A327988 -id:A327988 - cp's OEIS frontend

A327987 a(n) = Sum_{d|n} d & (n/d), where & is the bitwise AND operator, with a(0) = 0.

0, 1, 0, 2, 2, 2, 4, 2, 0, 5, 0, 2, 4, 2, 4, 4, 4, 2, 4, 2, 12, 8, 4, 2, 8, 7, 0, 4, 12, 2, 16, 2, 0, 8, 0, 12, 10, 2, 4, 4, 0, 2, 16, 2, 4, 10, 4, 2, 8, 9, 0, 4, 12, 2, 8, 4, 8, 8, 0, 2, 24, 2, 4, 6, 8, 12, 8, 2, 4, 8, 16, 2, 24, 2, 0, 14, 4, 8, 16, 2, 24, 17

Offset: 0

Peter Luschny, Oct 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A327988 (zeros), A327989, A016754.

using IntegerSequences
vcat([0], [V327987(n) for n in 1:81]) |> println  # Peter Luschny, Sep 25 2021

[0] cat [&+[BitwiseAnd(d,n div d):d in Divisors(n)]:n in [1..90]]; // Marius A. Burtea, Oct 11 2019

A327987 := n -> add(Bits:-And(d, n/d), d=numtheory:-divisors(n)):
seq(A327987(n), n=0..81);

divisors[0] := {}; divisors[n_] := Divisors[n];
a[n_] := Total[Table[BitAnd[d , n/d], {d, divisors[n]}]] ;
Table[a[n], {n, 0, 81}]

a(n) = if (n, sumdiv(n, d, bitand(d, n/d)), 0); \\ Michel Marcus, Oct 11 2019

def a(n): return sum(d & n//d for d in divisors(n)) if n > 0 else 0
print([a(n) for n in (0..81)])

a(n) is odd if and only if n is an odd square (A016754).

A328176 a(n) is the maximal value of the expression d AND (n/d) where d runs through the divisors of n and AND denotes the bitwise AND operator.

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 1, 0, 3, 0, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 3, 2, 1, 4, 5, 0, 1, 4, 1, 4, 1, 0, 3, 0, 5, 6, 1, 2, 1, 0, 1, 6, 1, 2, 3, 2, 1, 4, 7, 0, 1, 4, 1, 2, 1, 4, 3, 0, 1, 4, 1, 2, 1, 8, 5, 2, 1, 2, 3, 4, 1, 8, 1, 0, 5, 2, 3, 4, 1, 8, 9, 0, 1, 6, 1, 2, 1

Offset: 1

Rémy Sigrist, Oct 06 2019

			For n = 12:
- we have the following values:
    d   12/d  d AND (12/d)
    --  ----  ------------
     1    12             0
     2     6             2
     3     4             0
     4     3             0
     6     2             2
    12     1             0
- hence a(12) = max({0, 2}) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..16384
Rémy Sigrist, Scatterplot of the first 2^16 terms

See A328177 and A328178 for similar sequences.

Cf. A327987, A327988.

a:= n-> max(map(d-> Bits[And](d, n/d), numtheory[divisors](n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 09 2019

a(n) = vecmax(apply(d -> bitand(d, n/d), divisors(n)))

a(n)^2 <= n with equality iff n is a square.
a(n) = 1 for any odd prime number p.
a(n) <= A327987(n).
a(n) = 0 iff n belongs to A327988.

A327989 Nonnegative integers k such that Sum_{d|k} k & (k/d) is an odd prime, where & is the bitwise AND operator.

Original entry on oeis.org

9, 25, 81, 121, 289, 625, 729, 841, 1681, 3249, 3481, 5041, 7225, 8281, 8649, 9025, 10201, 11449, 13689, 15129, 18769, 19881, 22201, 25281, 27225, 28561, 29241, 31329, 32041, 33489, 34225, 36481, 38025, 38809, 42849, 46225, 48841, 51529, 53361, 55225, 56169, 57121

Offset: 1

Peter Luschny, Oct 11 2019

nonn

A subsequence of the odd squares A016754.

Cf. A327987, A327988, A016754.

[k:k in [1..60000]|IsOdd(a) and IsPrime(a) where a is &+[BitwiseAnd(d,k div d):d in Divisors(k)]]; // Marius A. Burtea, Oct 11 2019

select(k -> (A327987(k) <> 2 and isprime(A327987(k))), [$0..60000]);

cp's OEIS Frontend

A327987 a(n) = Sum_{d|n} d & (n/d), where & is the bitwise AND operator, with a(0) = 0.

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A328176 a(n) is the maximal value of the expression d AND (n/d) where d runs through the divisors of n and AND denotes the bitwise AND operator.

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A327989 Nonnegative integers k such that Sum_{d|k} k & (k/d) is an odd prime, where & is the bitwise AND operator.

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Magma

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