A073904 -id:A073904 - cp's OEIS frontend

A076931 Smallest k such that n*k has n divisors.

1, 1, 3, 2, 125, 2, 16807, 3, 4, 8, 2357947691, 5, 1792160394037, 32, 135, 24, 2862423051509815793, 10, 5480386857784802185939, 12, 1701, 512, 39471584120695485887249589623, 15, 400, 2048, 972, 48, 3053134545970524535745336759489912159909

Offset: 1

Amarnath Murthy, Oct 18 2002

nonn

n=p_1^a_1*...*p_r^a_r => tau(p_1^(p_1^a_1-1)*...*p_r^(p_r^a_r-1))=n, so sequence is well-defined.

a(n)= A073904(n)/n.

f[n_] := Block[{k = 1, m = If[ PrimeQ[n], n^(n-2), 1]}, While[ DivisorSigma[0, k*m*n] != n, k++ ]; k*m]; Table[f[n], {n, 29}] (* Robert G. Wilson v, Sep 29 2005 *)

a(p)=p^(p-2), a(pq)=p^(q-2)*q^(p-2) for p

More terms from Sascha Kurz, Jan 21 2003

More terms from David Wasserman, Aug 19 2005

A182749 a(n) = the smallest multiple of n with n digits and exactly n divisors. a(n) = 0 if no such number exists.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 10000024, 100580841, 0, 25937424601, 100000000068, 0, 0, 0, 1000000000001152, 0, 100000000000000404, 0, 10000000000000000880, 0, 0, 0, 100000000000000000001184, 1000060001350013500050625, 0, 100000000015460000000597529, 1000000000000000000000022464

Offset: 1

Jaroslav Krizek, Nov 27 2010

Conjecture: a(n) > 0 for all nonsquarefree n except for n = 4. - Jon E. Schoenfield, Jun 01 2024

Jon E. Schoenfield, Table of n, a(n) for n = 1..100

Cf. A000005, A073904, A182750.

A000005(a(n)) = n for a(n) > 0.

a(n) <= A182750(n).

a(16), a(18), a(20) corrected by Jon E. Schoenfield, May 26 2024
a(21)-a(23) (implied by A182750) from Pontus von Brömssen, May 26 2024
a(24)-a(25) from Michael S. Branicky, May 26 2024
a(26)-a(28) from Jon E. Schoenfield, May 26 2024

A359964 Refactorable numbers (A033950) having more divisors than all smaller refactorable numbers.

Original entry on oeis.org

1, 2, 8, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 3360, 5040, 10080, 15120, 20160, 25200, 30240, 55440, 100800, 110880, 221760, 277200, 443520, 665280, 720720, 1108800, 1441440, 2494800, 2882880, 3603600, 5765760, 8648640, 12972960, 14414400, 25945920, 28828800

Offset: 1

Amiram Eldar, Jan 20 2023

nonn

The corresponding numbers of divisors are 1, 2, 4, 6, 8, 9, 12, 18, 20, 24, ... .

This sequence if infinite since there are refactorable numbers with arbitrarily large number of divisors. E.g., for any prime p, p^(p-1) is a refactorable number with p divisors.

Amiram Eldar, Table of n, a(n) for n = 1..60

Subsequence of A033950.
Cf. A000005, A073904, A110821.
Similar sequences: A002182, A335317, A356078, A359963.

seq[nmax_] := Module[{s = {}, dm = 0, d}, Do[d = DivisorSigma[0, n]; If[d > dm && Divisible[n, d], dm = d; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[10^6]

lista(nmax) = {my(dm = 0, d); for(n = 1, nmax, d = numdiv(n); if(d > dm && n%d == 0, dm = d; print1(n, ", "))); }

A136641 a(n) is the smallest positive integer that is coprime to n and has n divisors.

Original entry on oeis.org

1, 3, 4, 15, 16, 175, 64, 105, 100, 567, 1024, 1925, 4096, 3645, 784, 945, 65536, 13475, 262144, 6237, 1600, 295245, 4194304, 25025, 1296, 2657205, 4900, 40095, 268435456, 3776773, 1073741824, 10395, 25600, 215233605, 5184, 175175, 68719476736, 1937102445, 102400

Offset: 1

Leroy Quet, Apr 14 2008

nonn

Is this the same as the least index m where A090387(m) = n? - Michel Marcus, Mar 25 2022

For p prime, a(p) = 2^(p-1) for p > 2, a(2*p) = 3^(p-1)*5 for p > 5, a(3*p) = 2^(p-1)*25 for p > 3, a(5*p) = 2^(p-1)*3^4 for p >5, ... . - Michael S. Branicky, Mar 26 2022

			The sequence of positive integers each with 9 divisors starts: 36, 100, 196, 225, 256, ... Now 36 is not coprime to 9. But 100, the next bigger value with 9 divisors, is. So a(9) = 100.

Michael S. Branicky, Table of n, a(n) for n = 1..65

Cf. A000005, A073904, A090387.

a(n) = my(k=1); while ((gcd(n,k) != 1) || (numdiv(k) != n), k++); k; \\ Michel Marcus, Mar 25 2022

from math import gcd
from sympy import divisor_count
def a(n):
    k = 1
    while gcd(n, k) != 1 or divisor_count(k) != n: k += 1
    return k
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Mar 25 2022

a(11)-a(36) from Sean A. Irvine, May 03 2010

a(37) and beyond from Michael S. Branicky, Mar 26 2022

A182750 a(n) = the largest multiple of n with n digits and exactly n divisors; a(n) = 0 if no such number exists.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 99999992, 998117649, 0, 25937424601, 999999999924, 0, 0, 0, 9999999999999872, 0, 999999999999999252, 0, 99999999999999999440, 0, 0, 0, 999999999999999999999456, 9999450940359923567250625, 0, 999999999997420003910299449, 9999999999999999999999972416, 0, 0, 0

Offset: 1

Jaroslav Krizek, Nov 27 2010

Conjecture: a(n) > 0 for all nonsquarefree n except for n = 4. - Jon E. Schoenfield, Jun 01 2024

Cf. A000005, A073904, A182749.

A000005(a(n)) = n if a(n) > 0.

a(n) >= A182749(n).

More terms from Harvey P. Dale, Mar 02 2018
a(24)-a(25) and a(27) from Michael S. Branicky, May 26 2024
a(26), a(28) added, and a(16), a(18), a(20) corrected by Jon E. Schoenfield, May 26 2024
a(29)-a(31) from David A. Corneth, May 27 2024

A076961 Smallest multiple of n with prime(n) divisors, or 0 if no such number exists.

Original entry on oeis.org

2, 4, 81, 64, 9765625, 0, 33232930569601, 262144, 31381059609, 0, 17449402268886407318558803753801, 0, 361188648084531445929920877641340156544317601, 0, 0, 4503599627370496

Offset: 1

Amarnath Murthy, Oct 21 2002

nonn

a(n) is zero if n is not a prime power. - Franklin T. Adams-Watters, May 14 2014

Cf. A076962, A073904.

More terms from Sascha Kurz, Jan 22 2003

A109429 Rearrange terms of A050376 so that a(2^j)=2^(2^j) for j>=0.

Original entry on oeis.org

2, 4, 3, 16, 5, 7, 9, 256, 11, 13, 17, 19, 23, 25, 29, 65536, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 4294967296, 89, 97

Offset: 1

Thomas Ordowski, Aug 26 2005

nonn

A073904(2^n) is the product of the first n members of this sequence. Generalization: for any prime p, we may consider the analogous permutation of numbers of the form q^(p^k) such that a(p^j)=p^(p^j); then A073904(p^n)=(product of the first n members)^(p-1). - David Wasserman and Thomas Ordowski. Corrected by Thomas Ordowski, Jun 06 2015

			Numbers: 2, 3, 2^2, 5, 7, 3^2, 11, 13, 2^(2^2), 17, ..., 2^(2^3), ...
Permutation: 2, 2^2, 3, 2^(2^2), 5, 7, 3^2, 2^(2^3), 11, 13, 17, ...
If n=4 then A073904(16)=2*4*3*16=384.

Cf. A050376.

a(2^j)=2^(2^j). So a(1)=2 for j=0; a(2)=4 for j=1; a(4)=16 for j=2.

A073904(2^n)=2*4*3*...*a(n) for every n.

Definition edited by N. J. A. Sloane, Oct 27 2014

More terms from Thomas Ordowski, Jun 05 2015

A359965 a(n) is the least arithmetic number (A003601) having exactly n divisors.

Original entry on oeis.org

1, 3, 49, 6, 14641, 20, 594823321, 30, 8281, 304, 41426511213649, 60, 491258904256726154641, 832, 717409, 168, 160470643909878751793805444097921, 612, 114445997944945591651333831028437092270721, 432, 87616, 44032, 6111571184724799803076702357055363809, 420, 13521270961

Offset: 1

Amiram Eldar, Jan 20 2023

nonn

a(n) is the least number k such that A000005(k) = n and n | A000203(k).

a(n) exists for all n: for example, if p is a prime such that p == 1 (mod n), then p^(n-1) has n divisors and n | A000203(p^(n-1)).

Subsequence of A003601.
Cf. A000005, A000203, A359963.
Similar sequences: A005179, A073904.

A366918 a(n) is the number of multiples of n that have exactly n divisors, or -1 if there are infinitely many such numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, -1, -1, 2, 1, -1, 1, 2, 2, -1, 1, -1, 1, -1, 2, 2, 1, -1, -1, 2, -1, -1, 1, 6, 1, -1, 2, 2, 2, -1, 1, 2, 2, -1, 1, 6, 1, -1, -1, 2, 1, -1, -1, -1, 2, -1, 1, -1, 2, -1, 2, 2, 1, -1, 1, 2, -1, -1, 2, 6, 1, -1, 2, 6, 1, -1, 1, 2, -1, -1, 2, 6, 1, -1, -1, 2, 1, -1, 2, 2, 2, -1, 1, -1, 2, -1, 2, 2, 2

Offset: 1

Jon E. Schoenfield, Oct 28 2023

sign

a(n) = -1 for each nonsquarefree n > 4 with n = Product_{j=1..w} p_j ^ e_j because there exist infinitely many multiples of n that have exactly n divisors. Included among such multiples are all numbers of the form c*q^(b-1) where b is the smallest prime factor of n, q is any prime that does not divide n, and c = Product_{j=1..w} p_j ^ (d_j - 1), with the d_j values chosen so that Product_{j=1..w} d_j = n/b and d_j > e_j for j = 1..w.

			a(1) = 1 because there is only 1 number that has exactly 1 divisor (namely, 1).
For every prime p, a(p) = 1 because the only multiple of p that has exactly p divisors is p^(p-1).
a(4) = 1 because the only multiple of 4 that has exactly 4 divisors is 8. (4 itself has only 3 divisors, and 4*k for every k > 2 has more than 4 divisors.)
If n is squarefree, then a(n) = w! where w is the number of prime factors of n, because every multiple of n that has exactly n divisors must be of the form p_1^e_1 * p_2^e_2 * ... * p_w^e_w where p_1, p_2, ..., p_w are the prime factors of n and the vector (e_1 + 1, e_2 + 1, ..., e_w + 1) is one of the w! permutations of the vector (p_1, p_2, ..., p_w).
If n > 4 is nonsquarefree then a(n) = -1 because there exist infinitely many multiples of n that have exactly n divisors. Examples (where p and q are distinct odd primes and e > 1):
  n = 2^q: 2^(2^(q-1) - 1) * p has 2^q divisors for every p
  n = q^e: q^(q^(e-1) - 1) * p^(q-1) has q^e divisors for every p
  n = 4*q: 2^(q-1)*q*p has 4*q divisors for every p.

Antti Karttunen, Table of n, a(n) for n = 1..30030

Cf. A000005, A001221, A073904, A080756.

Cf. A000142 (records), A002110 (and their positions).

a[n_]:=If[n==4,1,If[SquareFreeQ[n],PrimeNu[n]!,-1]]; Array[a,78] (* Stefano Spezia, Oct 28 2023 *)

A366918(n) = if(issquarefree(n), omega(n)!, if(4==n, 1, -1)); \\ Antti Karttunen, Jan 29 2025

a(n) = omega(n)! if n is squarefree (where omega(n) = A001221(n)),
       1 if n = 4, and
       -1 otherwise.

More terms from Antti Karttunen, Jan 29 2025

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cp's OEIS Frontend

A110821 SuperRefactorable numbers: m=A005179(n) such that k=m/n is an integer.

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A076931 Smallest k such that n*k has n divisors.

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A182749 a(n) = the smallest multiple of n with n digits and exactly n divisors. a(n) = 0 if no such number exists.

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A359964 Refactorable numbers (A033950) having more divisors than all smaller refactorable numbers.

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A136641 a(n) is the smallest positive integer that is coprime to n and has n divisors.

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A182750 a(n) = the largest multiple of n with n digits and exactly n divisors; a(n) = 0 if no such number exists.

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A076961 Smallest multiple of n with prime(n) divisors, or 0 if no such number exists.

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A109429 Rearrange terms of A050376 so that a(2^j)=2^(2^j) for j>=0.

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A359965 a(n) is the least arithmetic number (A003601) having exactly n divisors.

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