A030626 -id:A030626 - cp's OEIS frontend

A182676 a(n) is the largest n-digit number with exactly 8 divisors, a(n) = 0 if no such number exists.

0, 88, 999, 9994, 99995, 999994, 9999994, 99999994, 999999998, 9999999995, 99999999998, 999999999998, 9999999999998, 99999999999998, 999999999999995, 9999999999999998, 99999999999999998, 999999999999999987, 9999999999999999995, 99999999999999999985, 999999999999999999995

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(n) is the largest 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.

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

Cf. A000005, A030626, A182675.

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

a(n)=forstep(k=10^n-1,10^(n-1),-1,numdiv(k)==8 & return(k)) \\ M. F. Hasler, Nov 27 2010

a(n) = max {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
Given terms double-checked with given PARI code by M. F. Hasler, Nov 27 2010
a(20)-a(21) from Amiram Eldar, Apr 09 2024

A330809 Triangular numbers having exactly 8 divisors.

Original entry on oeis.org

66, 78, 105, 136, 190, 231, 351, 406, 435, 465, 561, 595, 741, 861, 903, 946, 1378, 1431, 1653, 2211, 2278, 2485, 3081, 3655, 3741, 4371, 4465, 5151, 5253, 5995, 6441, 7021, 7503, 8515, 8911, 9453, 9591, 10011, 10153, 10585, 11026, 12561, 13366, 14878, 15051

Offset: 1

Jon E. Schoenfield, Jan 11 2020

nonn

Terms may be categorized as belonging to the following types:
type 1: products of 3 distinct primes p,q,r such that 2*p*q + 1 = r: 78, 406, 465, ... (27108 of the first 100000 terms);
type 2: products of 3 distinct primes p,q,r such that 2*p*q - 1 = r: 66, 190, 435, ... (26848 of the first 100000 terms);
type 3: products of 3 distinct primes p,q,r such that p*q + 1 = 2*r: 231, 561, 1653, ... (23050 of the first 100000 terms);
type 4: products of 3 distinct primes p,q,r such that p*q - 1 = 2*r: 105, 595, 741, ... (22983 of the first 100000 terms);
type 5: products of the cube of a prime p and a distinct prime q such that 2*p^3 + 1 = q: 136, 31375, 3544453, ... (6 of the first 100000 terms);
type 6: products of the cube of a prime p and a distinct prime q such that 2*p^3 - 1 = q: 1431, 1774977571, 12642646591, ... (4 of the first 100000 terms);
type 7: products of the cube of a prime p and a distinct prime q such that p^3 - 1 = 2*q: the only term of this type is 351 = 3^3 * 13.
(No term is a product of the cube of a prime p and a distinct prime q such that p^3 + 1 = 2*q.)

			Type
(see
cmts)  Initial terms             Notes
-----  ------------------------  -----------------------------
  1    78, 406, 465, ...         p*q*r such that 2*p*q + 1 = r
  2    66, 190, 435, ...         p*q*r such that 2*p*q - 1 = r
  3    231, 561, 1653, ...       p*q*r such that p*q + 1 = 2*r
  4    105, 595, 741, ...        p*q*r such that p*q - 1 = 2*r
  5    136, 31375, 3544453, ...  p^3*q such that 2*p^3 + 1 = q
  6    1431, 1774977571, ...     p^3*q such that 2*p^3 - 1 = q
  7    351 (only)                p^3*q such that p^3 - 1 = 2*q

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A000217 (triangular numbers) and A030626 (8 divisors).

Cf. A063440 (number of divisors of n-th triangular number), A292989 (triangular numbers having exactly 6 divisors).

[k:k in [1..16000]| IsSquare(8*k+1) and NumberOfDivisors(k) eq 8]; // Marius A. Burtea, Jan 12 2020

select(t -> numtheory:-tau(t) = 8, [seq(i*(i+1)/2, i=1..1000)]); # Robert Israel, Jan 13 2020

Select[PolygonalNumber@ Range[180], DivisorSigma[0, #] == 8 &] (* Michael De Vlieger, Jan 11 2020 *)

isok(k) = ispolygonal(k, 3) && (numdiv(k) == 8); \\ Michel Marcus, Jan 11 2020

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

A139416 a(n) is the smallest positive integer k such that d(k) = d(k+2n) = 2n, where d(m) (A000005) is the number of positive divisors of m, or 0 if no such k exists.

Original entry on oeis.org

3, 6, 12, 70, 600281, 60, 1458, 264, 450, 266875, 12670498046853, 480, 3998684814453099, 11968722, 299538, 3640, 49921080474853515591, 1980, 6107691150665283203125, 14960, 575047378296833, 4068173828125, 13936286848094463348388671875, 6552, 5035427051913

Offset: 1

Leroy Quet, Apr 20 2008

nonn

Does this sequence have a term for every positive integer n, or are there no solutions for some n?
There is no solution for any odd positive integer n to d(k) = d(k+n) = n.
If n is prime, then a(n) exists if and only if there exist either three primes p, q, r such that p^(2*n-1) +- 2*n = q^(n-1)*r or four primes p_1, q_1, p_2, q_2 such that p_1^(n-1)*q_1 + 2*n = p_2^(n-1)*q_2. - Vladimir Shevelev, Jul 14 2015

			For a(4) we want the smallest integer m such that d(m) = d(m+8) = 8. The positive integers that have 8 divisors each form the sequence: 24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 102, 104, 105, 110, ... (A030626)
The first (not necessarily adjacent) pair of integers with 8 divisors each that is separated by exactly 8 is (70,78). So a(4) is the least element of this pair, which is 70.
Let n=5, a(5) = 600281. According to our comment, 600281 is the smallest number such that there exist either three primes p, q, r such that p^9 +- 10 = q^4*r or four primes p_1, q_1, p_2, q_2 such that p_1^4*q_1 + 10 = p_2^4*q_2. Here p_1 = 11, q_1 = 41, p_2 = 3, q_2 = 7411. - _Vladimir Shevelev_, Jul 14 2015

Cf. A000005, A137532.

A subsequence of A175304.

f[n_] := Block[{a = {}, d, k, lim = 1000000}, d[x_] := DivisorSigma[0, x]; Do[k = 1; While[Nand[d[k] == d[k + d[k]], d[k] == 2 i] && k <= lim, k++]; If[k > lim, AppendTo[a, 0], AppendTo[a, k]], {i, n}]; a]; f@ 10 (* Michael De Vlieger, Jul 13 2015 *)

A_simple(n)=local(m=2);n*=2;until(numdiv(m)==n&numdiv(m+n)==n,m++);m
A_try_pair(p,q,n,limit)=
{
/* Helper for A_prime() */
/* Look for solution which is 0 mod p^(n-1) and -n*2 mod q^(n-1) */
local(m = chinese(Mod(0,p^(n-1)), Mod(-n*2,q^(n-1))));
forstep(x=lift(m), limit, component(m,1),
if(isprime(x\p^(n-1)) & isprime((x+n*2)\q^(n-1)), return(x)));
limit
}
A_try_above_below(m,n)=
{
/* Helper for A_prime() */
/* Function presumes that numdiv(m)==n*2 */
if(numdiv(m-n*2)==n*2, limit=m-n*2,
if(numdiv(m+n*2)==n*2, limit=m,
0))
}
A_prime(n,limit,pairmax=30)=
{
if (n%2==0 || !isprime(n), error("Only works for odd primes"));
if (default(primelimit) < limit\nextprime(pairmax+1)^(n-1),
default(primelimit, limit\nextprime(pairmax+1)^(n-1));
);
/* Evens with numdiv==n*2 are {2^(n*2-1)} u {2*p^(n-1)} u {2^(n-1)*p} */
/* Potential solutions must come from different sets */
/* Try above and below first two sets */
A_try_above_below(2^(n*2-1),n);
forprime(p=3, (limit\2)^(1/(n-1)),
if (A_try_above_below(2*p^(n-1),n), break));
/* Odd numbers with numdiv==n*2 are {p^(n*2-1)} u {p^(n-1)*q} */
/* Try where a(n) and a(n)+n*2 are (small prime)^(n-1)*(big prime) */
forprime(p=3, pairmax, forprime(q=3, pairmax,
if (p!=q, limit = A_try_pair(p,q,n,limit))));
/* Try above and below all other odd numbers with numdiv==n*2 */
forprime(p=pairmax+1, (limit\3)^(1/(n-1)),
forprime(q=3, limit\p^(n-1),
if (p!=q & A_try_above_below(p^(n-1)*q,n), break)));
forprime(p=3, limit^(1/21),
if (A_try_above_below(p^21,n), break));
limit
} /* Martin Fuller, Apr 20 2008 */

First 10 terms calculated by M. F. Hasler
a(11)-a(20) from Martin Fuller, Apr 20 2008
a(21)-a(25) from Jinyuan Wang, Sep 24 2021

A297401 Non-sphenic numbers with exactly 8 divisors.

Original entry on oeis.org

24, 40, 54, 56, 88, 104, 128, 135, 136, 152, 184, 189, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 459, 472, 488, 513, 536, 568, 584, 621, 632, 664, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029, 1048, 1096, 1107, 1112, 1161, 1192

Offset: 1

G. L. Honaker, Jr., Dec 29 2017

nonn

These are the numbers of the form p^3*q (with primes p and q distinct) or p^7. Thus it is the union of A065036 and A092759, and this can be used for direct enumeration. - Alex Meiburg, Dec 31 2017

Robert Israel, Table of n, a(n) for n = 1..10000
Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios!
Wikipedia, Sphenic number

Subsequence of A030626.

Cf. A000005, A030626, A065036, A092759.

N:= 1000: # to get all terms <= N
P:= select(isprime, [2,seq(i,i=3..N)]):
R:= NULL:
for p in P do
  if p^7 <= N then R:= R, p^7 fi;
  if p^3 > N then break fi;
  for q in P while p^3*q <= N do if q <> p then R:= R, p^3*q fi od:
od:
sort([R]); # Robert Israel, Dec 31 2017

Select[Range@ 1200, And[DivisorSigma[0, #] == 8, Nand[PrimeNu[#] == 3, PrimeOmega[#] == 3]] &] (* Michael De Vlieger, Dec 29 2017 *)

isok(n) = !((bigomega(n)==3) && (omega(n)==3)) && (numdiv(n) == 8); \\ Michel Marcus, Dec 29 2017

from sympy import primepi, primerange, integer_nthroot
def A297401(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-sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,4)[0])-primepi(integer_nthroot(x,7)[0]))
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Equals {A030626} \ {A007304}. - Omar E. Pol, Dec 30 2017

More terms from Michel Marcus, Dec 29 2017

A350767 a(1)=1. Thereafter, a(n+1) is the least unused number k such that either d(j(n)) properly divides d(k) or d(k) properly divides d(j(n)), where j(n) = a(n)+1 and d is the divisor counting function A000005.

Original entry on oeis.org

1, 6, 8, 12, 10, 14, 2, 15, 48, 18, 20, 3, 28, 21, 5, 7, 11, 4, 22, 24, 32, 13, 17, 9, 19, 23, 26, 29, 27, 25, 30, 33, 31, 37, 40, 34, 41, 35, 49, 43, 47, 16, 38, 42, 39, 46, 44, 53, 51, 59, 45, 54, 56, 60, 50, 61, 66, 52, 55, 57, 67, 71, 58, 62, 72, 63, 192, 65

Offset: 1

David James Sycamore, Jan 14 2022

nonn

If d(j(n)) is prime p then d(a(n+1)) must be properly divisible by p. In practice the proper divisor for computation of a(n+1) toggles between d(j(n)) and d(k).
Conjecture: This is a permutation of the positive integers. Numbers with the same number (tau) of divisors appear in their natural orders (e.g., primes, semiprimes, squares).
The plot, after the first few terms, resolves itself into points tightly packed on and around a straight line of slope 1, with exceptional points appearing as significant upward or downward "spikes".
When d(j(n)) is prime p appearing for the first time in the sequence J = {d(j(a(n)), n>=1}, then a(n+1) is the smallest number with 2p divisors, which produces a significantly large upward spike above the straight line (6, 12, 48, 192, 3072, 12288, ...).
When d(j(a(n)) is 2p, seen for the first time in J, then a(n+1) is the smallest number with p divisors, which produces a large downward spike, below the straight line (2, 4, 16, 64, 1024, 4096, ...).
The sequence of fixed points starts: 1, 46, 69, 74, 110, 140, 142, 152, 154, 178, ... apparently becoming denser as n increases.

			a(1)=1, so j(1)=2, d(j(1))=2, a prime, so we need the smallest unused k such that d(k) is properly divisible by 2, hence a(2)=6.
a(2)=6, j(2)=4, d(j(2))=3, a prime so we need the smallest unused k such that d(k) is properly divisible by 3, hence a(3)=8.

Michael De Vlieger, Table of n, a(n) for n = 1..10001
Michael De Vlieger, Scatterplot of a(n), n = 1..256, color coded to show primes in red, evens in blue, 2 in magenta, odd composites in black. A gold highlight indicates terms such that (a(n)+1) | a(n+1). Outlying points are annotated.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16. Terms in A3680(p) such that p is prime appear annotated in red, and those in A061286 appear in blue.

Cf. A000040, A001248, A030514, A054753, A030516, A030626, A085986, A178739, A030629, A030630, A030631, A030632, A030633, A030634, A005179, A061286, A003680.

Block[{c, d, j, k, u, nn}, nn = 120; j = u = 2; c[] = 0; c[1] = 1; Do[d[i] = DivisorSigma[0, i], {i, 2^(Ceiling@ Log2[nn] + 3)}]; {1}~Join~Reap[Do[k = u; While[Nand[Or[Divisible[d[j], d[k]], Divisible[d[k], d[j]]], d[j] != d[k], c[k] == 0], k++]; Sow[k]; c[k] = i; j = k + 1; If[k == u, While[c[u] != 0, u++]], {i, nn}] ][[-1, -1]] ] (* _Michael De Vlieger, Jan 14 2022 *)

isok(k, last, set) = if (!setsearch(set, k), my(ndk=numdiv(k), ndl=numdiv(last+1)); (ndl != ndk) && ((!(ndk % ndl)) || (!(ndl % ndk))));
lista(nn) = {my(last = 1, list = List(last), set = Set(list)); for (n=2, nn, my(k=1); while (!isok(k, last, set), k++); listput(list, k); set = Set(list); last = k;); Vec(list);} \\ Michel Marcus, Jan 15 2022

More terms from Michael De Vlieger, Jan 14 2022

cp's OEIS Frontend

A182676 a(n) is the largest n-digit number with exactly 8 divisors, a(n) = 0 if no such number exists.

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A330809 Triangular numbers having exactly 8 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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A139416 a(n) is the smallest positive integer k such that d(k) = d(k+2*n) = 2*n, where d(m) (A000005) is the number of positive divisors of m, or 0 if no such k exists.

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A297401 Non-sphenic numbers with exactly 8 divisors.

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A350767 a(1)=1. Thereafter, a(n+1) is the least unused number k such that either d(j(n)) properly divides d(k) or d(k) properly divides d(j(n)), where j(n) = a(n)+1 and d is the divisor counting function A000005.

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A139416 a(n) is the smallest positive integer k such that d(k) = d(k+2n) = 2n, where d(m) (A000005) is the number of positive divisors of m, or 0 if no such k exists.