003877 Degrees of irreducible representations of symmetric group S_13.

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