A030627 -id:A030627 - cp's OEIS frontend

A137487 Numbers with 24 divisors.

360, 420, 480, 504, 540, 600, 630, 660, 672, 756, 780, 792, 864, 924, 936, 990, 1020, 1050, 1056, 1092, 1120, 1140, 1152, 1170, 1176, 1188, 1224, 1248, 1350, 1368, 1380, 1386, 1400, 1404, 1428, 1470, 1500, 1530, 1540, 1596, 1632, 1638, 1650, 1656, 1710

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^23, p^2*q^7, p*q^2*r^3 (like 360, 504), p*q*r^5 (like 480, 672), p*q*r*s^2 (like 420, 660), p^3*q^5 (like 864) or p*q^11, where p, q, r and s are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[3000],DivisorSigma[0,#]==24&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==24 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=24.

A137489 Numbers with 26 divisors.

Original entry on oeis.org

12288, 20480, 28672, 45056, 53248, 69632, 77824, 94208, 118784, 126976, 151552, 167936, 176128, 192512, 217088, 241664, 249856, 274432, 290816, 299008, 323584, 339968, 364544, 397312, 413696, 421888, 438272, 446464, 462848, 520192, 536576

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^25 (5th powers of A050997, subset of A010813) or p*q^12, where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[900000],DivisorSigma[0,#]==26&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==26 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=26.

A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 8, 16, 11, 49, 10, 81, 12, 13, 121, 14, 625, 18, 64, 17, 169, 15, 2401, 20, 729, 24, 19, 289, 21, 14641, 28, 15625, 30, 36, 23, 361, 22, 28561, 32, 117649, 40, 100, 48, 29, 529, 26, 83521, 44, 1771561, 42, 196, 80, 1024, 31, 841, 27

Offset: 1

Leroy Quet, May 31 2006

From Peter Munn, May 17 2023: (Start)
As a square array A(n,m), n, m >= 1, read by ascending antidiagonals, A(n,m) is the n-th positive integer with m+1 divisors.
Thus both formats list the numbers with m+1 divisors in their m-th column. For the corresponding sequences giving numbers with a specific number of divisors see the index entries link.
(End)

			Looking at the 4th row, 7 is the 4th positive integer with 2 divisors, 25 is the 3rd positive integer with 3 divisors, 8 is the 2nd positive integer with 4 divisors and 16 is the first positive integer with 5 divisors. So the 4th row is (7,25,8,16).
The triangle T(n,m) begins:
  n\m:    1     2     3     4     5     6     7
  ---------------------------------------------
   1 :    2
   2 :    3     4
   3 :    5     9     6
   4 :    7    25     8    16
   5 :   11    49    10    81    12
   6 :   13   121    14   625    18    64
   7 :   17   169    15  2401    20   729    24
  ...
Square array A(n,m) begins:
  n\m:     1      2      3       4      5  ...
  --------------------------------------------
   1 :     2      4      6      16     12  ...
   2 :     3      9      8      81     18  ...
   3 :     5     25     10     625     20  ...
   4 :     7     49     14    2401     28  ...
   5 :    11    121     15   14641     32  ...
  ...

Index entries for sequences related to divisors of numbers

Columns: A000040, A001248, A007422, A030514, A030515, A030516, A030626, A030627, A030628, ... (see the index entries link for more).
Cf. A073915.
Diagonals (equivalently, rows of the square array) start: A005179\{1}, A161574.
Cf. A091538.

t[n_, m_] := Block[{c = 0, k = 1}, While[c < n + 1 - m, k++; If[DivisorSigma[0, k] == m + 1, c++ ]]; k]; Table[ t[n, m], {n, 11}, {m, n}] // Flatten (* Robert G. Wilson v, Jun 07 2006 *)

More terms from Robert G. Wilson v, Jun 07 2006

A079836 First column of the triangle in which the n-th row contains n numbers with n divisors that lie between A079835(n) and A079835(n+1).

Original entry on oeis.org

1, 3, 9, 51, 81, 28577, 117649, 594823330, 595067236, 596971504

Offset: 1

Amarnath Murthy, Feb 15 2003

1
3 5
9 25 49
51 55 57 58
81 625 2401 14641 28561
...
The 4th row consists of 4 consecutive elements of A030513, the 5th row 5 consecutive elements of A030514, the 6th and 7th rows consecutive elements of A030515 and A030516, the 8th of A030626, the 9th of A030627 etc. - R. J. Mathar, Mar 29 2007

Cf. A079835, A079837.

Cf. A030513, A030514, A030515, A030516, A030626, A030627.

a(6)-a(7) from R. J. Mathar, Mar 29 2007
a(8)-a(10) from Lambert Herrgesell (zero815(AT)googlemail.com), Feb 08 2008
a(2) and a(9) corrected by Pontus von Brömssen, Jan 14 2024

A137490 Numbers with 27 divisors.

Original entry on oeis.org

900, 1764, 2304, 4356, 4900, 6084, 6400, 10404, 11025, 12100, 12544, 12996, 16900, 19044, 23716, 26244, 27225, 28900, 30276, 30976, 33124, 34596, 36100, 38025, 43264, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 73984, 74529

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^26 (subset of A089081), p^2*q^2*r^2 (like 900, 1764, 4356, squares of A007304) or p^2*q^8 (like 2304, 6400, subset of the squares of A030628) where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000005, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[150000],DivisorSigma[0,#]==27&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

isA137490(n) = numdiv(n)==27 \\ Michael B. Porter, Apr 10 2010

A000005(a(n)) = 27.

Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 + P(2)*P(8) - P(10) + P(26) = 0.00453941..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A307980 Numbers k whose number of divisors is the square of the number of decimal digits of k.

Original entry on oeis.org

1, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 196, 225, 256, 441, 484, 676, 1000, 1026, 1032, 1064, 1110, 1122, 1128, 1144, 1155, 1160, 1190, 1218, 1230, 1240, 1242, 1254, 1272, 1288, 1290, 1302, 1326, 1330, 1365, 1408

Offset: 1

Bernard Schott, May 08 2019

The terms with an odd number of digits are squares.
The terms with 2 digits are squarefree semiprimes (cf. A006881) Union {27}. The terms with 3 digits belong to A030627 (numbers with 9 divisors) and the ones with 4 digits belong to A030634 (numbers with 16 divisors).
The number of terms b(n) with n digits begins with: 1, 30, 7, 753, 3, 11409, 2, ... When there are an odd number of digits, the number of terms decreases from b(3) = 7, b(5) = 3, b(7) = 2. Is there a 2q+1 such that b(2q+1) = 0?
The sequence is infinite because 10^k is the term for each k. We have tau(10^k) = tau(2^k)*tau(5^k) = (k + 1)^2 and 10^k has k + 1 digits. - Marius A. Burtea, May 09 2019
a(n) >= 1, for any n, so b(2q+1)>= 1 for any q. - Marius A. Burtea, May 09 2019

			65 is a term with 2 digits and 4 divisors: {1, 5, 13, 65}.
484 is a term with 3 digits and 9 divisors: {1, 2, 4, 11, 22, 44, 121, 242, 484}.

Sean A. Irvine, Java program (github)

Cf. A095862 (number of decimal digits = number of divisors).
Cf. A006881 (squarefree semiprimes).
Cf. A030513 (numbers with 4 divisors), A030627 (numbers with 9 divisors), A030634 (numbers with 16 divisors).
Cf. A011557 (subsequence).

[n:n in [1..1500]|NumberOfDivisors(n) eq (#Intseq(n))^2]; // Marius A. Burtea, May 09 2019

is(n) = numdiv(n) == #digits(n)^2 \\ David A. Corneth, May 08 2019

A331234 Triangular numbers having exactly 9 divisors.

Original entry on oeis.org

36, 1225, 1413721, 7885505171090778556470578126753302097454601, 67594562493730400324395236678194231988753004665644702944024074038452164931450549832074201

Offset: 1

Jon E. Schoenfield, Jan 12 2020

nonn

Any number having an odd number of divisors is a square, so each term in this sequence is a term of A001110 (numbers that are both triangular and square). Since A001110(k) = (A000129(k)*A001333(k))^2, A001110(k) will have exactly 9 divisors iff A000129(k) and A001333(k) are both prime (i.e., k is in both A096650 and A099088); the first 5 values of k at which this occurs are 2, 3, 5, 29, and 59.

Conjecture: a(5) is the final term of this sequence.

			Writing the k-th triangular number A000217(k) as T(k):
a(1) = T(8) = 8*9/2 = 36 = 2^2 * 3^2;
a(2) = T(49) = 49*50/2 = 1225 = 5^2 * 7^2;
a(3) = T(1681) = 1681*1682/2 = 1413721 = 29^2 * 41^2.
Factorization of larger known terms:
a(4) = 44560482149^2 * 63018038201^2;
a(5) = 13558774610046711780701^2 * 19175002942688032928599^2.

Intersection of A000217 (triangular numbers) and A030627 (numbers with exactly 9 divisors).
Triangular numbers having exactly k divisors: A068443 (k=4), A292989 (k=6), A330809 (k=8).
Cf. A063440 (number of divisors of n-th triangular number), A242585 (number of divisors of the n-th positive number that is both triangular and square).
Cf. A001110, A000129, A001333, A096650, A099088.

A369209 Numbers whose number of divisors has the largest prime factor 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset: 1

Amiram Eldar, Jan 16 2024

Subsequence of A059269 and first differs from it at n = 36: A059269(136) = 44 has 15 = 3 * 5 divisors and thus is not a term of this sequence.
Numbers k such that A000005(k) is in A065119.
Numbers k such that A071188(k) = 3.
Equals the complement of A354181, without the terms of A036537 (i.e., complement(A354181) \ A036537).
The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (Sum_{k>=1} 1/p^(A003586(k)-1)) - A327839 = 0.26087647470200496716... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A003586, A006530, A036537, A065119, A336595, A071188, A211337, A211338, A327839, A354181.
Subsequence of A013929 and A059269.
Subsequences: A001248, A030627, A050997, A054753, A062503, A067259, A079395, A085986, A085987, A086975, A095990, A096156, A138032, A162143, A179643, A179645.

gpf[n_] := FactorInteger[n][[-1, 1]]; Select[Range[300], gpf[DivisorSigma[0, #]] == 3 &]

gpf(n) = if(n == 1, 1, vecmax(factor(n)[, 1]));
is(n) = gpf(numdiv(n)) == 3;

cp's OEIS Frontend

A137487 Numbers with 24 divisors.

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A137489 Numbers with 26 divisors.

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A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.

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A079836 First column of the triangle in which the n-th row contains n numbers with n divisors that lie between A079835(n) and A079835(n+1).

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A137490 Numbers with 27 divisors.

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A307980 Numbers k whose number of divisors is the square of the number of decimal digits of k.

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A331234 Triangular numbers having exactly 9 divisors.

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A369209 Numbers whose number of divisors has the largest prime factor 3.

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