A035035 -id:A035035 - cp's OEIS frontend

A056763 Number of integers in the range (2^(n-1), 2^n] for which d(k)^3 > k holds, i.e., the cube of the number of divisors of k exceeds the number k.

1, 2, 4, 6, 11, 24, 30, 60, 110, 137, 248, 399, 491, 801, 1146, 1386, 1988, 2525, 2914, 3637, 4081, 4334, 4649, 4579, 4305, 3867, 3211, 2467, 1730, 1119, 592, 272, 104, 28, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Aug 16 2000

nonn

a(n) = 0 for n >= 36 since A056757 is finite and its last term is 27935107200 < 2^35. - Amiram Eldar, Jun 02 2024

			a(5) = 11 because 11 integers, {18,20,21,22,24,25,26,27,28,30,32} occur between 1+2^4 = 17 and 2^5 = 32 for which the cube of number of divisors exceeds the number itself.
Between 2^28 and 2^29, 1730 such numbers occur, so a(29) = 1730.

Cf. A000005, A029837, A034884, A035033-A035035, A056757-A056767.

With[{s = Import["https://oeis.org/A056757/b056757.txt", "Table"][[;; , 2]]}, a[n_] := Count[s, ?(2^(n-1) < # <= 2^n &)]; Table[a[n], {n, 1, 35}]] (* _Amiram Eldar, Jun 02 2024 *)

a(30)-a(32) from Sean A. Irvine, May 06 2022

More terms from Amiram Eldar, Jun 02 2024

A056764 Number of integers k not exceeding 2^n such that the cube of number of divisors [A000005(k)] is larger than k.

Original entry on oeis.org

1, 3, 7, 13, 24, 48, 78, 138, 248, 385, 633, 1032, 1523, 2324, 3470, 4856, 6844, 9369, 12283, 15920, 20001, 24335, 28984, 33563, 37868, 41735, 44946, 47413, 49143, 50262, 50854, 51126, 51230, 51258, 51261, 51261, 51261, 51261, 51261, 51261, 51261, 51261, 51261

Offset: 1

Labos Elemer, Aug 16 2000

a(n) = 51261 for n >= 35 since A056757 is finite with 51261 terms. - Amiram Eldar, Jun 02 2024

			Below 2^29 = 536870912 in A056757 altogether 49143 terms occur, so a(29) = 49143.

Index entries for linear recurrences with constant coefficients, signature (1).

Number of entries in A056757 not exceeding 2^n.

Cf. A000005, A029837, A034884, A035033-A035035, A056757-A056767, A056781.

With[{s = Import["https://oeis.org/A056757/b056757.txt", "Table"][[;; , 2]]}, a[n_] := Count[s, ?(# <= 2^n &)]; Table[a[n], {n, 1, 35}]] (* _Amiram Eldar, Jun 02 2024 *)

More terms from Amiram Eldar, Jun 02 2024

A056765 Number of integers from A056757 (defined by A000005(x)^3 > x) not exceeding 2^n.

Original entry on oeis.org

0, 1, 3, 5, 8, 16, 21, 33, 52, 65, 90, 121, 143, 176, 204, 221, 245, 257, 263, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267, 267

Offset: 1

Labos Elemer, Aug 16 2000

nonn

			The finite sequence A056757 has 267 entries of which the following 8 occur below 32 = 2^5: {3, 5, 7, 9, 15, 21, 25, 27}. So a(5)=8.

Cf. A000005, A029837, A035033-A035035, A034884, A056757-A056767.

More terms from Sean A. Irvine, May 05 2022

A056766 Smallest term of A056757 (numbers for which the cube of the number of divisors exceeds the number) between 2^(n-1) and 2^n.

Original entry on oeis.org

2, 3, 5, 9, 18, 33, 66, 130, 258, 516, 1026, 2052, 4100, 8200, 16400, 32800, 65550, 131100, 262200, 524400, 1048800, 2097600, 4195200, 8390400, 16783200, 33566400, 67132800, 134265600, 268606800, 537213600, 1074427200, 2148854400, 4297708800, 8627018400, 18897278400

Offset: 1

Labos Elemer, Aug 16 2000

Smallest k so that 2^(n-1) < k <= 2^n and A000005(k)^3 > k.

			For n=7, 64 < a(7) = 66 < 128, A000005(66)^3 = 8^3 = 512 > 66, and no other such number occurs between 64 and 66.
For n=31, a(31) = 1074427200, 2^30 < a(31) < 2^31; a(31) has 1344 divisors and 1344^3 = 2427715584 > 1074427200. Between 2^30 and a(31) no other numbers occur with this property.

Cf. A000005, A035033, A035034, A035035, A034884, A029837, A056757-A056767.

a(33)-a(35) from Amiram Eldar, Aug 15 2024

A215138 Numbers n such that n > d(n)^2/2 where d = A000005.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Gerasimov Sergey, Aug 04 2012

nonn

n <= A000005(n)^2/2 begins: 2, 4, 6, 8, 12, 18, 24, 30, 36, 48, 60, 72, 120.

			a(1)=1 because 1 > A000005(1)^2/2 = 1^2/2 = 1/2;
a(2)=3 because 3 > A000005(3)^2/2 = 2^2/2 = 2;
a(3)=5 because 5 > A000005(5)^2/2 = 2^2/2 = 2;
a(4)=7 because 7 > A000005(7)^2/2 = 2^2/2 = 2;
a(5)=9 because 9 > A000005(9)^2/2 = 3^2/2 = 9/2;
a(6)=10 because 10 > A000005(1)^2/2 = 4^2/2 =8.

Cf. A035033, A035035.

PARI
```
is(n)=numdiv(n)^2/2
				
```

cp's OEIS Frontend

A056763 Number of integers in the range (2^(n-1), 2^n] for which d(k)^3 > k holds, i.e., the cube of the number of divisors of k exceeds the number k.

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A056764 Number of integers k not exceeding 2^n such that the cube of number of divisors [A000005(k)] is larger than k.

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A056765 Number of integers from A056757 (defined by A000005(x)^3 > x) not exceeding 2^n.

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A056766 Smallest term of A056757 (numbers for which the cube of the number of divisors exceeds the number) between 2^(n-1) and 2^n.

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A215138 Numbers n such that n > d(n)^2/2 where d = A000005.

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