A072273 -id:A072273 - cp's OEIS frontend

A357905 a(n) = log_3(A060839(n)).

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

Offset: 1

Jianing Song, Oct 19 2022

3-rank of the multiplicative group of integers modulo n.

			a(63) = 2 since (Z/63Z)* = C_6 X C_6 has 3-rank 2.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A060839, A072273, A357906.

f[p_, e_] := If[Mod[p, 3] == 1, 1, 0]; f[3, e_] := 1; f[3, 1] = 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 05 2023 *)

a(n)=my(f=factor(n)); sum(i=1, #f~, if(f[i, 1]==3, min(f[i, 2]-1, 1), if(f[i, 1]%3==1, 1, 0)))

from sympy import factorint
def A357905(n): return sum(1 for p, e in factorint(n).items() if (p!=3 or e!=1) and p%3!=2) # Chai Wah Wu, Oct 19 2022

Additive with a(3) = 0, a(3^e) = 1, e >= 2; a(p^e) = 1 for p == 1 (mod 3), 0 for p == 2 (mod 3).

A357906 a(n) = log_2(A073103(n)).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 3, 3, 2, 1, 1, 3, 2, 1, 1, 3, 2, 2, 1, 2, 2, 3, 1, 3, 2, 2, 3, 2, 2, 1, 3, 4, 2, 2, 1, 2, 3, 1, 1, 4, 1, 2, 3, 3, 2, 1, 3, 3, 2, 2, 1, 4, 2, 1, 2, 3, 4, 2, 1, 3, 2, 3, 1, 3, 2, 2, 3, 2, 2, 3, 1, 5, 1, 2, 1, 3, 4, 1, 3, 3, 2, 3, 3, 2, 2, 1, 3, 4, 2, 1, 2, 3

Offset: 1

Jianing Song, Oct 19 2022

			a(16) = 3 since x^4 == 1 (mod 16) has 2^3 = 8 solutions.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A073103, A072273, A357905.

f[p_, e_] := If[Mod[p, 4] == 1, 2, 1]; f[2, e_] := Switch[e, 1, 0, 2, 1, 3, 2, , 3]; a[1] = 0; a[n] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 05 2023 *)

a(n)=my(f=factor(n)); sum(i=1, #f~, if(f[i, 1]==2, min(f[i, 2]-1, 3), if(f[i, 1]%4==1, 2, 1))) \\ after Charles R Greathouse IV's program for A073103

Additive with a(2) = 0, a(4) = 1, a(8) = 2, a(2^e) = 3, e >= 4; a(p^e) = 2 for p == 1 (mod 4), 1 for p == 3 (mod 4).

A332761 Exponents m such that the number of nonnegative k <= n, possessing the property that n + n*k - k is a square, is equal to 2^m.

Original entry on oeis.org

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

Offset: 0

Juri-Stepan Gerasimov, Feb 23 2020

nonn

Where records occur gives 0, 1, 9, 25, 121, 841, 9241, ...

			a(0) = 0 because 0 + 0*0 - 0 = 1 = 1^2 and 1 = 2^0.
a(1) = 1 because 1 + 1*0 - 0 = 1 = 1^2, 1 + 1*1 - 1 = 1^2 and 2 = 2^1.
a(9) = 2 because 9 + 9*0 - 0 = 9 = 3^2, 9 + 9*2 - 2 = 25 = 5^2, 9 + 9*8 - 8 = 64 = 8^2, 9 + 9*9 - 9 = 81 = 9^2 and 4 = 2^2.

Cf. A000290, A072273, A060594, A332802.

[[m: m in [0..n] | #[k: k in [0..n] | IsSquare(n+n*k-k)] eq 2^m]: n in [0..100]];

a(n+2) is the exponent r if 2^r is equal to the number of squares of the form k + k*n - n, 0 <= k <= n.

a(n) = A072273(n-1). - Jinyuan Wang, Feb 25 2020

a(70) corrected by Jinyuan Wang, Feb 25 2020

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A357905 a(n) = log_3(A060839(n)).

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A357906 a(n) = log_2(A073103(n)).

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A332761 Exponents m such that the number of nonnegative k <= n, possessing the property that n + n*k - k is a square, is equal to 2^m.

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