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 1-10 of 10 results.

A197863 Smallest powerful number that is a multiple of n.

Original entry on oeis.org

1, 4, 9, 4, 25, 36, 49, 8, 9, 100, 121, 36, 169, 196, 225, 16, 289, 36, 361, 100, 441, 484, 529, 72, 25, 676, 27, 196, 841, 900, 961, 32, 1089, 1156, 1225, 36, 1369, 1444, 1521, 200, 1681, 1764, 1849, 484, 225, 2116, 2209, 144, 49, 100, 2601, 676, 2809, 108, 3025
Offset: 1

Views

Author

Franklin T. Adams-Watters, Oct 18 2011

Keywords

Links

Crossrefs

Cf. A055231, A087320, A001694, A086463.
Cf. A027748, A124010, A007948.

Programs

  • Haskell
    a197863 n = product $
   zipWith (^) (a027748_row n) (map (max 2) $ a124010_row n)
-- Reinhard Zumkeller, Jan 06 2012
  • Mathematica
    With[{pwrnos=Join[{1},Select[Range[5000],Min[Transpose[ FactorInteger[#]] [[2]]]>1&]]},Flatten[Table[Select[pwrnos,Divisible[#,n]&,1],{n,60}]]] (* Harvey P. Dale, Aug 14 2012 *)
f[p_, e_] := p^Max[e, 2]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)
  • PARI
    a(n)=local(fm=factor(n));prod(k=1,matsize(fm)[1],fm[k,1]^max(fm[k,2],2))

Formula

Multiplicative with a(p^e) = p^max(e,2).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (2*p-1)/(p^2*(p-1))) = 2.71098009471568319328... . - Amiram Eldar, Jul 29 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 2/p^2 + 2/p^4 - 1/p^5) = 0.2165355664... . - Amiram Eldar, Nov 19 2022
a(n) = n * A055231(n). - Amiram Eldar, Sep 01 2023

A356191 a(n) is the smallest exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 32, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 32, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 96, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65
Offset: 1

Views

Author

Amiram Eldar, Jul 29 2022

Keywords

Links

Crossrefs

Cf. A064549, A007913, A268335, A336643, A350390.
Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356192, A356193, A356194.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], p^e, p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^f[i,2], f[i,1]^(f[i,2]+1)))};
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

Formula

Multiplicative with a(p^e) = p^e if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A268335.
a(n) = A064549(n)/A007913(n).
a(n) = n*A336643(n).
a(n) = n^2/A350390(n).
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(6-5*s) + p^(7-5*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(3-3*s) - p^(4-3*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(2) * n^2 / 24 * (log(n) + 3*gamma - 1/2 + 12*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
f'(2) = f(2) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.7165968208567630786609552448708126340725121316268495170070986645608062483...
and gamma is the Euler-Mascheroni constant A001620. (End)

A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 32, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336
Offset: 1

Views

Author

Amiram Eldar, Jul 29 2022

Keywords

Comments

First differs from A053149 and A356193 at n=16.

Links

Crossrefs

Cf. A001694, A013664, A268335, A335988.
Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356191, A356193, A356194.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], p^Max[e, 3], p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
  • PARI
    a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^max(f[i,2],3), f[i,1]^(f[i,2]+1)))};

Formula

Multiplicative with a(p^e) = p^max(e,3) if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A335988.
a(n) = A356191(n) iff n is a powerful number (A001694).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p^2-1)/(p^3*(p^2-1))) = 1.69824776889117043774... .
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(6)/4) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^8 + 1/p^9 - 1/p^10) = 0.1559368144... . - Amiram Eldar, Nov 13 2022

A356193 a(n) is the smallest cubefull number (A036966) that is a multiple of n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 16, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336
Offset: 1

Views

Author

Amiram Eldar, Jul 29 2022

Keywords

Comments

First differs from A053149 and A356192 at n=16.

Links

Crossrefs

Cf. A002117, A036966, A360541.
Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356194.

Programs

  • Mathematica
    f[p_, e_] := p^Max[e, 3]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
  • PARI
    a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^max(f[i,2],3))};

Formula

Multiplicative with a(p^e) = p^max(e,3).
a(n) = n iff n is in A036966.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p-2)/(p^3*(p-1))) = 1.76434793373691907811... . - Amiram Eldar, Jul 29 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(3)/4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.1559111567... . - Amiram Eldar, Nov 13 2022
a(n) = n * A360541(n). - Amiram Eldar, Sep 01 2023

A356194 a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 25, 26, 81, 28, 29, 30, 31, 256, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 162, 55, 112, 57, 58, 59, 60, 61, 62, 63, 256, 65, 66, 67
Offset: 1

Views

Author

Amiram Eldar, Jul 29 2022

Keywords

Links

Crossrefs

Cf. A138302, A353897.
Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193.

Programs

  • Mathematica
    f[p_, e_] := p^(2^Ceiling[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    s(n) = {my(e=logint(n,2)); if(n == 2^e, n, 2^(e+1))};
a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^s(f[i,2]))};

Formula

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e))).
a(n) = n iff n is in A138302.

A203025 Largest perfect power divisor of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1
Offset: 1

Views

Author

Antonio Roldán, Dec 28 2011

Keywords

Comments

This sequence shares many elements with A057521, but is not identical: A057521(72)=72 but a(72)=36.
Not multiplicative: a(49)=49; a(125)=125, a(49*125) = 1225 <> 49*125.

Examples

			a(40)=a(2^3*5)=2^3=8.

Links

Crossrefs

Cf. A057521, A087320, A274006.

Programs

  • Maple
    A203025:=proc(n)
    local a,Le,d,i,k,pe;
    pe := ifactors(n)[2];
    Le := {seq(i[2],i=pe)} minus {1};
    a := 1;
    for k in Le do
        d := mul(i[1]^(k*floor(i[2]/k)), i=pe) ;
        a:=max(a,d);
    end do;
    a
end proc:
seq(A203025(n),n=1..10000); # Felix Huber, Jun 01 2025
  • Mathematica
    Table[If[SquareFreeQ[n], 1, s = FactorInteger[n]; Max[Table[Times @@ Cases[s, {p_, ep_} :> p^i /; (ep >= i)], {i, 2, Max[s[[All, 2]]]}]]], {n, 100}] (* Olivier Gerard, Jun 03 2016 *)
  • PARI
    a(n)=my(f=factor(n),mx=1);for(e=2,if(n>1,vecmax(f[,2])), mx=max(mx,prod(i=1,#f[,1],f[i,1]^(f[i,2]\e*e))));mx \\ Charles R Greathouse IV, Dec 28 2011

Formula

a(n) = max{ A001597(k) : A001597(k)|n }. - R. J. Mathar, Jun 09 2016

Extensions

Values matching definition restored by Franklin T. Adams-Watters, Jun 06 2016

A087321 Smallest multiple of n which is a perfect power (at least a square) of a squarefree number.

Original entry on oeis.org

1, 4, 9, 4, 25, 36, 49, 8, 9, 100, 121, 36, 169, 196, 225, 16, 289, 36, 361, 100, 441, 484, 529, 216, 25, 676, 27, 196, 841, 900, 961, 32, 1089, 1156, 1225, 36, 1369, 1444, 1521, 1000, 1681, 1764, 1849, 484, 225, 2116, 2209, 1296, 49, 100, 2601, 676, 2809, 216
Offset: 1

Views

Author

Amarnath Murthy, Sep 03 2003

Keywords

Examples

			a(12) = 3*12 = 36 = 6^2.
a(24) = 9*24 = 216 = 6^3.

Links

Crossrefs

Cf. A007947, A051903, A072774, A087320.

Programs

  • Mathematica
    a[n_] := (Times @@ First@#)^(Max[Max @@ Last@#, 2]) &@ Transpose @ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 23 2021 *)

Formula

If n = p^a*q^b*r^c... p, q, r are primes and max(a, b, c, ...) = K then a(n) = p^K*q^K*r^K...
a(n) = A007947(n)^max(A051903(n), 2). - David Wasserman, May 04 2005

Extensions

More terms from David Wasserman, May 04 2005

A160400 a(n) is the smallest positive integer such that a(n)*n = j^k, for some j (j>=1) and k (k>=2).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 6, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 4, 55, 14, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2, 73, 74, 3, 19, 77, 78, 79, 5
Offset: 1

Views

Author

Leroy Quet, May 12 2009

Keywords

Links

Crossrefs

Cf. A087320, A001597, A007913.

Programs

  • Maple
    isA001597 := proc(n) local e,p ; if n = 1 then RETURN(true) ; fi; p := [] ; for e in ifactors(n)[2] do p := [op(p), op(2,e) ] ; od: if igcd(op(p)) > 1 then true; else false; fi; end: A160400 := proc(n) local a; for a from 1 do if isA001597(a*n) then RETURN(a) ; fi; od: end: seq(A160400(n),n=1..120) ; # R. J. Mathar, May 26 2009
  • Mathematica
    a[n_] := SelectFirst[Range[n], GCD @@ FactorInteger[n*#][[;; , 2]] > 1 &]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

Formula

a(n) = A087320(n)/n.
a(n) = A007913(n) for n cubefree, a(n) = 1 for n in A001597. - Charlie Neder, Dec 26 2018

Extensions

More terms from R. J. Mathar, May 26 2009

A372329 a(n) is the smallest multiple of n whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 128, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 128, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 384, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65
Offset: 1

Views

Author

Amiram Eldar, Apr 28 2024

Keywords

Links

Crossrefs

Cf. A036537, A372328.
Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193, A356194.
Differs from A102631 at n = 8, 24, 27, 32, 40, 54, 56, 64, ... .

Programs

  • Mathematica
    f[p_, e_] := p^(2^Ceiling[Log2[e + 1]] - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    s(n) = {my(e=logint(n + 1, 2)); if(n + 1 == 2^e, n, 2^(e+1) - 1)};
a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i, 1]^s(f[i, 2]))};

Formula

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e+1)) - 1).
a(n) = n * A372328(n).
a(n) = n if and only if n is in A036537.
a(n) <= n^2, with equality if and only if n = 1.

A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 18, 13, 14, 15, 2, 17, 12, 19, 50, 21, 22, 23, 54, 5, 26, 3, 98, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 7, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 2, 65, 66, 67
Offset: 1

Views

Author

Gus Wiseman, May 18 2018

Keywords

Comments

Image is the weak numbers A052485, on which n -> a(n) is an involution whose fixed points are the squarefree numbers A005117.

Links

Crossrefs

Cf. A001597, A001694, A005117, A007916, A007947, A013929, A051903, A052410, A062759, A066638, A072774, A087320, A303554.

Programs

  • Mathematica
    acj[n_]:=Module[{f,m},f=FactorInteger[n];m=Max[Last/@f];Times@@Table[p[[1]]^(m-p[[2]]+1),{p,f}]];
Array[acj,100]
  • PARI
    a(n) = {if(n==1, 1, my(f = factor(n), e = vecmax(f[,2]) + 1); prod(i = 1, #f~, f[i,1]^e) / n);} \\ Amiram Eldar, Feb 12 2023

Formula

If n = Product_{i = 1..k} prime(x_i)^y_i, then a(n) = Product_{i = 1..k} prime(x_i)^(max{y_1,...,y_k} - y_i + 1).
Showing 1-10 of 10 results.

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