cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    f[n_]:=MemberQ[Last/@FactorInteger[n],3]; Select[Range[6!],f]
  • PARI
    isok(n) = vecsearch(vecsort(factor(n)[,2]), 3); \\ Michel Marcus, Jan 11 2015
  • Python
    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

A176313 First of two consecutive numbers with at least one 3 in their prime signature.

Original entry on oeis.org

135, 296, 343, 375, 999, 1160, 1431, 1592, 1624, 2295, 2375, 2456, 2727, 2888, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4887, 4913, 5048, 5144, 5319, 5480, 5831, 6183, 6344, 6375, 6615, 6776, 6858, 6859, 7479, 7624, 7640, 7749, 7911, 8072, 8375, 8775, 8936, 9125, 9207, 9368, 9624, 10071, 10232, 10375, 10503, 10632, 10664, 10984, 11124, 11319, 11367, 11528, 11624, 11799, 11960
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

Keywords

Examples

			135 is a term since 135 = 3^3 * 5 and 136 = 2^3 * 17.

Links

Crossrefs

A068140 lists the smallest of two consecutive numbers such that each is divisible by a cube greater than 1. See also A000578, A001235, A176297, A176350.

Programs

  • Mathematica
    f[n_]:=MemberQ[Last/@FactorInteger[n], 3]; Select[Range[8!],f[#]&&f[#+1]&]

Extensions

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A000578, A001235, A176297, A176313, A176350.
A176359 is a subsequence.

Programs

  • Mathematica
    f[n_]:=Count[Last/@FactorInteger[n],3]>1; Select[Range[8!],f]
  • PARI
    is(n)=#select(e->e==3, factor(n)[,2])>1 \\ Charles R Greathouse IV, Oct 19 2015

Extensions

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A000578, A001235, A176297, A176313, A176350.
Subsequence of A109399.

Programs

  • Mathematica
    f[n_]:=Count[Last/@FactorInteger[n],3]>2; Select[Range[10!],f]
  • PARI
    is(n) = #select(x -> x == 3, factor(n)[, 2]) > 2; \\ Amiram Eldar, Jul 22 2024

Extensions

Edited by Matthew Vandermast, Dec 09 2010
Showing 1-4 of 4 results.

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