A340682 -id:A340682 - 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)

A352780 Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, such that the row product is n and column k contains only (2^k)-th powers of squarefree numbers.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14

Offset: 1

Antti Karttunen and Peter Munn, Apr 02 2022

This is well-defined because positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2.
Each (infinite) row is the lexicographically earliest with product n and terms that are a (2^k)-th power for all k.
For all k, column k is column k+1 of A060176 conjugated by A225546.

			The top left corner of the array:
  n/k |   0   1   2   3   4   5   6
------+------------------------------
    1 |   1,  1,  1,  1,  1,  1,  1,
    2 |   2,  1,  1,  1,  1,  1,  1,
    3 |   3,  1,  1,  1,  1,  1,  1,
    4 |   1,  4,  1,  1,  1,  1,  1,
    5 |   5,  1,  1,  1,  1,  1,  1,
    6 |   6,  1,  1,  1,  1,  1,  1,
    7 |   7,  1,  1,  1,  1,  1,  1,
    8 |   2,  4,  1,  1,  1,  1,  1,
    9 |   1,  9,  1,  1,  1,  1,  1,
   10 |  10,  1,  1,  1,  1,  1,  1,
   11 |  11,  1,  1,  1,  1,  1,  1,
   12 |   3,  4,  1,  1,  1,  1,  1,
   13 |  13,  1,  1,  1,  1,  1,  1,
   14 |  14,  1,  1,  1,  1,  1,  1,
   15 |  15,  1,  1,  1,  1,  1,  1,
   16 |   1,  1, 16,  1,  1,  1,  1,
   17 |  17,  1,  1,  1,  1,  1,  1,
   18 |   2,  9,  1,  1,  1,  1,  1,
   19 |  19,  1,  1,  1,  1,  1,  1,
   20 |   5,  4,  1,  1,  1,  1,  1,

Antti Karttunen, Table of n, a(n) for n = 1..33153; the first 257 antidiagonals

Sequences used in a formula defining this sequence: A000188, A007913, A060176, A225546.
Cf. A007913 (column 0), A335324 (column 1).
Range of values: {1} U A340682 (whole table), A005117 (column 0), A062503 (column 1), {1} U A113849 (column 2).
Row numbers of rows:
- with a 1 in column 0: A000290\{0};
- with a 1 in column 1: A252895;
- with a 1 in column 0, but not in column 1: A030140;
- where every 1 is followed by another 1: A337533;
- with 1's in all even columns: A366243;
- with 1's in all odd columns: A366242;
- where every term has an even number of distinct prime factors: A268390;
- where every term is a power of a prime: A268375;
- where the terms are pairwise coprime: A138302;
- where the last nonunit term is coprime to the earlier terms: A369938;
- where the last nonunit term is a power of 2: A335738.
Number of nonunit terms in row n is A331591(n); their positions are given (in reversed binary) by A267116(n); the first nonunit is in column A352080(n)-1 and the infinite run of 1's starts in column A299090(n).

up_to = 105;
A352780sq(n, k) = if(k==0, core(n), A352780sq(core(n, 1)[2], k-1)^2);
A352780list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, forstep(col=a-1,0,-1, i++; if(i > up_to, return(v)); v[i] = A352780sq(a-col,col))); (v); };
v352780 = A352780list(up_to);
A352780(n) = v352780[n];

A(n,0) = A007913(n); for k > 0, A(n,k) = A(A000188(n), k-1)^2.
A(n,k) = A225546(A060176(A225546(n), k+1)).
A331591(A(n,k)) <= 1.

A340675 Exponential of Mangoldt function conjugated by Tek's flip: a(n) = A225546(A014963(A225546(n))).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 1, 4, 2, 2, 1, 2, 2, 2, 16, 2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 16, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 4, 2, 2, 2, 1, 2

Offset: 1

Antti Karttunen and Peter Munn, Feb 01 2021

nonn

Nonunit squarefree numbers take the value 2, other nonsquares take the value 1, and squares take the square of the value taken by their square root.

Sequences used in a definition of this sequence: A014963, A048298, A225546, A267116, A297108, A340676.
Cf. A051903, A051144, A331591, A340673.
Positions of 1's: {1} U A340681, 2's: A005117 \ {1}, of 4's: A062503 \ {1}, of 16's: A113849.
Positions of terms > 1: A340682, of terms > 2: A340674.
Sequences used to express relationship between terms of this sequence: A003961, A331590.

A340675(n) = if(1==n,n,if(issquarefree(n), 2, if(!issquare(n), 1, A340675(sqrtint(n))^2)));

a(n) = A225546(A340673(n)) = A225546(A014963(A225546(n))).
a(n) = 2^A048298(A267116(n)).
If A340673(n) = 1, then a(n) = 1, otherwise a(n) = 2^A297108(A340673(n)).
If A340676(n) = 0, then a(n) = 1, otherwise a(n) = 2^(2^(A340676(n)-1)).
If n = s^(2^k), s squarefree >= 2, k >= 0, then a(n) = 2^(2^k), otherwise a(n) = 1.
For n, k > 1, if a(n) = a(k) then a(A331590(n, k)) = a(n), otherwise a(A331590(n, k)) = 1.
a(n^2) = a(n)^2.
a(A003961(n)) = a(n).
a(A051144(n)) = 1.
a(n) = 1 if and only if A331591(n) <> 1, otherwise a(n) = 2^A051903(n).

A340674 Numbers of the form s^(2^e), where s is a squarefree number, and e >= 1.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 81, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 625, 676, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2401, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900

Offset: 1

Antti Karttunen, Jan 31 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000079, A005117, A050376, A209229, A302777.
Positions of terms larger than 2 in A340673 (also in A340675), and of terms larger than 1 in A340676.
Subsequence of A072777 and of A340682.

Select[Range[10^4], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] > 1 && u[[1]] == 2^IntegerExponent[u[[1]], 2] &] (* Amiram Eldar, Feb 08 2021 *)

A209229(n) = (n && !bitand(n,n-1));
isA340674(n) = { my(b,e); (((e=ispower(n,,&b))>0)&&issquarefree(b)&&A209229(e)); };

from math import isqrt
from sympy import mobius, integer_nthroot
def A340674(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n+x-sum(g(integer_nthroot(x,1<Chai Wah Wu, Jun 01 2025

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2^k)/zeta(2^(k+1))-1) = 0.6018231854... - Amiram Eldar, Feb 08 2021

A340681 The closure under squaring of A051144, the nonsquarefree nonsquares.

Original entry on oeis.org

8, 12, 18, 20, 24, 27, 28, 32, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208

Offset: 1

Antti Karttunen and Peter Munn, Feb 07 2021

Numbers not of the form s^(2^e), where s is a squarefree number, and e >= 0.
The categorization provided by this sequence and its complement, A340682, is an alternative extension (to all integers greater than 1) of the 2-way distinction between squarefree and nonsquarefree as it applies to nonsquares.
All positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. This sequence lists the numbers where this factorization has more than one term, that is numbers m such that A331591(m) > 1.
Presence in the sequence is determined by prime signature (A101296). The set of represented signatures starts: {{3}, {2,1}, {3,1}, {2,1,1}, {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1}, {6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,1,1,1}, {2,2,1,1}, {2,1,1,1,1}, {7}, ...}.
Gives positions of 1's in A340675 after its initial one.

			24 = 6 * 4 = 6^1 * 2^2 = 6^(2^0) * 2^(2^1), which is the factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. As this factorization has 2 terms, 24 is in the sequence.
The equivalent factorization for 100 is 100 = 10^2 = 10^(2^1). As this factorization has only 1 term, 100 is not in the sequence.

Index entries for sequences related to prime signature

Cf. A101296, A331591, A340675.
Cf. A340682 (complement, apart from 1 which is in neither).
Cf. subsequences: A051144, A059404.
Subsequence of A013929.

isA340681(n) = if(!issquare(n), !issquarefree(n), (n>1)&&isA340681(sqrtint(n)));

from math import isqrt
from sympy import mobius, integer_nthroot
def A340681(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+1+sum(g(integer_nthroot(x,1<Chai Wah Wu, Jun 01 2025

A340673 If n is of the form s^(2^e), where s is a squarefree number, and e >= 0, then a(n) is the (1+e)-th prime, otherwise a(n) = 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 2, 5, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 5, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 3, 2, 2, 2, 1, 2

Offset: 1

Antti Karttunen, Jan 31 2021

nonn

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

Cf. A000040, A008578, A014963, A225546, A340675, A340676.
Positions of 1's: {1} U A340681, of 2's: A005117 \ {1}, of 3's: A062503 \ {1}, of 5's: A113849.
Positions of terms > 1: A340682, of terms > 2: A340674.

A008578(n) = if(1==n,n,prime(n-1));
A340673(n) = A008578(1+A340676(n)); \\ Uses also code from A340676.

a(n) = A014963(A225546(n)).
a(n) = A225546(A340675(n)).
a(n) = A008578(1+A340676(n)).
If n is of the form s^(2^e), where s is a squarefree number, and e >= 0, then a(n) = A000040(1+e), otherwise a(n) = 1.

A340676 If n is of the form s^(2^e), where s is a squarefree number, and e >= 0, then a(n) = 1+e, otherwise a(n) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 3, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 3, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 0, 1

Offset: 1

Antti Karttunen, Feb 01 2021

nonn

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

Cf. A005117, A104117, A225546, A267116, A297109, A340673, A340675.
Positions of zeros: {1} U A340681, of 1's: A005117 \ {1}, of 2's: A062503 \ {1}, of 3's: A113849.
Positions of nonzero terms: A340682, of terms > 1: A340674.

a[1] = 0; a[n_] := If[Length[(u = Union[FactorInteger[n][[;; , 2]]])] == 1 && u[[1]] == 2^(e = IntegerExponent[u[[1]], 2]), e + 1, 0]; Array[a, 100] (* Amiram Eldar, Feb 10 2021 *)

A001511(n) = 1+valuation(n,2);
A209229(n) = (n && !bitand(n,n-1));
A104117(n) = (A209229(n)*A001511(n));
A267116(n) = if(n>1, fold(bitor, factor(n)[, 2]), 0);
A340676(n) = if(1==n,0,A104117(A267116(n)));

a(n) = A297109(A225546(n)).

For n > 1, a(n) = A104117(A267116(n)). - Peter Munn, Feb 05 2021

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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A352780 Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, such that the row product is n and column k contains only (2^k)-th powers of squarefree numbers.

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A340675 Exponential of Mangoldt function conjugated by Tek's flip: a(n) = A225546(A014963(A225546(n))).

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A340674 Numbers of the form s^(2^e), where s is a squarefree number, and e >= 1.

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A340681 The closure under squaring of A051144, the nonsquarefree nonsquares.

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A340673 If n is of the form s^(2^e), where s is a squarefree number, and e >= 0, then a(n) is the (1+e)-th prime, otherwise a(n) = 1.

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A340676 If n is of the form s^(2^e), where s is a squarefree number, and e >= 0, then a(n) = 1+e, otherwise a(n) = 0.

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