A070777 -id:A070777 - cp's OEIS frontend

A070003 Numbers divisible by the square of their largest prime factor.

4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 75, 81, 98, 100, 108, 121, 125, 128, 144, 147, 150, 162, 169, 196, 200, 216, 225, 242, 243, 245, 250, 256, 288, 289, 294, 300, 324, 338, 343, 361, 363, 375, 392, 400, 432, 441, 450, 484, 486, 490, 500, 507

Offset: 1

Labos Elemer, May 07 2002

nonn

Numbers n such that P(phi(n)) - phi(P(n)) = 1, where P(x) is the largest prime factor of x. P(phi(n)) - phi(P(n)) = A006530(A000010(n)) - A000010(A006530(n)).
Numbers n such that the value of the commutator of phi and P functions at n is -1.
Equivalently, n such that n and phi(n) have the same largest prime factor since Phi(p) = p-1 if p is prime. - Benoit Cloitre, Jun 08 2002
Since n is divisible by P(n)^2, n cannot divide P(n)! and so A057109 is a supersequence. Hence all A002034(a(n)) are composite. - Jonathan Sondow, Dec 28 2004
A225546 defines a self-inverse bijection between this sequence and A335740, considered as sets. - Peter Munn, Jul 19 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355. [alternate link]
Paul Erdős and Ilias Kastanas, Solution 6674:The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.
A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n."
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Eric Weisstein's World of Mathematics, Totient Function

Subsequence of A057109, A122145.
Complement within A020725 of A102750.
Cf. A000010, A006530, A068211, A070777, A070812, A070002, A070004, A007283, A070813, A070814, A070815, A070816, A002034, A102067, A102068.
Related to A335740 via A225546.
A195212 is a subsequence.
Cf. A319988 (characteristic function). Positions of odd terms > 1 in A122111.

isA070003 := proc(n)
    if modp(n,A006530(n)^2) = 0 then # code re-use
        true;
    else
        false;
    end if;
end proc:
A070003 := proc(n)
    option remember ;
    if n =1 then
        4;
    else
        for a from procname(n-1)+1 do
            if isA070003(a) then
                return a
            end if;
        end do:
    end if;
end proc:
seq( A070003(n),n=1..80) ; # R. J. Mathar, Jun 27 2024

p[n_] := FactorInteger[n][[-1, 1]]; ep[n_] := EulerPhi[n]; fQ[n_] := p[ep[n]] == 1 + ep[p[n]]; Select[ Range[ 510], fQ] (* Robert G. Wilson v, Mar 26 2012 *)
Select[Range[500], FactorInteger[#][[-1,2]] > 1 &] (* T. D. Noe, Dec 06 2012 *)

for(n=3,1000,if(component(component(factor(n),1),omega(n))==component(component(factor(eulerphi(n)),1),omega(eulerphi(n))),print1(n,",")))

is(n)=my(f=factor(n)[,2]);f[#f]>1 \\ Charles R Greathouse IV, Mar 21 2012

sm(lim,mx)=if(mx==2,return(vector(log(lim+.5)\log(2)+1,i,1<<(i-1))));my(v=[1]);forprime(p=2,min(mx,lim),v=concat(v,p*sm(lim\p,p)));vecsort(v)
list(lim)=my(v=[]);forprime(p=2,sqrt(lim),v=concat(v,p^2*sm(lim\p^2,p)));vecsort(v) \\ Charles R Greathouse IV, Mar 27 2012

from sympy import factorint
def ok(n): f = factorint(n); return f[max(f)] >= 2
print(list(filter(ok, range(4, 508)))) # Michael S. Branicky, Apr 08 2021

Erdős proved that there are x * exp(-(1 + o(1))sqrt(log x log log x)) members of this sequence up to x. - Charles R Greathouse IV, Mar 26 2012

New name from Jonathan Sondow and Charles R Greathouse IV, Mar 27 2012

A070812 a(n) = phi(gpf(n)) - gpf(phi(n)) = A000010(A006530(n)) - A006530(A000010(n)).

Original entry on oeis.org

0, -1, 2, 0, 3, -1, -1, 2, 5, 0, 9, 3, 2, -1, 14, -1, 15, 2, 3, 5, 11, 0, -1, 9, -1, 3, 21, 2, 25, -1, 5, 14, 3, -1, 33, 15, 9, 2, 35, 3, 35, 5, 1, 11, 23, 0, -1, -1, 14, 9, 39, -1, 5, 3, 15, 21, 29, 2, 55, 25, 3, -1, 9, 5, 55, 14, 11, 3, 63, -1, 69, 33, -1, 15, 5, 9, 65, 2, -1, 35, 41, 3, 14, 35, 21, 5, 77, 1, 9, 11, 25, 23, 15, 0, 93, -1, 5

Offset: 3

Labos Elemer, May 09 2002

sign

Value of commutator[A000010, A006530] at n.

			Cases of n when a(n) = 1, -1, 2 or 0 are listed in A070002, A070003, A070004, A007283 respectively. Further regular solutions: if a(n)=3, then n=7k, where k has prime divisors < 7; if a(n)=5, then n=11k, where k has no prime divisors >=11; if a(n)=25, then mostly (not always!) n=31k ...

Michael De Vlieger, Table of n, a(n) for n = 3..10000
Michael De Vlieger, Log log scatterplot of a(n)+2, n = 3..10^5.

Cf. A000010, A006530, A070777, A068211, A070002, A070003, A070004, A007283.

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Table[EulerPhi[pf[u]]-pf[EulerPhi[u]], {u, 3, 128}]

gpf(n)=my(f=factor(n)[,1]);f[#f]
a(n)=gpf(n)-gpf(eulerphi(n))-1 \\ Charles R Greathouse IV, Feb 19 2013

a(n) = A070777(n) - A068211(n).

A070004 Numbers of the form 52^n or 532^n; a(n) = 5A029744(n).

Original entry on oeis.org

5, 10, 15, 20, 30, 40, 60, 80, 120, 160, 240, 320, 480, 640, 960, 1280, 1920, 2560, 3840, 5120, 7680, 10240, 15360, 20480, 30720, 40960, 61440, 81920, 122880, 163840, 245760, 327680, 491520, 655360, 983040, 1310720, 1966080, 2621440

Offset: 1

Labos Elemer, May 07 2002

Old name was: Numbers n such that phi(P(n)) - P(phi(n)) = 2, where P(x)=largest prime factor of x, or A000010(A006530(n))-A006530(A000010(n))=2.

Solutions to phi(P(x))-P(phi(x))=c, presence or absence of special prime factors in x are usually derivable.

Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000010, A006530, A068211, A070777, A070812, A070002, A070003, A007283, A070813-A070816.

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2]; Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, 2], Print[n]], {n, 3, 1000000}]
Union[Flatten[Table[2^n {5,15},{n,0,20}]]] (* or *) Join[ {5}, LinearRecurrence[ {0,2},{10,15},40]] (* Harvey P. Dale, Dec 23 2014 *)

gpf(n)=if(n>1,my(f=factor(n)[,1]);f[#f],1)
is(n)=eulerphi(gpf(n))-gpf(eulerphi(n))==2 \\ Charles R Greathouse IV, Feb 19 2013

a(n) = 5*A029744(n); numbers of the forms 5*2^n and 15*2^n.
G.f.: 5*x*(x+1)^2/(1-2*x^2). - Ralf Stephan, Jul 15 2013
Sum_{n>=1} 1/a(n) = 8/15. - Amiram Eldar, Jan 02 2021

Simpler name by Joerg Arndt, Jul 16 2013

A070002 Numbers k such that phi(P(k)) - P(phi(k)) = 1, where P(k) is the largest prime factor of k.

Original entry on oeis.org

45, 90, 135, 175, 180, 270, 350, 360, 405, 525, 540, 700, 720, 810, 875, 1050, 1080, 1215, 1400, 1440, 1573, 1575, 1620, 1750, 2100, 2160, 2430, 2625, 2800, 2880, 3146, 3150, 3240, 3500, 3645, 4200, 4320, 4375, 4719, 4725, 4860, 5250, 5491, 5600, 5760

Offset: 1

Labos Elemer, May 07 2002

nonn

phi(P(k)) - P(phi(k)) = A000010(A006530(k)) - A006530(A000010(k)) = 1, where P(k) = largest prime factor of k. Value of commutator of phi and P functions at k equals 1.

Many but not all terms are divisible by 5.

			m = 77077 = 7*7*11*11*13*13 is here because P(m) = 13, phi(P(13)) = 12, phi(m) = 55440 = 2*2*2*2*3*3*5*7*11 with P(Phi(55440)) = 13 and the difference is 13 - 12 = 1.

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

Cf. A000010, A006530, A068211, A070777, A070812, A070003, A070004, A007283, A070813, A070814, A070815, A070816.

pf[n_] := FactorInteger[n][[-1, 1]];
Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, 1], Print[n]], {n, 3, 100000}]

A070816 Solutions to phi(gpf(x)) - gpf(phi(x)) = 65534 = c are special multiples of 65537, x=65537*k, where the largest prime factors of factor k were observed in {2, 3, 5, 17, 257}.

Original entry on oeis.org

65537, 131074, 196611, 262148, 327685, 393222, 524296, 655370, 786444, 983055, 1048592, 1114129, 1310740, 1572888, 1966110, 2097184, 2228258, 2621480, 3145776, 3342387, 3932220, 4194368, 4456516, 5242960, 5570645, 6291552

Offset: 1

Labos Elemer, May 09 2002

nonn

See solutions to other even cases of c [=A070813]: A007283 for 0, A070004 for 2, A070814 for 14, A070815 for 254.

			For n = 572662306 = 2*17*257*65537, gpf(n) = 65537, phi(n) = 268435456, commutator[572662306] = phi(65537) - gpf(268435456) = 65536 - 2 = 65534.

Cf. A000010, A000215, A006530, A007283, A070002, A070003, A070004, A070777, A070812, A070813.

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, 65534], Print[{n, n/65537, pf[n/65537]}]], {n, 3, 1000000}] (* Terms of sequence are n *)

A070814 Solutions to phi(gpf(x)) - gpf(phi(x)) = 14 = c are special multiples of 17, x = 17k, where greatest prime factors of factor k were observed from {2, 3, 5}, i.e., it is smaller than 17. See solutions to other even cases of c (=A070813): A007283 for 0, A070004 for 2, A070815 for 254, A070816 for 65534. Gpf = greatest prime factor.

Original entry on oeis.org

17, 34, 51, 68, 85, 102, 136, 170, 204, 255, 272, 340, 408, 510, 544, 680, 816, 1020, 1088, 1360, 1632, 2040, 2176, 2720, 3264, 4080, 4352, 5440, 6528, 8160, 8704, 10880, 13056, 16320, 17408, 21760, 26112, 32640, 34816, 43520, 52224, 65280

Offset: 1

Labos Elemer, May 09 2002

nonn

For n > 10, a(n) = 2a(n-4). First, it is easy to show that with i >= 0 and k,m in {0,1}, a(n) are of the form 2^i*3^k*5^m. Factoring this sequence reveals the regular pattern 2^i, 2^(i-2)*5, 2^(i-1)*3, 2^(i-3)*3*5, 2^(i+1), ... which obviously has the property a(n) = 2a(n-4) for n > 10. - Lambert Herrgesell (lambert.herrgesell(AT)googlemail.com), Jan 09 2007

			For n = 32640 = 128*3*5*17, gpf(n) = 17, phi(n) = 16384, commutator[32640] = phi(17) - gpf(16384) = 16 - 2 = 14.

Cf. A000010, A000215, A006530, A007283, A070002, A070003, A070004, A070777, A070812, A070813.

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, 14], Print[{n, n/17, pf[n/17]}]], {n, 3, 1000000}] (* Terms of sequence are n *)

For n > 10, a(n) = 2a(n-4) (conjectured). - Ralf Stephan, May 09 2004

A070815 Solutions to phi(gpf(x)) - gpf(phi(x)) = 254 = c are special multiples of 257, x = 257k, where largest prime factors of factor k were observed from {2, 3, 5, 17}. See solutions to other even cases of c (=A070813): A007283 for 0, A070004 for 2, A070814 for 14, A070816 for 65534.

Original entry on oeis.org

257, 514, 771, 1028, 1285, 1542, 2056, 2570, 3084, 3855, 4112, 4369, 5140, 6168, 7710, 8224, 8738, 10280, 12336, 13107, 15420, 16448, 17476, 20560, 21845, 24672, 26214, 30840, 32896, 34952, 41120, 43690, 49344, 52428, 61680, 65535, 65792

Offset: 1

Labos Elemer, May 09 2002

nonn

			For n = 87380 = 4*5*17*257, gpf(n) = 257, phi(n) = 65536, commutator[87380] = phi(257) - gpf(65536) = 256 - 2 = 254.

Cf. A000010, A000215, A006530, A007283, A070002, A070002, A070004, A070777, A070812, A070813.

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, 254], Print[{n, n/257, pf[n/257]}]], {n, 3, 1000000}] (* Terms of sequence are n *)

A130027 A130026 * A054525.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 1, 2, 0, 1, 4, 0, 0, 0, 1, -2, 4, 3, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 1, 2, 0, 4, 0, 0, 0, 1, 2, 0, 6, 0, 0, 0, 0, 0, 1, -4, 8, 0, 0, 5, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, May 02 2007

Row sums = (1, 2, 3, ...). Left column = signed version of A070777.

			First few rows of the triangle:
   1;
   1, 1;
   2, 0, 1;
   1, 2, 0, 1;
   4, 0, 0, 0, 1;
  -2, 4, 3, 0, 0, 1;
   6, 0, 0, 0, 0, 0, 1;
   1, 2, 0, 4, 0, 0, 0, 1;
  ...

Cf. A130027, A070777.

A130026 * A054525 as infinite lower triangular matrices.

A088659 a(n) = n*(p-1) where p is the largest prime factor of n.

Original entry on oeis.org

2, 6, 4, 20, 12, 42, 8, 18, 40, 110, 24, 156, 84, 60, 16, 272, 36, 342, 80, 126, 220, 506, 48, 100, 312, 54, 168, 812, 120, 930, 32, 330, 544, 210, 72, 1332, 684, 468, 160, 1640, 252, 1806, 440, 180, 1012, 2162, 96, 294, 200, 816, 624, 2756, 108, 550, 336, 1026

Offset: 2

Benoit Cloitre, Nov 21 2003

nonn

It is conjectured that sequence gives period length of the periodic sequence {A088957(k) mod n}_{k>n}.

The records of this sequence are given by A036689 (product of a prime and the previous number). - Michel Marcus, May 19 2015

Ivan Neretin, Table of n, a(n) for n = 2..1000

seq(n*(max(numtheory:-factorset(n))-1), n=2..100); # Robert Israel, May 19 2015

Table[n*(FactorInteger[n][[-1, 1]] - 1), {n, 2, 57}] (* Ivan Neretin, May 19 2015 *)

PARI
```
a(n)=n*(component(factor(n),1)-1)
```

For p the k-th prime, a(p) = A036689(k). - Michel Marcus, May 19 2015

a(n) = n*A070777(n). - Michel Marcus, May 19 2015

A128980 A054525 * A129691(unsigned).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 29 2007

Row sums = A070777: (1, 1, 2, 1, 4, 2, 6, 1, 2, 4, ...). A129691 = the unsigned inverse of A054523.

			First few rows of the triangle:
  1;
  0, 1;
  1, 0, 1;
  0, 0, 0, 1;
  3, 0, 0, 0, 1;
  0, 1, 0, 0, 0, 1;
  5, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 1, 0, 0, 0, 0, 0, 1;
  ...

Cf. A054525, A129691, A070777, A054523.

Moebius transform of A129691

cp's OEIS Frontend

A070003 Numbers divisible by the square of their largest prime factor.

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A070812 a(n) = phi(gpf(n)) - gpf(phi(n)) = A000010(A006530(n)) - A006530(A000010(n)).

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A070004 Numbers of the form 5*2^n or 5*3*2^n; a(n) = 5*A029744(n).

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A070002 Numbers k such that phi(P(k)) - P(phi(k)) = 1, where P(k) is the largest prime factor of k.

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A070816 Solutions to phi(gpf(x)) - gpf(phi(x)) = 65534 = c are special multiples of 65537, x=65537*k, where the largest prime factors of factor k were observed in {2, 3, 5, 17, 257}.

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A070815 Solutions to phi(gpf(x)) - gpf(phi(x)) = 254 = c are special multiples of 257, x = 257k, where largest prime factors of factor k were observed from {2, 3, 5, 17}. See solutions to other even cases of c (=A070813): A007283 for 0, A070004 for 2, A070814 for 14, A070816 for 65534.

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A130027 A130026 * A054525.

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A070004 Numbers of the form 52^n or 532^n; a(n) = 5A029744(n).