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 1-10 of 20 results. Next

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

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

A225729 Numbers n such that n < d(n)^(21/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 168, 180, 192, 210, 216, 240, 252, 264, 270, 280, 288, 300, 312, 330, 336, 360, 396
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write n^10 < d(n)^21. The last odd number is a(10) = 15.

Links

Crossrefs

Cf. A034884 (n < d(n)^2), A175495 (n < 2^d(n)), A056757 (n < d(n)^3).
Cf. A225730-A225738.

Programs

  • Mathematica
    t = {}; Do[If[n < DivisorSigma[0, n]^(21/10), AppendTo[t, n]], {n, 10^4}]; t

A225738 Number of numbers k such that k < d(k)^(n/10), where d(k) is the number of divisors of k.

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 7, 14, 21, 29, 52, 89, 155, 284, 528, 1018, 2046, 4282, 9272, 21466, 50967
Offset: 10

Views

Author

T. D. Noe, May 14 2013

Keywords

Crossrefs

Cf. A034884 (n < d(n)^2), A056757 (n < d(n)^3), A225729-A225737.

Programs

  • Mathematica
    Table[f = 0; Do[If[k < DivisorSigma[0, k]^(n/10), f++], {k, 10^4}]; f, {n, 10, 20}]

A056767 Largest number of binary size n (i.e., between (n-1)-th and n-th powers of 2) with the following property: cube of its number of divisors is larger than the number itself.

Original entry on oeis.org

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2046, 4095, 8190, 16380, 32760, 65520, 131040, 262080, 524160, 1048320, 2097144, 4193280, 8386560, 16773900, 33547800, 67095600, 134191200, 268382400, 536215680, 1073709000, 2144142000, 4288284000, 8527559040, 16908091200, 27935107200
Offset: 1

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Examples

			These maximal terms are usually "near" to 2^n. For n=1..10 they are equal to 2^n. At n=21, a(21)=2097144, 1048576 < a(21) < 2097144 = 8*27*7*19*73 has d=128 divisors, of which the cube is d^3d=2097152. So this maximum is near to but still less than d^3.

Crossrefs

Cf. A000005, A029837, A034884, A035033, A035034, A035035, A056757-A056767, A056781.

Programs

  • Mathematica
    Table[Last@ Select[Range @@ (2^{n - 1, n}), DivisorSigma[0, #]^3 > # &], {n, 22}] (* Michael De Vlieger, Dec 31 2016 *)
  • PARI
    a(n) = {k = 2^n; while(numdiv(k)^3 <= k, k--); k;} \\ Michel Marcus, Dec 11 2013

Formula

Largest terms of A056757 between 2^(n-1) and 2^n.

Extensions

a(32) from Michel Marcus, Dec 11 2013
a(33)-a(35) and keyword "full" added by Amiram Eldar, Feb 23 2025

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

A225730 Numbers k such that k < d(k)^(22/10), where d(k) is the number of divisors of k.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 168, 180, 192, 198, 200, 204, 210, 216, 220, 224, 228, 234
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write k^5 < d(k)^11. The last odd number is a(23) = 45.

Links

Crossrefs

Cf. A000005, A034884 (k < d(k)^2), A175495 (k < 2^d(k)), A056757 (k < d(k)^3).
Cf. A225729-A225738, A353448.

Programs

  • Mathematica
    t = {}; Do[If[n < DivisorSigma[0, n]^(22/10), AppendTo[t, n]], {n, 10^5}]; t
Select[Range[250],#Harvey P. Dale, Apr 10 2024 *)
  • PARI
    for (k=2, 20000, if (k^5 < numdiv(k)^11, print1(k,", "))) \\ Hugo Pfoertner, Apr 25 2023

A225737 Numbers n such that n < d(n)^(29/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write n^10 < d(n)^29. The last odd term is a(6362) = 225225.

Links

Crossrefs

Cf. A034884 (n < d(n)^2), A175495 (n < 2^d(n)), A056757 (n < d(n)^3).
Cf. A225729-A225738.

Programs

  • Mathematica
    t = {}; Do[If[n < DivisorSigma[0, n]^(29/10), AppendTo[t, n]], {n, 10^7}]; t

A225731 Numbers n such that n < d(n)^(23/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write n^10 < d(n)^23. The last odd number is a(44) = 105.

Links

Crossrefs

Cf. A034884 (n < d(n)^2), A175495 (n < 2^d(n)), A056757 (n < d(n)^3).
Cf. A225729-A225738.

Programs

  • Mathematica
    t = {}; Do[If[n < DivisorSigma[0, n]^(23/10), AppendTo[t, n]], {n, 10^5}]; t
Showing 1-10 of 20 results. Next

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