A333576 -id:A333576 - cp's OEIS frontend

A384046 Triangle in which the n-th row gives the numbers from 1 to n whose largest divisor that is a unitary divisor of n is 1.

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 3, 7, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 5, 7, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 3, 5, 9, 11, 13, 1, 2, 4, 7, 8, 11, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 1

Amiram Eldar, May 18 2025

			Triangle begins:
  1,
  1,
  1, 2,
  1, 2, 3,
  1, 2, 3, 4,
  1, 5,
  1, 2, 3, 4, 5, 6,
  1, 2, 3, 4, 5, 6, 7,
  1, 2, 3, 4, 5, 6, 7, 8,
  1, 3, 7, 9

Amiram Eldar, Table of n, a(n) for n = 1..10204 (first 170 rows flattened)
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.

The unitary analog of A038566.

Cf. A047994 (row lengths), A333576 (row sums), A077610, A225174, A384047.

udiv[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; uGCD[n_, k_] := Max[Intersection[udiv[n], Divisors[k]]]; row[n_] := Select[Range[n], uGCD[n, #] == 1 &]; Array[row, 10] // Flatten

udiv(n) = select(x -> gcd(x, n/x) == 1, divisors(n));
ugcd(n, k) = vecmax(setintersect(udiv(n), divisors(k)));
row(n) = select(x -> ugcd(n, x) == 1, vector(n, i, i));

T(n, 1) = 1.

A384248 The sum of the integers from 1 to n whose largest divisor that is an infinitary divisor of n is 1.

Original entry on oeis.org

1, 1, 3, 6, 10, 6, 21, 16, 36, 20, 55, 36, 78, 42, 60, 120, 136, 72, 171, 120, 126, 110, 253, 96, 300, 156, 243, 252, 406, 120, 465, 256, 330, 272, 420, 432, 666, 342, 468, 320, 820, 252, 903, 660, 720, 506, 1081, 720, 1176, 600, 816, 936, 1378, 486, 1100, 672

Offset: 1

Amiram Eldar, May 23 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Row sums of A384246.
Cf. A138302, A268335, A384247.
Analogous sequences: A023896, A200723, A333576.

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); a[1] = 1; a[n_] := (n/2) * Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = if(n == 1, 1, my(f = factor(n)); n^2 * prod(i = 1, #f~, (1 - 1/f[i,1]^(1 << valuation(f[i,2], 2)))) / 2);

a(n) = n * A384247(n) / 2, for n >= 2.
a(n) <= A333576(n), with equality if and only if n is in A138302.
a(n) >= A023896(n), with equality if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) ~ c * n^2 / 6, where c = Product_{p prime} f(1/p) = 0.66718130416373472394..., and f(x) = 1 - (1-x)*Sum_{k>=1} x^(2^k)/(1-x^(2^k)).

cp's OEIS Frontend

A384046 Triangle in which the n-th row gives the numbers from 1 to n whose largest divisor that is a unitary divisor of n is 1.

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