A003680 -id:A003680 - cp's OEIS frontend

A140605 a(n) = A003680(n)/A005179(n), or 0 if quotient is not an integer.

2, 3, 3, 4, 3, 5, 3, 5, 5, 5, 3, 6, 3, 5, 5, 7, 3, 7, 3, 7, 5, 5, 3, 7, 5, 5, 7, 7, 3, 7, 3, 9, 5, 5, 5, 8, 3, 5, 5, 9, 3, 7, 3, 7, 7, 5, 3, 11, 5, 7, 5, 7, 3, 8, 5, 9, 5, 5, 3, 11, 3, 5, 7, 11, 5, 7, 3, 7, 5, 7, 3, 11, 3, 5, 7, 7, 5, 7, 3, 11, 8, 5, 3, 11, 5, 5, 5, 9, 3, 11, 5, 7, 5, 5, 5, 12, 3, 7, 7, 11, 3

Offset: 1

J. Lowell, Jul 07 2008

nonn

a(810) = 0; the quotient is the non-integer 144/11.
n=810 is the only such case among the first 1000 terms. - Antti Karttunen, Oct 21 2017
a(n) = 0 for n in {810, 10206, 13608, 18225, 24300, 32400, ...}. - David A. Corneth, Oct 05 2022

			a(10) = 5 because A003680(10) = 240 and A005179(10) = 48 and 240/48=5.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Antti Karttunen)

Cf. A003680, A005179.

More terms from R. J. Mathar, Feb 19 2009

A061286 Smallest integer for which the number of divisors is the n-th prime.

Original entry on oeis.org

2, 4, 16, 64, 1024, 4096, 65536, 262144, 4194304, 268435456, 1073741824, 68719476736, 1099511627776, 4398046511104, 70368744177664, 4503599627370496, 288230376151711744, 1152921504606846976

Offset: 1

Labos Elemer, May 22 2001

Seems to be the same as "Even numbers with prime number of divisors" - Jason Earls, Jul 04 2001

Except for the first term, smallest number == 1 (mod prime(n)) having n divisors (by Fermat's little theorem). - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 20 2003

Karl V. Keller, Jr., Table of n, a(n) for n = 1..460
Index to divisibility sequences

Cf. A000005, A005179, A003680, A061283, A061286, A006093, A005097, A006254, A119523, A196202.

Table[2^(p-1),{p,Table[Prime[n],{n,1,18}]}] (* Geoffrey Critzer, May 26 2013 *)

forstep(n=2,100000000,2,x=numdiv(n); if(isprime(x),print(n)))

a(n)=2^(prime(n)-1) \\ Charles R Greathouse IV, Apr 08 2012

from sympy import isprime, divisor_count as tau
[2] + [2**(2*n) for n in range(1, 33) if isprime(tau(2**(2*n)))] # Karl V. Keller, Jr., Jul 10 2020

a(n) = 2^(prime(n)-1) = 2^A006093(n).
a(n) = A005179(prime(n)). - R. J. Mathar, Aug 09 2019
Sum_{n>=1} 1/a(n) = A119523. - Amiram Eldar, Aug 11 2020

A137492 Numbers with 29 divisors.

Original entry on oeis.org

268435456, 22876792454961, 37252902984619140625, 459986536544739960976801, 144209936106499234037676064081, 15502932802662396215269535105521, 28351092476867700887730107366063041

Offset: 1

R. J. Mathar, Apr 22 2008

Maple implementation: see A030513.

28th 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. - Omar E. Pol, May 06 2008

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

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

Cf. A000040.

Prime[Range[13]]^28 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^28 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=29.

a(n)=A000040(n)^(29-1)=A000040(n)^(28). - Omar E. Pol, May 06 2008

A061283 Smallest number with exactly 2n-1 divisors.

Original entry on oeis.org

1, 4, 16, 64, 36, 1024, 4096, 144, 65536, 262144, 576, 4194304, 1296, 900, 268435456, 1073741824, 9216, 5184, 68719476736, 36864, 1099511627776, 4398046511104, 3600, 70368744177664, 46656, 589824, 4503599627370496, 82944

Offset: 1

Labos Elemer, May 22 2001

nonn

The terms are always squares (because the divisors of a nonsquare N come in pairs, d and N/d, and so their number is always even - N. J. A. Sloane, Dec 26 2018).

			For n=15, a(15)=144 with 15 divisors: 1,2,3,4,6,8,9,12,16,18,24,36,48,72 and 144.

Amiram Eldar, Table of n, a(n) for n = 1..1661 (terms 1..1000 from T. D. Noe)

Cf. A000005, A000290, A005408, A003680, A016017, A037992, A055079, A048691.

Second bisection of A005179.

mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m < n, {}, {{n}}]; mp[n_, m_] := Join @@ Table[Map[Prepend[#, d] &, mp[n/d, d]], {d, Select[Rest[Divisors[n]], # <= m &]}]; mp[n_] := mp[n, n]; Table[mulpar = mp[2*n-1] - 1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]], {n, 1, 100}] (* Vaclav Kotesovec, Apr 04 2021 *)

a(n) = Min{k | A000005(k)=2n-1}.
a((p+1)/2) = 2^(p-1) for odd prime p. [Corrected by Jianing Song, Aug 30 2021]
From Jianing Song, Aug 30 2021: (Start)
a(n) = A016017(n)^2.
a(n) <= 2^(2n-2), where the equality holds if and only if n=1 or 2n-1 is prime. (End)

A016017 Smallest k such that 1/k can be written as a sum of exactly 2 unit fractions in n ways.

Original entry on oeis.org

1, 2, 4, 8, 6, 32, 64, 12, 256, 512, 24, 2048, 36, 30, 16384, 32768, 96, 72, 262144, 192, 1048576, 2097152, 60, 8388608, 216, 768, 67108864, 288, 1536, 536870912, 1073741824, 120, 576, 8589934592, 6144, 34359738368, 68719476736, 180, 864

Offset: 1

Robert G. Wilson v

nonn

From Jianing Song, Aug 30 2021: (Start)
a(n) is the smallest number whose square has exactly 2n-1 divisors.
a(n) is the earliest occurrence of 2n-1 in A048691. (End)

			a(1)=1 and a(2)=2 because 1/2 = 1/3 + 1/6 = 1/4 + 1/4.
a(3)=4 because 1/4 = 1/5 + 1/20 = 1/6 + 1/12 = 1/8 + 1/8.
a(4)=8 because 1/8 = 1/9 + 1/72 = 1/10 + 1/40 = 1/12 + 1/24 = 1/16 + 1/16.
a(5)=6 because 1/6 = 1/7 + 1/42 = 1/8 + 1/24 = 1/9 + 1/18 = 1/10 + 1/15 = 1/12 + 1/12.

David W. Wilson, Table of n, a(n) for n = 1..1000 (terms 122, 365 and 608 corrected by _Amiram Eldar_, Nov 08 2024)

Identical to A071571 shifted right.

Cf. A000005, A000290, A005408, A005179, A003680, A037992, A055079, A048691.

f[j_, n_] := (Times @@ (j(Last /@ FactorInteger[n]) + 1) + j - 1)/j; t = Table[0, {50}]; Do[a = f[2, n]; If[a < 51 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 2^30}] (* Robert G. Wilson v, Aug 03 2005 *)

a(n) = {k = 1; while (numdiv(k^2) != (2*n-1), k++); return (k); }; \\ Amiram Eldar, Jan 07 2019 after Michel Marcus at A071571

a(n+1) <= 2^n.
From Labos Elemer, May 22 2001: (Start)
a(n) = sqrt(A061283(n)).
a(n) = sqrt(Min{k| A000005(k)=2n-1}).
a((p+1)/2) = 2^((p-1)/2) = 2^A005097(i) if p is the i-th odd prime. [Corrected by Jianing Song, Aug 30 2021] (End)
a(n) is the least k such that (tau(k^2) + 1)/2 = n. - Vladeta Jovovic, Aug 01 2001

Entry revised by N. J. A. Sloane, Aug 14 2005

Offset corrected by David W. Wilson, Dec 27 2018

A137488 Numbers with 25 divisors.

Original entry on oeis.org

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

A061234 Smallest number with prime(n)^2 divisors where prime(n) is the n-th prime.

Original entry on oeis.org

6, 36, 1296, 46656, 60466176, 2176782336, 2821109907456, 101559956668416, 131621703842267136, 6140942214464815497216, 221073919720733357899776, 10314424798490535546171949056, 13367494538843734067838845976576

Offset: 1

Labos Elemer, Jun 01 2001

nonn

			1296 = 2*2*2*2*3*3*3*3 is the smallest number with 25 divisors.

Amiram Eldar, Table of n, a(n) for n = 1..208

Cf. A000005, A000040, A001248, A003680, A005179, A037992, A061148, A061149, A061283, A061286.

a(n) = Min_{x : d(x) = A000005(x) = p(n)^2} = 6^(p(n)-1) because x = 2^(pp-1) > 2^(p-1)3^(p-1) holds if p > 1.

a(n) = A005179(A001248(n)). - Amiram Eldar, Jun 21 2024

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.

A292580 T(n,k) is the start of the first run of exactly k consecutive integers having exactly 2n divisors. Table read by rows.

Original entry on oeis.org

5, 2, 6, 14, 33, 12, 44, 603, 242, 10093613546512321, 24, 104, 230, 3655, 11605, 28374, 171893, 48, 2511, 7939375, 60, 735, 1274, 19940, 204323, 368431323, 155385466971, 18652995711772, 15724736975643, 2973879756088065948, 9887353188984012120346

Offset: 1

Jon E. Schoenfield, Sep 19 2017

The number of terms in row n is A119479(2n).
Düntsch and Eggleton (1989) has typos for T(3,5) and T(10,3) (called D(6,5) and D(20,3) in their notation). Letsko (2015) and Letsko (2017) both have a wrong value for T(7,3).
The first value required to extend the data is T(6,13) <= 586683019466361719763403545; the first unknown value that may exist is T(12,19). See the a-file for other known values and upper bounds up to T(50,7).

			T(1,1) = 5 because 5 is the start of the first "run" of exactly 1 integer having exactly 2*1=2 divisors (5 is the first prime p such that both p-1 and p+1 are nonprime);
T(1,2) = 2 because 2 is the start of the first run of exactly 2 consecutive integers having exactly 2*1=2 divisors (2 and 3 are the only consecutive integers that are prime);
T(3,4) = 242 because the first run of exactly 4 consecutive integers having exactly 2*3=6 divisors is 242 = 2*11^2, 243 = 3^5, 244 = 2^2*61, 245 = 5*7^2.
Table begins:
   n  T(n,1), T(n,2), ...
  ==  ========================================================
   1  5, 2;
   2  6, 14, 33;
   3  12, 44, 603, 242, 10093613546512321;
   4  24, 104, 230, 3655, 11605, 28374, 171893;
   5  48, 2511, 7939375;
   6  60, 735, 1274, 19940, 204323, 368431323, 155385466971, 18652995711772, 15724736975643, 2973879756088065948, 9887353188984012120346, 120402988681658048433948, T(6,13), ...;
   7  192, 29888, 76571890623;
   8  120, 2295, 8294, 153543, 178086, 5852870, 17476613;
   9  180, 6075, 959075, 66251139635486389922, T(9,5);
  10  240, 5264, 248750, 31805261872, 1428502133048749, 8384279951009420621, 189725682777797295066519373;
  11  3072, 2200933376, 104228508212890623;
  12  360, 5984, 72224, 2919123, 15537948, 973277147, 33815574876, 1043710445721, 2197379769820, 2642166652554075, 17707503256664346, T(12,12), ...;
  13  12288, 689278976, 1489106237081787109375;
  14  960, 156735, 23513890624, 4094170438109373, 55644509293039461218749, 4230767238315793911295500109374, 273404501868270838132985214432619890621;
  15  720, 180224, 145705879375, 10868740069638250502059754282498, T(15,5);
  16  840, 21735, 318680, 6800934, 57645182, 1194435205, 14492398389;
  ...

Hugo van der Sanden, Table of n, a(n) for n = 1..32
Ivo Düntsch and Roger B. Eggleton, Equidivisible consecutive integers, 1989.
Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.
Vladimir A. Letsko, Table of a(n) for all even n such that exact value of a(n) is proved, 2017.
Carlos Rivera, Problem 20: k consecutive numbers with the same number of divisors, The Prime Puzzles and Problems Connection.
Hugo van der Sanden, calculation of T(6,11).
Hugo van der Sanden, calculation of T(6,12).
Hugo van der Sanden, A-file with known values and bounds up to T(50,7)

Cf. A000005, A006558, A072507, A119479.
Cf. A005237, A005238, A006601, A049051, A049052, A049053.
Cf. A003680, A075036, A075040.

T(n,2) = A075036(n). - Jon E. Schoenfield, Sep 23 2017

a(1)-a(25) from Düntsch and Eggleton (1989) with corrections by Jon E. Schoenfield, Sep 19 2017
a(26)-a(27) from Giovanni Resta, Sep 20 2017
a(28)-a(29) from Hugo van der Sanden, Jan 12 2022
a(30) from Hugo van der Sanden, Sep 03 2022
a(31) added by Hugo van der Sanden, Dec 05 2022; see "calculation of T(6,11)" link for a list of the people involved.
a(32) added by Hugo van der Sanden, Dec 18 2022; see "calculation of T(6,12)" link for a list of the people involved.

cp's OEIS Frontend

A140605 a(n) = A003680(n)/A005179(n), or 0 if quotient is not an integer.

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A061286 Smallest integer for which the number of divisors is the n-th prime.

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A137492 Numbers with 29 divisors.

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A061283 Smallest number with exactly 2n-1 divisors.

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A016017 Smallest k such that 1/k can be written as a sum of exactly 2 unit fractions in n ways.

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A137488 Numbers with 25 divisors.

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A061234 Smallest number with prime(n)^2 divisors where prime(n) is the n-th prime.

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A137485 Numbers with 22 divisors.