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.

Previous Showing 11-20 of 20 results.

A225732 Numbers n such that n < d(n)^(24/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138, 140
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write n^5 < d(n)^12. The last odd number is a(95) = 315.

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]^(24/10), AppendTo[t, n]], {n, 10^6}]; t

A225733 Numbers n such that n < d(n)^(5/2), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write n^2 < d(n)^5. The last odd number is a(206) = 945.

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]^(5/2), AppendTo[t, n]], {n, 10^6}]; t

A225734 Numbers n such that n < d(n)^(26/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write n^5 < d(n)^13. The last odd number is a(473) = 3465.

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]^(26/10), AppendTo[t, n]], {n, 10^7}]; t

A225735 Numbers n such that n < d(n)^(27/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 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, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write n^10 < d(n)^27. The last odd term is a(995) = 10395.

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]^(27/10), AppendTo[t, n]], {n, 10^7}]; t

A225736 Numbers n such that n < d(n)^(28/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 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, 52, 54, 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, 112, 114
Offset: 1

Views

Author

T. D. Noe, May 14 2013

Keywords

Comments

Alternatively, we could write n^5 < d(n)^14. The last odd term is a(2447) = 45045.

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]^(28/10), AppendTo[t, n]], {n, 10^7}]; t

A374793 a(n) is the largest k such that tau(k)^n >= k.

Original entry on oeis.org

2, 1260, 27935107200, 29564884570506808579056000
Offset: 1

Views

Author

Jon E. Schoenfield, Jul 20 2024

Keywords

Comments

Let prime(j)# denote the product of the first j primes, A002110(j); then
a(1) = prime(1)# = 2,
a(2) = 6*prime(4)# = 1260,
a(3) = 2880*prime(8)# = 2.7935...*10^10,
a(4) = 907200*prime(16)# = 2.9564...*10^25,
a(5) >= 259459200*prime(30)# = 8.2015...*10^54,
a(6) >= 3238237626624000*prime(52)# = 3.4403...*10^111,
a(7) >= 248818180782850398720000*prime(91)# = 5.4351...*10^218.

Examples

			27935107200 = 2^7 * 3^3 * 5^2 * 7^1 * 11^1 * 13^1 * 17^1 * 19^1,
so tau(27935107200) = (7+1)*(3+1)*(2+1)*(1+1)*(1+1)*(1+1)*(1+1)*(1+1) = 8*4*3*2*2*2*2*2 = 3072; 3072^3 = 28991029248 > 27935107200, and there is no larger number k such that tau(k)^3 >= k, so a(3) = 27935107200.

Crossrefs

Cf. A000005, A035033, A056757, A056758.

A056759 The 17 prime powers k = p^w such that d(p^w)^3 > p^w where d = A000005().

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 16, 25, 27, 32, 64, 81, 128, 256, 512, 1024
Offset: 1

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

For all divisors of the LCM of the terms of this sequence (14515200) the defining relation d(x)^3 > x is also satisfied.

Examples

			Differences d(x)^3 - x of the 17 entries of this sequence are 6, 5, 23, 3, 1, 56, 18, 109, 2, 37, 184, 279, 44, 384, 473, 488, 307.

Crossrefs

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

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

A225422 Largest number k such that k < d(k)^(n/10), where d(k) is the number of divisors of k.

Original entry on oeis.org

2, 2, 6, 12, 12, 24, 60, 180, 360, 1260, 5040, 15120, 55440, 166320, 831600, 4324320, 36756720, 367567200, 2327925600, 27935107200
Offset: 11

Views

Author

T. D. Noe, May 15 2013

Keywords

Comments

Each of these numbers is the product of small primes. For example, a(30) = 2^7 2^3 5^2 7 11 13 17 19. - T. D. Noe, May 16 2013

Crossrefs

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

Programs

  • Mathematica
    Table[last = 0; Do[If[n < DivisorSigma[0,n]^(i/10), last = n], {n, 10^4}]; last, {i, 11, 20}]
Previous Showing 11-20 of 20 results.

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