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.

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A055742 Numbers k such that k and EulerPhi(k) have same number of prime factors, counted without multiplicity.

Original entry on oeis.org

1, 3, 4, 5, 8, 14, 16, 17, 18, 21, 22, 26, 28, 32, 33, 35, 36, 38, 39, 44, 45, 46, 50, 52, 54, 55, 56, 57, 58, 63, 64, 65, 69, 72, 74, 75, 76, 82, 87, 88, 91, 92, 94, 95, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119, 123, 128, 133, 135, 141, 144, 145, 146, 148
Offset: 1

Views

Author

Labos Elemer, Jul 11 2000

Keywords

Examples

			Known Fermat primes 3 and 5 are terms because their phi value is divisible only by 2. Several composites are also here, such as {50, 999, 1000} with prime factors (2,5), (3,37) and (2,5); their phi values, {20, 648, 400}, also have 2 prime factors: (2,5), (2,3), (2,5).

Links

Crossrefs

Cf. A001221, A000010, A055744, A070418.

Programs

  • Haskell
    a055742 n = a055742_list !! (n-1)
a055742_list = [x | x <- [1..], a001221 x == a001221 (a000010 x)]
-- Reinhard Zumkeller, Apr 14 2015
  • Mathematica
    Select[Range[200],PrimeNu[#]==PrimeNu[EulerPhi[#]]&] (* Harvey P. Dale, Sep 12 2014 *)
  • PARI
    is(n)=my(f=factor(n)); #f~ == omega(eulerphi(f)) \\ Charles R Greathouse IV, Mar 01 2017

Formula

A001221(A000010(n)) = A001221(n).

Extensions

Definition clarified by Harvey P. Dale, Sep 12 2014

A081381 Numbers n such that n and tau(n) = A000005(n) have the same prime factors (ignoring multiplicity).

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 72, 80, 96, 108, 128, 288, 448, 486, 625, 720, 768, 864, 972, 1152, 1200, 1250, 1620, 1944, 2000, 2025, 2560, 4032, 4050, 5000, 5625, 6144, 6561, 6912, 7500, 7776, 8748, 9408, 10800, 11250, 11264, 12960, 13122, 16200, 18000, 18432, 19440
Offset: 1

Views

Author

Labos Elemer, Mar 26 2003

Keywords

Examples

			n = 5000 = 2*2*2*5*5*5*5, tau(5000) = 20 = 2*2*5, common prime factors: {2,5}

Links

Crossrefs

Cf. A000005, A027598, A055744, A065642, A081377-A081380.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Do[s=ba[DivisorSigma[0, n]]; If[Equal[s, ba[n]], Print[n]], {n, 1, 10000}]
  • PARI
    is(n)=my(f=factor(n)); factor(numdiv(f))[,1]==f[,1] \\ Charles R Greathouse IV, Oct 19 2017

A116002 n*phi(n)*phi(phi(n)) is a square.

Original entry on oeis.org

1, 8, 27, 32, 108, 128, 243, 250, 432, 512, 686, 972, 1000, 1331, 1728, 2048, 2187, 2197, 2744, 3375, 3888, 4000, 5324, 6174, 6250, 6912, 8192, 8748, 8788, 9826, 10976, 13500, 15552, 16000, 19683, 19773, 21296, 24696, 25000, 27648, 30375
Offset: 1

Views

Author

Giovanni Resta, Feb 13 2006

Keywords

Examples

			15552*phi(15552)*phi(phi(15552))=373248^2.

Links

Crossrefs

Cf. A055744.

Programs

  • Mathematica
    esQ[n_]:=Module[{e=EulerPhi[n]},IntegerQ[Sqrt[n e EulerPhi[e]]]]; Select[Range[31000],esQ] (* Harvey P. Dale, Dec 16 2011 *)

A230400 Numbers n such that n = abc = 2(ab+ac+bc) for some positive integers a,b,c.

Original entry on oeis.org

216, 250, 256, 288, 400, 432, 450, 486, 576, 882
Offset: 1

Views

Author

M. F. Hasler, Oct 18 2013

Keywords

Comments

Otherwise said: Volumes of integer-sided cubes equal to their surface area (assuming dimensionless unit of length).
The sequence is a finite subsequence of A055744, A069167, A073539, A090779 and A137845.

Examples

			The triples (a,b,c) ordered by largest member(s) are (6,6,6), (8,8,4), (10,5,5), (12,6,4), (12,12,3), (15,10,3), (18,9,3), (20,5,4), (24,8,3), (42,7,3).

Links

Crossrefs

Cf. A229941.

Programs

  • PARI
    L=[];for(a=1,99,for(b=1,a,for(c=1,b,a*b*c==2*(a*b+b*c+a*c)&&!printf("(%d,%d,%d), ",a,b,c)&&L=concat(L,a*b*c))));vecsort(L)

A201009 Numbers m such that the set of distinct prime divisors of m is equal to the set of distinct prime divisors of the arithmetic derivative m'.

Original entry on oeis.org

1, 4, 16, 27, 108, 144, 256, 432, 500, 784, 972, 1323, 1728, 2700, 2916, 3125, 3456, 5292, 8788, 11664, 12500, 13068, 15376, 16875, 19683, 20736, 23328, 25000, 27648, 28125, 31212, 34300, 47916, 54000, 57132, 65536, 72000, 78732, 97556, 102400, 103788, 104544
Offset: 1

Views

Author

Paolo P. Lava, Jan 09 2013

Keywords

Comments

A027748(n,k) = A027748(A003415(n),k) for k=1..A001221(n). - Reinhard Zumkeller, Jan 16 2013
A051674 is a subsequence of this sequence.

Examples

			n = 1728 = 2^6*3^3, n' = 6912 = 2^8*3^3 have the same prime factors 2 and 3.

Links

Crossrefs

Cf. A001221, A003415, A027748, A051674, A055744, A081377, A110751, A110819.

Programs

  • Haskell
    a201009 = a201009_list
a201009_list = 1 : filter
   (\x -> a027748_row x == a027748_row (a003415 x)) [2..]
-- Reinhard Zumkeller, Jan 16 2013
  • Maple
    with(numtheory);
A201009:=proc(q)
local a,b,k,n;
for n from 1 to q do
  a:=ifactors(n)[2]; b:=ifactors(n*add(op(2,p)/op(1,p),p=ifactors(n)[2]))[2];
  if nops(a)=nops(b) then
    if product(a[k][1],k=1..nops(a))=product(b[k][1],k=1..nops(a)) then print(n);
fi; fi; od; end:
A201009(100000); # Paolo P. Lava, Jan 09 2013
  • Python
    from sympy import primefactors, factorint
A201009 = [n for n in range(1,10**5) if primefactors(n) == primefactors(sum([int(n*e/p) for p,e in factorint(n).items()]) if n > 1 else 0)] # Chai Wah Wu, Aug 21 2014

A214266 Numbers n such that n, phi(n) and sigma(n) have same set of prime factors.

Original entry on oeis.org

1, 103654150315463023813006470, 207308300630926047626012940, 414616601261852095252025880, 518270751577315119065032350, 829233202523704190504051760, 982794906694760522078876160, 1036541503154630238130064700
Offset: 1

Views

Author

Michel Marcus, Jul 09 2012

Keywords

Examples

			For n = 207308300630926047626012940 = 2^2*3^4*5*7*11*13^3*17^3*29^2*31^3*37^2*67^2
phi(n) = 2^18*3^9*5^2*7*11*13^2*17^2*29*31^2*37*67 and
sigma(n) = 2^16*3^6*5^2*7^5*11^2*13^2*17*29*31*37*67^2
have the same prime factors.

Links

Crossrefs

Cf. A027598, A055744, A081377.

A306478 The phi-radical numbers: composite numbers m such that rad(phi(m)) = rad(m-1).

Original entry on oeis.org

1729, 2431, 6601, 9605, 10585, 12801, 15211, 30889, 46657, 69751, 88561, 92929, 105001, 159895, 272323, 368641, 460801, 534061, 610051, 622909, 950797, 992251, 1047619, 1372801, 1374895, 1745701, 1902691, 2210671, 2628073, 2704801, 3225601, 5629339, 5690251, 6840001, 9738751
Offset: 1

Views

Author

Amiram Eldar and Thomas Ordowski, Feb 18 2019

Keywords

Comments

Problem: are there infinitely many such numbers?
These numbers are odd squarefree. They contain many Carmichael numbers.
The smallest such semiprime is 1525781251 = 19531*78121, see A306479.

Links

Crossrefs

Cf. A000010, A002997, A007947, A055744 (rad(phi(n)) = rad(n)), A306479.

Programs

  • Mathematica
    rad[n_] := Times @@ (First@# & /@ FactorInteger@n); Select[Range[100000], CompositeQ[#] && rad[EulerPhi[#]] == rad[# - 1] &]
  • PARI
    rad(n) = factorback(factorint(n)[, 1]); \\ A007947
isok(m) = !isprime(m) && (rad(eulerphi(m)) == rad(m-1)); \\ Michel Marcus, Feb 18 2019

A329859 Numbers k such that k and uphi(k) have the same set of prime divisors, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

1, 12, 36, 168, 240, 504, 702, 720, 1176, 1200, 1344, 1404, 1620, 3528, 3600, 4032, 4050, 6480, 8100, 9408, 14880, 19656, 22680, 23250, 28080, 28224, 32400, 44640, 46500, 53460, 63882, 65280, 69750, 74400, 113400, 127764, 132678, 133650, 137592, 139500
Offset: 1

Views

Author

Amiram Eldar, Nov 22 2019

Keywords

Examples

			12 is in the sequence since 12 = 2^2 * 3 and uphi(12) = 6 = 2 * 3 both have the same set of prime divisors, {2, 3}.

Links

Crossrefs

The unitary version of A055744.
Cf. A007947, A027598, A047994.

Programs

  • Mathematica
    rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); Select[Range[10^5], rad[#] == rad[uphi[#]] &]

A329878 Numbers k such that k and psi(k) have the same set of prime divisors, where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 150, 162, 192, 216, 288, 294, 300, 324, 384, 432, 450, 486, 576, 588, 600, 648, 726, 750, 768, 864, 882, 900, 972, 1152, 1176, 1200, 1296, 1350, 1452, 1458, 1500, 1536, 1728, 1734, 1764, 1800, 1944, 2058, 2178
Offset: 1

Views

Author

Amiram Eldar, Nov 23 2019

Keywords

Comments

Numbers k such that rad(psi(k)) = rad(k), where rad(k) is the squarefree kernel of k (A007947).

Examples

			6 is in the sequence since 6 = 2 * 3 and psi(6) = 12 = 2^2 * 3 have the same set of prime divisors, {2, 3}.

Links

Crossrefs

Cf. A001615, A007947, A027598, A055744.

Programs

  • Mathematica
    rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[2000], rad[psi[#]] == rad[#] &]

A116003 n*phi(n)*phi(phi(n)) is a fourth power.

Original entry on oeis.org

1, 32, 512, 3888, 8192, 25000, 62208, 131072, 314928, 400000, 759375, 995328, 2097152, 2839714, 3037500
Offset: 1

Views

Author

Giovanni Resta, Feb 13 2006

Keywords

Examples

			3888*phi(3888)*phi(phi(3888))=216^4.

Crossrefs

Cf. A055744, A116002.
Previous Showing 11-20 of 21 results. Next

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