A030626 -id:A030626 - cp's OEIS frontend

A137488 Numbers with 25 divisors.

1296, 10000, 38416, 50625, 194481, 234256, 456976, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 6765201, 9150625, 10556001, 11316496, 14776336, 16777216, 17850625, 22667121, 29986576, 35153041, 45212176, 52200625

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^24 (24th powers of A000040, subset of A010812) or p^4*q^4 (A189991), 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. A000005, A010812, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283, A135581, A175755.

a137488 n = a137488_list !! (n-1)
a137488_list = m (map (^ 24) a000040_list) (map (^ 4) a006881_list) where
   m xs'@(x:xs) ys'@(y:ys) | x < y = x : m xs ys'
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Nov 29 2011

lst = {}; Do[If[DivisorSigma[0, n] == 25, Print[n]; AppendTo[lst, n]], {n, 55000000}]; lst (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)
Select[Range[5221*10^4],DivisorSigma[0,#]==25&] (* Harvey P. Dale, Mar 11 2019 *)

is(n)=numdiv(n)==25 \\ Charles R Greathouse IV, Jun 19 2016

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A137488(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(y:=integer_nthroot(x,4)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)))-primepi(integer_nthroot(x,24)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

A000005(a(n)) = 25.

Sum_{n>=1} 1/a(n) = (P(4)^2 - P(8))/2 + P(24) = 0.000933328..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A137485 Numbers with 22 divisors.

Original entry on oeis.org

3072, 5120, 7168, 11264, 13312, 17408, 19456, 23552, 29696, 31744, 37888, 41984, 44032, 48128, 54272, 60416, 62464, 68608, 72704, 74752, 80896, 84992, 91136, 99328, 103424, 105472, 109568, 111616, 115712, 118098, 130048, 134144, 140288

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^21 or p*q^10, 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.

A137485=proc(q) local n;
for n from 1 to q do if tau(n)=22 then print(n); fi; od; end:
A137485(10^10);

Select[Range[200000],DivisorSigma[0,#]==22&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==22 \\ Charles R Greathouse IV, Jun 19 2016

from sympy import primepi, integer_nthroot, primerange
def A137485(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p**10) for p in primerange(integer_nthroot(x,10)[0]+1))+primepi(integer_nthroot(x,11)[0])-primepi(integer_nthroot(x,21)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A000005(a(n))=22.

A137491 Numbers with 28 divisors.

Original entry on oeis.org

960, 1344, 1728, 2112, 2240, 2496, 3264, 3520, 3648, 4160, 4416, 4928, 5440, 5568, 5824, 5832, 5952, 6080, 7104, 7290, 7360, 7616, 7872, 8000, 8256, 8512, 9024, 9152, 9280, 9920, 10176, 10206, 10304, 11328, 11712, 11840, 11968, 12864, 12992, 13120

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^27 (subset of A122968), p*q^13, p*q*r^6 (A179672) or p^3*q^6 (A179694), where p, q and r 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.

with(numtheory): A137491:=n->`if`(tau(n)=28,n,NULL): seq(A137491(n), n=1..15000); # Wesley Ivan Hurt, Sep 18 2014

Select[Range[30000],DivisorSigma[0,#]==28&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

is(n)=numdiv(n)==28 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n)) = 28.

A166546 Natural numbers n such that d(n) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 87, 89, 90

Offset: 1

Giovanni Teofilatto, Oct 16 2009

nonn

Natural numbers n such that d(d(n)+1)= 2. - Giovanni Teofilatto, Oct 26 2009

The complement is the union of A001248, A030514, A030516, A030626, A030627, A030629, A030631, A030632, A030633 etc. - R. J. Mathar, Oct 26 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..6955

Cf. A000005.

Cf. A073915. - R. J. Mathar, Oct 26 2009

[n: n in [1..100] | IsPrime(NumberOfDivisors(n)+1)]; // Vincenzo Librandi, Jan 20 2019

Select[Range@90, PrimeQ[DivisorSigma[0, #] + 1] &] (* Vincenzo Librandi, Jan 20 2019 *)

isok(n) = isprime(numdiv(n)+1); \\ Michel Marcus, Jan 20 2019

{1} U A000040 U A030513 U A030515 U A030628 U A030630 U A030634 U A030636 U A137485 U A137491 U A137493 U ... . - R. J. Mathar, Oct 26 2009

A137487 Numbers with 24 divisors.

Original entry on oeis.org

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.

A139588 Nonprime numbers with Fibonacci number of divisors.

Original entry on oeis.org

1, 4, 9, 16, 24, 25, 30, 40, 42, 49, 54, 56, 66, 70, 78, 81, 88, 102, 104, 105, 110, 114, 121, 128, 130, 135, 136, 138, 152, 154, 165, 169, 170, 174, 182, 184, 186, 189, 190, 195, 222, 230, 231, 232, 238, 246, 248, 250, 255, 258, 266, 273, 282, 285, 286, 289

Offset: 1

Omar E. Pol, May 09 2008

A000005(a(n)) is a Fibonacci number.

The union of {1}, A001248, A030514, A030626, A030631, A137484, etc. [From R. J. Mathar, Oct 26 2009]

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A000045, A018252, A123193, A139589, A139590.

Module[{fibs=Fibonacci[Range[20]]},Select[Range[300],!PrimeQ[#]&&MemberQ[ fibs,DivisorSigma[0,#]]&]] (* Harvey P. Dale, Jan 20 2023 *)

A123193 \ A000040. [From R. J. Mathar, Oct 23 2009]

More terms from R. J. Mathar, Oct 23 2009

A276045 Primes p such that d(p*(2p+1)) = 8 where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

7, 13, 17, 19, 43, 47, 59, 61, 71, 79, 101, 107, 109, 149, 151, 163, 167, 197, 223, 257, 263, 271, 311, 317, 347, 349, 353, 383, 389, 401, 421, 439, 449, 461, 479, 503, 521, 523, 557, 569, 599, 601, 613, 631, 673, 677, 691, 701, 811, 821, 827, 839, 853, 863, 881, 919

Offset: 1

Anthony Hernandez, Aug 17 2016

Primes p such that 2p+1 is in A030513. - Robert Israel, Aug 17 2016
From Anthony Hernandez, Aug 29 2016: (Start)
Conjecture: this sequence is infinite.
It appears that the prime numbers in this sequence which have 7 for as final digit form the sequence A104164.
Conjecture: this sequence contains infinitely many twin primes. The first few twin primes in this sequence are 17,19,59,61,107,109,521,523,599,601,... (End)
From Bernard Schott, Apr 28 2020: (Start)
This sequence equals the union of {13} and A234095; proof by double inclusion:
-> 1st inclusion: {13} Union A234095 is included in A276045.
1) if p = 13, then 13*27 = 351 = 3^3 * 13, hence d(351) = 8 and 13 belongs to A276045.
2) if p is in A234095, then p*(2*p+1) = p*r*s (p,r,s primes) and d(p*r*s) = 8, hence p is in 276045.
-> 2nd inclusion: A276045 is included in {13} Union A234095.
If p is in A276095, then m=p*(2*p+1) has 8 divisors and there are only three possibilities: m = u*v*w, or m = u^3*v or m = u^7 with u, v, w are distinct primes.
1st case: if p*(2*p+1) = u*v*w then u=p, and 2p+1=v*w is semiprime; hence, p is in A234095 Union {13}.
2nd case: if p*(2p+1) = u^3*v then  p=v and 2*p+1=u^3 ==> 2*p = u^3-1 = (u-1)*(u^2+u+1) with 2 and p are primes; then (u-1=2, u^2+u+1=p) so u=3, and p=3^2+3+1=13; hence p = 13 belongs to {13} Union A234095.
3rd case: p*(2p+1) = u^7 is impossible.
Conclusion: this sequence = {13} Union A234095. (End)

			d(7*(2*7+1))=d(105)=8 so 7 is a term.

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

Cf. A000005, A030513, A030626.

Equals {13} Union A234095.

select(n -> isprime(n) and numtheory:-tau(n*(2*n+1))=8,
[seq(i, i=3..1000, 2)]); # Robert Israel, Aug 17 2016

Select[Prime@ Range@ 160, DivisorSigma[0, # (2 # + 1)] == 8 &] (* Michael De Vlieger, Aug 28 2016 *)

lista(nn) = forprime(p=2, nn, if (numdiv(p*(2*p+1))==8, print1(p, ", "))); \\ Michel Marcus, Aug 17 2016

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

A182675 a(n) is the smallest n-digit number with exactly 8 divisors, a(n) = 0 if no such number exists.

Original entry on oeis.org

0, 24, 102, 1001, 10002, 100006, 1000002, 10000005, 100000006, 1000000003, 10000000001, 100000000006, 1000000000001, 10000000000001, 100000000000018, 1000000000000002, 10000000000000006, 100000000000000007, 1000000000000000001, 10000000000000000007, 100000000000000000003

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(n) is the smallest n-digit number of the form p^7, p^3*q or p*q*r (p, q, r = distinct primes), a(n) = 0 if no such number exists.

There is a large overlap with A180922 because the candidates p*q*r are also of the 3-almost-primes format required there. - R. J. Mathar, Apr 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000005, A030626, A182676, A180922.

with (numtheory):
a:= proc(n) local k;
     if n<2 then 0
   else for k from 10^(n-1) while tau(k)<>8
        do od; k
     fi
    end:
seq (a(n), n=1..20);

a(n) = for(k = 10^(n-1), 10^n-1, if(numdiv(k)==8, return(k))); 0; \\ Amiram Eldar, Apr 09 2024

a(n) = min {10^(n-1) <= k < 10^n : A000005(k)=8} if set is nonempty, else a(n) = 0.

Edited by Alois P. Heinz, Nov 27 2010

a(20)-a(21) from Amiram Eldar, Apr 09 2024

cp's OEIS Frontend

A137488 Numbers with 25 divisors.

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A137485 Numbers with 22 divisors.

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A137491 Numbers with 28 divisors.

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A166546 Natural numbers n such that d(n) + 1 is prime.

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A137487 Numbers with 24 divisors.

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A137489 Numbers with 26 divisors.

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A139588 Nonprime numbers with Fibonacci number of divisors.

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