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.

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A173533 Numbers with 53 divisors.

Original entry on oeis.org

4503599627370496, 6461081889226673298932241, 2220446049250313080847263336181640625, 88124787089723195184393736687912818113311201, 1420429319844313329730664601483335671261683881745483121, 8415003868347247618489696679505181495471801448798649088081
Offset: 1

Views

Author

Omar E. Pol, Oct 16 2010

Keywords

Comments

52nd powers of primes.
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575.

Programs

Formula

a(n) = A000040(n)^(53-1) = A000040(n)^52.

A183062 Numbers with 59 divisors.

Original entry on oeis.org

288230376151711744, 4710128697246244834921603689, 34694469519536141888238489627838134765625, 10367793076318844190248738727596255138212949486449
Offset: 1

Views

Author

Omar E. Pol, Jul 31 2011

Keywords

Comments

Also, 58th powers of primes.
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533.

Programs

Formula

a(n) = A000040(n)^(59-1) = A000040(n)^58.
A000005(a(n)) = 59.

A183085 Numbers with 61 divisors.

Original entry on oeis.org

1152921504606846976, 42391158275216203514294433201, 867361737988403547205962240695953369140625, 508021860739623365322188197652216501772434524836001
Offset: 1

Views

Author

Omar E. Pol, Jul 31 2011

Keywords

Comments

Also, 60th powers of primes.
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062.

Programs

  • PARI
    a(n)=prime(n)^60

Formula

a(n) = A000040(n)^(61-1) = A000040(n)^60.
A000005(a(n)) = 61.

A030636 Numbers with 18 divisors.

Original entry on oeis.org

180, 252, 288, 300, 396, 450, 468, 588, 612, 684, 700, 768, 800, 828, 882, 972, 980, 1044, 1100, 1116, 1280, 1300, 1332, 1452, 1476, 1548, 1568, 1575, 1692, 1700, 1792, 1900, 1908, 2028, 2124, 2156, 2178, 2196, 2205, 2300, 2412, 2420, 2450
Offset: 1

Views

Author

Jeff Burch

Keywords

Comments

Numbers of the form p^17, p*q^8 (A179668), p*q^2*r^2 (A179643) or p^2*q^5 (A179646), where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

Links

Crossrefs

Cf. A030635 (17 divs), A030637 (19 divs).

Programs

Extensions

882 inserted by N. J. A. Sloane, Mar 26 2007

A369938 Numbers whose maximal exponent in their prime factorization is a power of 2.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77
Offset: 1

Views

Author

Amiram Eldar, Feb 06 2024

Keywords

Comments

First differs from its subsequence A138302 \ {1} at n = 378: a(378) = 432 = 2^4 * 3^3 is not a term of A138302.
First differs from A096432, A220218 \ {1}, A335275 \ {1} and A337052 \ {1} at n = 56, and from A270428 \ {1} at n = 113.
Numbers k such that A051903(k) is a power of 2.
The asymptotic density of this sequence is 1/zeta(3) + Sum_{k>=2} (1/zeta(2^k+1) - 1/zeta(2^k)) = 0.87442038669659566330... .

Links

Crossrefs

Cf. A000079, A002117, A051903.
Subsequences: A001146, A001248, A005117, A030514, A030635, A050376, A062503, A113849, A138302 \ {1}, A179645.
Similar sequences: A368714, A369937, A369939.
Cf. A096432, A220218, A270428, A335275, A337052.

Programs

  • Mathematica
    pow2Q[n_] := n == 2^IntegerExponent[n, 2];
Select[Range[2, 100], pow2Q[Max[FactorInteger[#][[;; , 2]]]] &]
Select[Range[2,80],IntegerQ[Log2[Max[FactorInteger[#][[;;,2]]]]]&] (* Harvey P. Dale, Nov 06 2024 *)
  • PARI
    ispow2(n) = n >> valuation(n, 2) == 1;
is(n) = n > 1 && ispow2(vecmax(factor(n)[, 2]));

A261700 Numbers with 101 divisors.

Original entry on oeis.org

1267650600228229401496703205376, 515377520732011331036461129765621272702107522001, 7888609052210118054117285652827862296732064351090230047702789306640625, 3234476509624757991344647769100216810857203198904625400933895331391691459636928060001
Offset: 1

Views

Author

Omar E. Pol, Aug 28 2015

Keywords

Comments

Also, 100th powers of primes.
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

Examples

			a(1) = 2^100, a(2) = 3^100, a(3) = 5^100, a(4) = 7^100.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085.

Programs

Formula

a(n) = A000040(n)^(101-1) = A000040(n)^100.
A000005(a(n)) = 101.

A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 7, 1, 32, 81, 125, 49, 11, 1, 64, 243, 625, 343, 121, 13, 1, 128, 729, 3125, 2401, 1331, 169, 17, 1, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 1, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 1, 1024, 19683, 390625, 823543, 1771561, 371293
Offset: 0

Views

Author

Omar E. Pol, Sep 09 2018

Keywords

Comments

If n = p - 1 where p is prime, then row n lists the numbers with p divisors.
The partial sums of column k give the column k of A319076.

Examples

			The corner of the square array is as follows:
         A000079 A000244 A000351  A000420    A001020    A001022     A001026
A000012        1,      1,      1,       1,         1,         1,          1, ...
A000040        2,      3,      5,       7,        11,        13,         17, ...
A001248        4,      9,     25,      49,       121,       169,        289, ...
A030078        8,     27,    125,     343,      1331,      2197,       4913, ...
A030514       16,     81,    625,    2401,     14641,     28561,      83521, ...
A050997       32,    243,   3125,   16807,    161051,    371293,    1419857, ...
A030516       64,    729,  15625,  117649,   1771561,   4826809,   24137569, ...
A092759      128,   2187,  78125,  823543,  19487171,  62748517,  410338673, ...
A179645      256,   6561, 390625, 5764801, 214358881, 815730721, 6975757441, ...
...

Crossrefs

Rows 0-13: A000012, A000040, A001248, A030078, A030514, A050997, A030516, A092759, A179645, A179665, A030629, A079395, A030631, A138031.
Other rows n: A030635 (n=16), A030637 (n=18), A137486 (n=22), A137492 (n=28), A139571 (n=30), A139572 (n=36), A139573 (n=40), A139574 (n=42), A139575 (n=46), A173533 (n=52), A183062 (n=58), A183085 (n=60), A261700 (n=100).
Columns 1-15: A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Main diagonal gives A093360.
Second diagonal gives A062457.
Third diagonal gives A197987.
Removing the 1's we have A182944/ A182945.
Cf. A006093, A319074, A319076.

Programs

  • PARI
    T(n, k) = prime(k)^n;

Formula

T(n,k) = A000040(k)^n, n >= 0, k >= 1.

A369937 Numbers whose maximal exponent in their prime factorization is square.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102
Offset: 1

Views

Author

Amiram Eldar, Feb 06 2024

Keywords

Comments

First differs from A366762 at n = 84, and from A197680, A361177 and A369210 at n = 95.
Numbers k such that A051903(k) is square.
The asymptotic density of this sequence is 1/zeta(2) + Sum_{k>=2} (1/zeta(k^2+1) - 1/zeta(k^2)) = 0.64939447949574562687... .

Links

Crossrefs

Cf. A000290, A013661, A051903.
Subsequences: A002416, A005117, A030514, A030635, A113849, A178739, A179665, A179692, A197680.
Similar sequences: A368714, A369938, A369939.
Cf. A361177, A366762, A369210.

Programs

  • Mathematica
    Select[Range[100], IntegerQ@ Sqrt[Max[FactorInteger[#][[;; , 2]]]] &]
  • PARI
    lista(kmax) = for(k = 1, kmax, if(k == 1 || issquare(vecmax(factor(k)[, 2])), print1(k, ", ")));

A280298 Numbers with 67 divisors.

Original entry on oeis.org

73786976294838206464, 30903154382632612361920641803529, 13552527156068805425093160010874271392822265625, 59768263894155949306790119265585619217025149412430681649, 539407797827634189900210968137750826278309533633974732577186113975161
Offset: 1

Views

Author

Omar E. Pol, Dec 31 2016

Keywords

Comments

Also, 66th powers of primes.
More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 67.

Examples

			a(1) = 2^66, a(2) = 3^66, a(3) = 5^66, a(4) = 7^66, a(5) = 11^66.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280299, A280301, A261700.

Programs

  • Mathematica
    Array[Prime[#]^66 &, {5}] (* Michael De Vlieger, Dec 31 2016 *)
  • PARI
    a(n)=prime(n)^66

Formula

a(n) = A000040(n)^(67-1) = A000040(n)^66.
A000005(a(n)) = 67.

A280299 Numbers with 71 divisors.

Original entry on oeis.org

1180591620717411303424, 2503155504993241601315571986085849, 8470329472543003390683225006796419620513916015625, 143503601609868434285603076356671071740077383739246066639249, 7897469567994392174328988784504809847540729881935024059662581894710332201
Offset: 1

Views

Author

Omar E. Pol, Dec 31 2016

Keywords

Comments

Also, 70th powers of primes.
More generally, the n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. In this case, p = 71.

Examples

			a(1) = 2^70, a(2) = 3^70, a(3) = 5^70, a(4) = 7^70, a(5) = 11^70.

Links

Crossrefs

Cf. A000005, A000040, A001248, A030514, A030516, A030629, A030631, A030635, A030637, A137486, A137492, A139571, A139572, A139573, A139574, A139575, A173533, A183062, A183085, A280298, A280301, A261700.

Programs

  • Mathematica
    Array[Prime[#]^70 &, {5}] (* Michael De Vlieger, Dec 31 2016 *)
  • PARI
    a(n)=prime(n)^70

Formula

a(n) = A000040(n)^(71-1) = A000040(n)^70.
A000005(a(n)) = 71.
Previous Showing 11-20 of 27 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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