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 1-8 of 8 results.

A081379 Duplicate of A081377.

Original entry on oeis.org

1, 3, 14, 35, 42, 70, 105, 119, 209, 210, 238, 248, 297, 357, 418, 477, 594, 595, 616, 627, 714, 744, 954, 1045, 1178, 1190, 1240, 1254, 1463, 1485, 1672, 1674, 1736, 1785, 1848, 1863, 2079, 2090, 2376, 2385, 2540, 2728, 2926, 2945, 2970
Offset: 1

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Author

Keywords

A055744 Numbers k such that k and phi(k) have the same prime factors.

Original entry on oeis.org

1, 4, 8, 16, 18, 32, 36, 50, 54, 64, 72, 100, 108, 128, 144, 162, 200, 216, 250, 256, 288, 294, 324, 400, 432, 450, 486, 500, 512, 576, 578, 588, 648, 800, 864, 882, 900, 972, 1000, 1014, 1024, 1152, 1156, 1176, 1210, 1250, 1296, 1350, 1458, 1600, 1728, 1764
Offset: 1

Views

Author

Labos Elemer, Jul 11 2000

Keywords

Comments

Contains products of suitable powers of 2 and Fermat primes. For x = 2^u*3^w, phi(x) = 2^u*3^(w-1) with suitable exponents. Analogous constructions are possible with {2,3,7} prime divisors, etc.
From Ivan Neretin, Mar 19 2015: (Start)
Also, numbers k that meet the following criteria for every prime p dividing k:
1. All prime divisors of p-1 must also divide k;
2. If k has no prime divisors of the form m*p+1, and k is divisible by p, then k must be divisible by p^2.
Also, numbers k for which {k, phi(k), phi(phi(k))} is a geometric progression.
(End)
All terms > 1 are even. An even number k is in the sequence iff 2*k is in the sequence. - Robert Israel, Mar 19 2015
For n > 1, the largest prime factor of a(n) has multiplicity >= 2. For all prime factors more than half of the largest prime factor of a(n), the multiplicity differs from 1.
If k = p1^a1 * p2^a2 * ... * pm^am is in the sequence, then so is p1^b1 * p2^b2 * ... * pm^bm for 1 <= i <= m and prime pi and bi >= ai.
If m * p^2 is not in the sequence, for a prime p and some m > 0, then neither is m * p^3. - David A. Corneth, Mar 22 2015
A027748(a(n),j) = A027748(A000010(a(n)),j) for j=1..A001221(a(n)); also numbers k such that k and phi(k) have the same squarefree kernel: A007947(a(n)) = A007947(A000010(a(n))). - Reinhard Zumkeller, Jun 01 2015
Pollack and Pomerance call these numbers "phi-perfect numbers". - Amiram Eldar, Jun 02 2020

Examples

			k = 578 = 2*17*17, phi(578) = 272 = 2*2*2*2*17 with 2 and 17 prime factors, so 578 is a term.
k = 588 = 2*2*3*7*7, phi(588) = 168 = 2*2*2*3*7, so 588 is a term.
k = 264196 = 2*2*257*257, phi(264196) = 512*257 = 131584, so 264196 is a term.

Links

Crossrefs

Intersection of A073539 and A151999. - Amiram Eldar, Jun 02 2020
Cf. A001221, A000010, A110751, A110819, A027598, A081377.
Cf. A007947, A027748, A055742, A173557, A256248, subsequence of A124240.

Programs

  • Haskell
    a055744 n = a055744_list !! (n-1)
a055744_list = 1 : filter f [2..] where
   f x = all ((== 0) . mod x) (concatMap (a027748_row . subtract 1) ps) &&
         all ((== 0) . mod (a173557 x))
             (map fst $ filter ((== 1) . snd) $ zip ps $ a124010_row x)
         where ps = a027748_row x
-- Reinhard Zumkeller, Jun 01 2015
  • Maple
    select(numtheory:-factorset = numtheory:-factorset @ numtheory:-phi,
[1, 2*i $ i=1..2000]); # Robert Israel, Mar 19 2015
isA055744 := proc(n)
    nfs := numtheory[factorset](n) ;
    phinfs := numtheory[factorset](numtheory[phi](n)) ;
    if nfs = phinfs then
        true;
    else
        false;
    end if;
end proc:
A055744 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA055744(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 23 2016
  • Mathematica
    Select[Range@ 1800,
First /@ FactorInteger@ # == First /@ FactorInteger@ EulerPhi@ # &] (* Michael De Vlieger, Mar 21 2015 *)
  • PARI
    is(n)=factor(n)[,1]==factor(eulerphi(n))[,1] \\ Charles R Greathouse IV, Oct 31 2011
  • PARI
    is(n)=my(f=factor(n)); f[,1]==factor(eulerphi(f))[,1] \\ Charles R Greathouse IV, May 26 2015

Extensions

Corrected and extended by James Sellers, Jul 11 2000

A027598 Numbers k such that the set of prime divisors of k is equal to the set of prime divisors of sigma(k).

Original entry on oeis.org

1, 6, 28, 120, 270, 496, 672, 1080, 1638, 1782, 3780, 8128, 18600, 20580, 24948, 26208, 30240, 32640, 32760, 35640, 41850, 44226, 55860, 66960, 164640, 167400, 185220, 199584, 273000, 293760, 401310, 441936, 446880, 502740, 523776, 614250, 707616, 802620, 819000
Offset: 1

Views

Author

Jud McCranie

Keywords

Comments

Multiplicities are ignored.
All even perfect numbers are in the sequence. It seems that 1 is the only odd term of the sequence. - Farideh Firoozbakht, Jul 01 2008
sigma() is the multiplicative sum-of-divisors function. - Walter Nissen, Dec 16 2009
Pollack and Pomerance call these "prime-perfect numbers" and show that there are << x^(1/3+e) of these up to x for any e > 0. - Charles R Greathouse IV, May 09 2013
Except for unity for the obvious reason, the primitive terms are the perfect numbers (A000396). - Robert G. Wilson v, Feb 19 2019
If an odd term > 1 exists, it is larger than 5*10^23. - Giovanni Resta, Jun 02 2020

Examples

			273000 = 2^3*3*5^3*7*13 and sigma(273000) = 1048320 = 2^8*3^2*5*7*13 so 273000 is in the sequence.

References

  • R. K. Guy, Unsolved Problems in Number Theory, B19.

Links

Crossrefs

Intersection of A105402 and A175200. - Amiram Eldar, Jun 02 2020
Cf. A110751, A110819, A055744, A081377, A000203.

Programs

  • GAP
    Filtered([1..1000000],n->Set(Factors(n))=Set(Factors(Sigma(n)))); # Muniru A Asiru, Feb 21 2019
  • Mathematica
    Select[Range[1000000], Transpose[FactorInteger[#]][[1]] == Transpose[FactorInteger[DivisorSigma[1, #]]][[1]] &] (* T. D. Noe, Dec 08 2012 *)
  • PARI
    a(n) = {for (i=1, n, fn = factor(i); fs = factor(sigma(i)); if (fn[,1] == fs[,1], print1(i, ", ")););} \\ Michel Marcus, Nov 18 2012
  • PARI
    is(n)=my(f=factor(n),fs=[],t);for(i=1,#f[,1], t=factor((f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))[,1]; fs=vecsort(concat(fs,t~),,8); if(#setminus(fs,f[,1]~), return(0))); fs==f[,1]~ \\ Charles R Greathouse IV, May 09 2013

Extensions

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar

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

A081378 Numbers k for which the sums of prime factors (ignoring multiplicity) of sigma(k) and phi(k) are equal but the sets of prime factors of sigma and phi are different.

Original entry on oeis.org

412, 1142, 1236, 1328, 1339, 1703, 2855, 2875, 2884, 3406, 3426, 3668, 3708, 3984, 4017, 5109, 5356, 5710, 5750, 5924, 6003, 6281, 6399, 6413, 6640, 6812, 7994, 8054, 8318, 8515, 8565, 8611, 8625, 8652, 8843, 8858, 9373, 9707, 9991
Offset: 1

Views

Author

Labos Elemer, Mar 26 2003

Keywords

Examples

			k = 412 = 2*2*103: sigma(412) = 728 = 2*2*2*7*13, phi(412) = 204 = 2*2*3*17, the sums of prime factors are identical (2 + 7 + 13 = 22 = 2 + 3 + 17) but the prime sets are different: {2,7,13} vs. {2,7,17}.

Links

Crossrefs

Cf. A000010, A000203, A065642, A076533, A081377.

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]}]; spf[x_] := Apply[Plus, ba[x]]; k=0; Do[s=ba[DivisorSigma[1, n]]; s1=ba[EulerPhi[n]]; s3=spf[DivisorSigma[1, n]]; s4=spf[EulerPhi[n]]; If[ !Equal[s, s1]&&Equal[s3, s4], k=k+1; Print[{n, s, s1, ba[n], s3}]], {n, 1, 10000}]
  • PARI
    is(n) = {my(f = factor(n), p1 = factor(sigma(f))[, 1], p2 = factor(eulerphi(f))[, 1]); p1 != p2 && vecsum(p1) == vecsum(p2) ;} \\ Amiram Eldar, Mar 25 2024

A081380 Numbers k such that the sets of prime factors (ignoring multiplicity) of A000203(k) = sigma(k) and of A001157(k) = sigma_2(k) are identical.

Original entry on oeis.org

1, 180, 1444, 12996, 23805, 36100, 52020, 60228, 64980, 68832, 95220, 301140, 324900, 344160, 481824, 1505700, 1718721, 1720800, 2275758, 2409120, 3755844, 6874884, 6879645, 7965153, 8593605, 11378790, 12045600, 15930306, 17405892
Offset: 1

Views

Author

Labos Elemer, Mar 26 2003

Keywords

Examples

			n = 1444 = 2^2*19^2, sigma(1444) = 2667 = 3*7*127, sigma_2(1444) = 2744343 = 3^2*7^4*127, common factor set = {3,7,127}.

References

  • Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 180, p. 56, Ellipses, Paris, 2008.

Links

Crossrefs

Cf. A000203, A001157, A081377, A081378.

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[1, n]]; s5=ba[DivisorSigma[2, n]]; If[Equal[s, s5], Print[n]], {n, 1, 1000000}]
  • PARI
    is(n)=factor(sigma(n))[,1]==factor(sigma(n,2))[,1] \\ Charles R Greathouse IV, Feb 19 2013

Extensions

More terms from Lekraj Beedassy, Jul 18 2008
a(16)-a(29) from Donovan Johnson, May 24 2009

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.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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