cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A056761 Odd numbers less than the cube of their number of divisors.

Original entry on oeis.org

3, 5, 7, 9, 15, 21, 25, 27, 33, 35, 39, 45, 51, 55, 57, 63, 75, 81, 99, 105, 117, 135, 147, 153, 165, 171, 175, 189, 195, 207, 225, 231, 255, 273, 285, 297, 315, 345, 351, 357, 375, 385, 399, 405, 429, 435, 441, 455, 459, 465, 483, 495, 525, 567, 585, 675, 693
Offset: 1

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

Last term is a(267) = 883575, confirming the author's conjecture. - Charles R Greathouse IV, Apr 27 2011

Examples

			14175 = 81*25*7 has 30 divisors, and 30^3 = 27000 > 14175.

Links

Crossrefs

Cf. A035033-A035035, A034884, A000005, A000265, A056757-A056767, A056781.

Programs

  • Mathematica
    Select[Range[1, 10^6 + 1, 2], DivisorSigma[0, #]^3 > # &] (* Michael De Vlieger, Oct 26 2017 *)
  • PARI
    isok(n) = (n % 2) && (numdiv(n)^3 > n); \\ Michel Marcus, Dec 19 2013

A056762 Number of terms in A056761 (i.e., odd numbers from A056757) between 2^(n-1) and 2^n.

Original entry on oeis.org

0, 1, 2, 2, 3, 8, 5, 12, 19, 13, 25, 31, 22, 33, 28, 17, 24, 12, 6, 4, 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, 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, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

A056761 is finite and has 267 terms.

Examples

			The last 4 terms arise between 524288 and 1048576: {675675, 765765, 855855, 883575}, thus here a(20)=4. For n > 20, a(n)=0.

Crossrefs

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

Formula

Occurrences of odd integers x such that 2^(n-1) < x <= 2^n and the cube of number of divisors is larger than the number: A000005(x)^3 > x.

Extensions

More terms from Sean A. Irvine, May 05 2022

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.

Original entry on oeis.org

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

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

Extensions

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

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

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

Examples

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

Links

Crossrefs

Number of entries in A056757 not exceeding 2^n.
Cf. A000005, A029837, A034884, A035033-A035035, A056757-A056767, A056781.

Programs

Extensions

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

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Examples

			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.

Crossrefs

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

Extensions

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

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Extensions

a(33)-a(35) from Amiram Eldar, Aug 15 2024
Showing 1-6 of 6 results.

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