A176313 -id:A176313 - cp's OEIS frontend

A176297 Numbers with at least one 3 in their prime signature.

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 432, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594, 600, 616, 621, 632, 648, 664, 675, 680, 686, 696, 702, 712

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

That is, if n = p1^e1 p2^e2 ... pr^er for distinct primes p1, p2,..., pr, then one of the exponents must be 3 for n to be in this sequence.

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/p^3 + 1/p^4) = 0.0952910730... - Amiram Eldar, Nov 14 2020

			8=2^3, 24=2^3*3, 27=3^3, 40=2^3*5, ...

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

Cf. A000578, A001235, A176313, A176350, A109399, A176359.

filter:= proc(x) local F; F:= map(t->t[2],ifactors(x)[2]);has(F,3) end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 11 2015
# alternative:
isA176297 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) = 3 then
            return true;
        end if;
    end do:
    false ;
end proc: # R. J. Mathar, Dec 08 2015

f[n_]:=MemberQ[Last/@FactorInteger[n],3]; Select[Range[6!],f]

isok(n) = vecsearch(vecsort(factor(n)[,2]), 3); \\ Michel Marcus, Jan 11 2015

from sympy import factorint
def ok(n): return 3 in [e for e in factorint(n).values()]
print(list(filter(ok, range(713)))) # Michael S. Branicky, Aug 24 2021

A176350 First of three consecutive numbers with at least one 3 in their prime signature.

Original entry on oeis.org

6858, 22625, 28375, 40472, 48248, 49624, 58374, 59750, 94471, 102248, 103624, 107702, 112374, 122823, 129623, 133623, 136214, 136375, 153063, 164295, 187623, 190375, 197910, 199624, 210248, 211624, 220374, 221750, 238248, 246616, 260874, 264248, 275750, 280231, 298375, 300806, 312471, 329623, 336824, 346086, 349623, 352375, 356375

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

			6858=2*3^3*127,6859=19^3,6860=2^2*5*7^3,..

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

Cf. A000578, A001235, A176297, A176313.

f[n_]:=MemberQ[Last/@FactorInteger[n], 3]; Select[Range[9!],f[#]&&f[#+1]&&f[#+2]&]
SequencePosition[Table[If[MemberQ[FactorInteger[n][[All,2]],3],1,0],{n,360000}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 09 2019 *)

Edited by Matthew Vandermast, Dec 09 2010

A109399 Numbers with at least two 3s in their prime signature.

Original entry on oeis.org

216, 1000, 1080, 1512, 2376, 2744, 2808, 3000, 3375, 3672, 4104, 4968, 5400, 6264, 6696, 6750, 7000, 7560, 7992, 8232, 8856, 9000, 9261, 9288, 10152, 10584, 10648, 11000, 11448, 11880, 12744, 13000, 13176, 13500, 13720, 14040, 14472, 15336, 15768, 16632, 17000, 17064, 17576, 17928, 18360, 18522, 19000, 19224, 19656, 20520, 20952, 21000, 21816, 22248, 23000, 23112, 23544, 23625, 24408, 24696, 24840, 25704, 26136, 27000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least two values of i.
Does not include all numbers with at least two unitary prime power divisors that are cubes (see example section).
The asymptotic density of this sequence is 1 - (1 + Sum_{p prime} ((p-1)/(p^4-p+1))) * Product_{p prime} (1-1/p^3+1/p^4) = 0.0024593812036570543518... . - Amiram Eldar, Jul 22 2024

			216 = 2^3*3^3, 1000 = 2^3*5^3, 1080 = 2^3*3^3*5, ...
On the other hand, 1728 = 2^6*3^3, 8000 = 2^6*5^3 and 21952 = 2^6*7^3 are not in the sequence.

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

Cf. A000578, A001235, A176297, A176313, A176350.

A176359 is a subsequence.

f[n_]:=Count[Last/@FactorInteger[n],3]>1; Select[Range[8!],f]

is(n)=#select(e->e==3, factor(n)[,2])>1 \\ Charles R Greathouse IV, Oct 19 2015

Edited by Matthew Vandermast, Dec 07 2010

A176359 Numbers with at least three 3s in their prime signature.

Original entry on oeis.org

27000, 74088, 189000, 287496, 297000, 343000, 351000, 370440, 459000, 474552, 513000, 621000, 783000, 814968, 837000, 963144, 999000, 1029000, 1061208, 1107000, 1157625, 1161000, 1259496, 1269000, 1323000, 1331000, 1407672, 1431000, 1437480, 1481544, 1593000, 1647000, 1704024, 1809000, 1852200, 1917000, 1971000, 2012472, 2079000, 2133000, 2148552

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least three values of i.

The asymptotic density of this sequence is 1 - (1 + s(1) + s(1)^2/2 - s(2)/2) * Product_{p prime} (1-1/p^3+1/p^4) = 0.000018992895371889141564..., where s(k) = Sum_{p prime} ((p-1)/(p^4-p+1))^k. - Amiram Eldar, Jul 22 2024

			27000 is a term since 27000 = 2^3 * 3^3 * 5^3.
74088 is a term since 74088 = 2^3 * 5^3 * 7^3.

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

Cf. A000578, A001235, A176297, A176313, A176350.

Subsequence of A109399.

f[n_]:=Count[Last/@FactorInteger[n],3]>2; Select[Range[10!],f]

is(n) = #select(x -> x == 3, factor(n)[, 2]) > 2; \\ Amiram Eldar, Jul 22 2024

Edited by Matthew Vandermast, Dec 09 2010

A372691 Numbers k such that k and k+1 are both nonsquarefree numbers whose number of divisors is a power of 2 (A175496).

Original entry on oeis.org

135, 296, 343, 375, 999, 1160, 1431, 1592, 1624, 2295, 2375, 2456, 2727, 2943, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4887, 4913, 5048, 5144, 5319, 5480, 6183, 6344, 6375, 6858, 7479, 7624, 7640, 7749, 7911, 8072, 8375, 8936, 9207, 9368, 9624, 9855, 10071, 10232

Offset: 1

Amiram Eldar, May 10 2024

First differs from A176313 at n = 14.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 0, 5, 43, 404, 4086, 40839, 408366, 4083039, 40830831, ... . Apparently, the asymptotic density of this sequence exists and equals 0.004083... .

			135 is a term since 135 = 3^3 * 5 and 136 = 2^3 * 17 are both nonsquarefree numbers, and the number of divisors of 135 and 136 are both 8 = 2^3.
343 is a term since 343 = 7^3 and 344 = 2^3 * 43 are both nonsquarefree numbers, the number of divisors of 343 is 4 = 2^2, and the number of divisors of 344 is 8 = 2^3.

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

Subsequence of A013929, A068781, A175496 and A372690.

Cf. A176313.

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; q[n_] := q[n] = Module[{e = FactorInteger[n][[;;, 2]]}, Max[e] > 1 && pow2Q[Times @@ (e+1)]]; Select[Range[500], q[#] && q[# + 1] &]

is(n) = {my(f = factor(n), d = numdiv(f)); n > 1 && vecmax(f[, 2]) > 1 && d >> valuation(d, 2) == 1;}
lista(kmax) = {my(is1 = is(1), is2); for(k = 2, kmax, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);}

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A176297 Numbers with at least one 3 in their prime signature.

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A176350 First of three consecutive numbers with at least one 3 in their prime signature.

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A109399 Numbers with at least two 3s in their prime signature.

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A176359 Numbers with at least three 3s in their prime signature.

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A372691 Numbers k such that k and k+1 are both nonsquarefree numbers whose number of divisors is a power of 2 (A175496).

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