A002618 -id:A002618 - cp's OEIS frontend

A115910 Numbers k such that phi(k)*k is a triangular number.

1, 3, 15, 84, 217, 255, 1710, 2967, 5363, 11242, 24066, 27370, 29697, 29953, 60977, 65535, 69324, 103257, 159761, 209305, 228333, 446516, 598559, 615410, 691410, 1231305, 1238358, 1365175, 1467732, 1841154, 1966220, 2081070, 2562370

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			phi(24066)*24066 = 164611440 = T(18144).

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A002618, A000217, A115905, A115909.

Select[Range[2.6*10^6],OddQ[Sqrt[8#*EulerPhi[#]+1]]&] (* Harvey P. Dale, Jul 11 2017 *)

isok(n) = ispolygonal(n*eulerphi(n), 3); \\ Michel Marcus, Jan 25 2014

A189393 a(n) = phi(n^4).

Original entry on oeis.org

1, 8, 54, 128, 500, 432, 2058, 2048, 4374, 4000, 13310, 6912, 26364, 16464, 27000, 32768, 78608, 34992, 123462, 64000, 111132, 106480, 267674, 110592, 312500, 210912, 354294, 263424, 682892, 216000, 893730, 524288, 718740, 628864, 1029000, 559872

Offset: 1

Vincenzo Librandi, Apr 21 2011

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..1000 (terms 1..680 from Vincenzo Librandi)

Cf. A002618 (phi(n^2)), A053191 (phi(n^3)), A238533 (phi(n^5)), A239442 (phi(n^7)), A239443 (phi(n^9)).

Magma
```
[ n^3*EulerPhi(n) : n in [1..100] ]
```

EulerPhi[Range[100]^4] (* T. D. Noe, Dec 27 2011 *)

vector(66,n,n^3*eulerphi(n))  /* Joerg Arndt, Apr 22 2011 */

a(n) = n^3*phi(n).
Dirichlet g.f.: zeta(s - 4) / zeta(s - 3). The n-th term of the Dirichlet inverse is n^3 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega=A001221. - Álvar Ibeas, Nov 24 2017
Sum_{k=1..n} a(k) ~ 6*n^5 / (5*Pi^2). - Vaclav Kotesovec, Feb 02 2019
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^5 - p^4 - p + 1)) = 1.15762316629211803144... - Amiram Eldar, Dec 06 2020

A374456 The Euler phi function values of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 4, 4, 10, 12, 6, 8, 16, 18, 12, 10, 22, 8, 12, 18, 28, 8, 30, 16, 20, 16, 24, 36, 18, 24, 16, 40, 12, 42, 22, 46, 32, 52, 18, 40, 24, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 40, 88, 72, 60, 46, 72, 32, 96

Offset: 1

Amiram Eldar, Jul 09 2024

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

Cf. A000010, A065463, A268335, A307868.
Similar sequences related to phi: A002618, A049200, A323333, A358039.
Similar sequences related to exponentially odd numbers: A366438, A366439, A366534, A366535, A367417, A368711, A374457.

f[p_, e_] := If[OddQ[e], (p-1) * p^(e-1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]-1) * f[i, 1]^(f[i, 2] - 1), 0));}
lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

a(n) = A000010(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A307868 / A065463^2 = 0.95051132596733153581... .

A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 3, 0, 6, 4, 4, 0, 8, 5, 0, 0, 0, 20, 6, 6, 12, 0, 0, 12, 7, 0, 0, 0, 0, 0, 42, 8, 8, 0, 16, 0, 0, 0, 32, 9, 0, 18, 0, 0, 0, 0, 0, 54, 10, 10, 0, 0, 40, 0, 0, 0, 0, 40

Offset: 1

Gary W. Adamson, Jan 15 2007

Mobius transform of A127481.

			First few rows of the triangle are:
1;
2, 2;
3, 0, 6;
4, 4, 0, 8;
5, 0, 0, 0, 20;
6, 6, 12, 0, 0, 12;
7, 0, 0, 0, 0, 0, 42;
8, 8, 0, 16, 0, 0, 0, 32;
...

Cf. A054522, A127093, A062952, A002618.

Cf. A127093, A127481, A054525.

A127466 := proc(n,k)
    add( A054525(n,j)*A127481(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 06 2013

Sum_{k=1..n} T(n,k) = n^2.

T(n,n) = A002618(n) = n*phi(n).

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A127649 A127648 * A054523 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 6, 0, 3, 8, 4, 0, 4, 20, 0, 0, 0, 5, 12, 12, 6, 0, 0, 6, 42, 0, 0, 0, 0, 0, 7, 32, 16, 0, 8, 0, 0, 0, 8, 54, 0, 18, 0, 0, 0, 0, 0, 9, 40, 40, 0, 0, 10, 0, 0, 0, 0, 10, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 48, 24, 24, 24, 0, 12, 0, 0, 0, 0, 0, 12, 156, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 84

Offset: 1

Gary W. Adamson, Jan 22 2007

Natural number transform of A054523.

Row sums = n^2, left column = A002618

			First few rows of the triangle are:
1;
2, 2;
6, 0, 3;
8, 4, 0, 4;
20, 0, 0, 0, 5;
12, 12, 6, 0, 0, 6;
42, 0, 0, 0, 0, 0, 7;
...

Cf. A127648, A002618, A054523.

A054523 := proc(n,k) if n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi ; end: A127649 := proc(n,k) A054523(n,k)*n ; end: for n from 1 to 20 do for k from 1 to n do printf("%d,",A127649(n,k)) ; od: od: # R. J. Mathar, Nov 01 2007

T(n,k)=n*A054523(n,k). - R. J. Mathar, Nov 01 2007

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x - y. - Mats Granvik, Oct 08 2023

More terms from R. J. Mathar, Nov 01 2007

A194507 a(n) = y is the unique solution to y*phi(y) = A082473(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 10, 7, 12, 9, 14, 18, 11, 15, 16, 13, 20, 24, 22, 30, 21, 17, 26, 28, 19, 36, 27, 25, 42, 23, 32, 34, 40, 33, 38, 48, 29, 35, 44, 31, 39, 60, 54, 50, 46, 45, 52, 66, 37, 56, 58, 51, 41, 70, 72, 43, 62, 78, 84, 64, 57, 49, 90, 47, 68, 55, 63, 80

Offset: 1

Franz Vrabec, Aug 27 2011

nonn

The permutation which rearranges the terms of A002618 into ascending order. - Antti Karttunen, Sep 28 2019

			a(6) = 5 because 5*phi(5) = 20 = A082473(6).

Antti Karttunen, Table of n, a(n) for n = 1..20001
Index entries for sequences that are permutations of the natural numbers

Cf. A002618, A082473, A327173 (inverse permutation).

Nonzero terms in A327172, in the order of appearance.

Block[{nn = 3000, s, t}, s = Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, nn]; t = TakeWhile[Union@ s, # <= nn &]; Map[Block[{y = 1}, While[y EulerPhi@ y != #, y++]; y] &, t]] (* Michael De Vlieger, Sep 29 2019, after Bill Gosper at A007913 *)

up_to = 105;
A327172(n) = { fordiv(n,d,if(eulerphi(d)*d == n, return(d))); (0); };
A194507list(up_to) = { my(v=vector(up_to),k=1); for(n=1,oo,if((v[k]=A327172(n))>0,k++); if(k>up_to, return(v))); };
v194507 = A194507list(up_to);
A194507(n) = v194507[n]; \\ Antti Karttunen, Sep 28 2019

From Antti Karttunen, Sep 28 2019: (Start)
a(n) = A327172(A082473(n)).
A002618(a(n)) = A082473(n).
(End)

A262747 Number of ordered ways to write n as x^2 + y^2 + phi(z^2) with 0 <= x <= y, z > 0, 2 | xyz, and phi(k^2) < n for all 0 < k < z, where phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 1, 2, 4, 2, 2, 1, 2, 2, 1, 2, 3, 2, 3, 4, 3, 1, 2, 3, 4, 2, 4, 2, 2, 3, 1, 4, 2, 1, 2, 5, 4, 1, 2, 2, 6, 3, 2, 4, 4, 3, 3, 4, 3, 3, 5, 3, 3, 4, 2, 6, 7, 3, 4, 4, 5, 2, 2, 5, 6, 6, 1, 5, 4, 4, 4, 6, 6, 1, 4, 4, 2, 4, 3, 5, 6, 3, 4, 5, 5, 4, 2, 2, 6, 5, 4, 6, 3, 3, 1, 5, 3, 2, 3

Offset: 1

Zhi-Wei Sun, Sep 30 2015

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 4, 5, 8, 13, 16, 23, 32, 35, 39, 68, 75, 96, 215, 219, 243, 363, 471, 723, 759, 923, 1443, 1551, 1839, 2739, 2883.
It is easy to see that all the numbers phi(n^2) = n*phi(n) (n = 1,2,3,...) are pairwise distinct. We have verified that a(n) > 0 for all n = 1,...,10^7.
See also A262311 for a related conjecture.

			a(5) = 1 since 5 = 0^2 + 2^2 + phi(1^2).
a(8) = 1 since 8 = 0^2 + 0^2 + phi(4^2) with 0*0*4 even, and phi(1^2) = 1, phi(2^2) = 2, phi(3^2) = 6 all smaller than 8.
a(13) = 1 since 13 = 1^2 + 2^2 + phi(4^2).
a(32) = 1 since 32 = 2^2 + 4^2 + phi(6^2).
a(68) = 1 since 68 = 0^2 + 6^2 + phi(8^2).
a(96) = 1 since 96 = 0^2 + 8^2 + phi(8^2).
a(363) = 1 since 363 = 0^2 + 19^2 + phi(2^2).
a(471) = 1 since 471 = 0^2 + 19^2 + phi(11^2).
a(723) = 1 since 723 = 17^2 + 18^2 + phi(11^2).
a(759) = 1 since 759 = 9^2 + 26^2 + phi(2^2).
a(923) = 1 since 923 = 16^2 + 25^2 + phi(7^2).
a(1443) = 1 since 1443 = 19^2 + 24^2 +phi(23^2).
a(1551) = 1 since 1551 = 18^2 + 35^2 + phi(2^2).
a(1839) = 1 since 1839 = 3^2 + 30^2 + phi(31^2).
a(2739) = 1 since 2739 = 1^2 + 24^2 + phi(47^2).
a(2883) = 1 since 2883 = 21^2 + 44^2 + phi(23^2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A001481, A002618, A262311, A262746, A262750, A262766.

f[n_]:=EulerPhi[n^2]
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[f[x]>n,Goto[aa]];Do[If[SQ[n-f[x]-y^2]&&(Mod[x*y,2]==0||Mod[Sqrt[n-f[x]-y^2],2]==0),r=r+1],{y,0,Sqrt[(n-f[x])/2]}];Continue,{x,1,n}];Label[aa];Print[n," ",r];Continue,{n,1,100}]

A262781 Number of ordered ways to write n as x^2 + phi(y^2) + phi(z^2) (x >= 0 and 0 < y <= z) with y or z prime, where phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

0, 0, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 2, 3, 2, 2, 1, 3, 3, 1, 2, 3, 4, 1, 1, 3, 2, 3, 2, 4, 1, 3, 2, 2, 3, 1, 3, 3, 4, 2, 2, 3, 5, 5, 1, 4, 4, 4, 2, 6, 4, 4, 4, 6, 3, 4, 5, 4, 5, 4, 4, 3, 6, 4, 2, 3, 3, 5, 4, 4, 4, 3, 1, 4, 5, 4, 3, 6, 3, 1, 2, 3, 4, 4, 5, 5, 3, 3, 2, 8, 5, 3, 4, 2, 4, 4, 2, 3, 7, 2

Offset: 1

Zhi-Wei Sun, Oct 01 2015

nonn

Conjecture: (i) a(n) > 0 for all n > 6, and a(n) = 1 only for n = 3, 5, 9, 10, 17, 20, 24, 25, 31, 36, 45, 73, 80, 101, 136, 145, 388, 649.
(ii) For any integer n > 4, we can write 2*n as phi(p^2) + phi(x^2) + phi(y^2) with p prime and p <= x <= y.
See also A262311 for a similar conjecture.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000290, A002618, A262311, A262746, A262747.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=EulerPhi[n^2]
Do[r=0;Do[If[f[z]>n,Goto[aa]];Do[If[SQ[n-f[z]-f[y]]&&(PrimeQ[y]||PrimeQ[z]),r=r+1],{y,1,z}];Label[aa];Continue,{z,1,n}];Print[n," ",r];Continue,{n,1,100}]

a(3) = 1 since 3 = 0^2 + phi(1^2) + phi(2^2) with 2 prime.
a(5) = 1 since 5 = 1^2 + phi(2^2) + phi(2^2) with 2 prime.
a(9) = 1 since 9 = 1^2 + phi(2^2) + phi(3^2) with 2 and 3 both prime.
a(10) = 1 since 10 = 0^2 + phi(2^2) + phi(4^2) with 2 prime.
a(17) = 1 since 17 = 3^2 + phi(2^2) + phi(3^2) with 2 and 3 both prime.
a(20) = 1 since 20 = 4^2 + phi(2^2) + phi(2^2) with 2 prime.
a(24) = 1 since 24 = 4^2 + phi(2^2) + phi(3^2) with 2 and 3 both prime.
a(25) = 1 since 25 = 2^2 + phi(1^2) + phi(5^2) with 5 prime.
a(31) = 1 since 31 = 3^2 + phi(2^2) + phi(5^2) with 2 and 5 both prime.
a(36) = 1 since 36 = 2^2 + phi(5^2) + phi(6^2) with 5 prime.
a(45) = 1 since 45 = 1^2 + phi(2^2) + phi(7^2) with 2 and 7 both prime.
a(73) = 1 since 73 = 5^2 + phi(3^2) + phi(7^2) with 3 and 7 both prime.
a(80) = 1 since 80 = 6^2 + phi(2^2) + phi(7^2) with 2 and 7 both prime.
a(101) = 1 since 101 = 7^2 + phi(5^2) + phi(8^2) with 5 prime.
a(136) = 1 since 136 = 5^2 + phi(1^2) + phi(11^2) with 11 prime.
a(145) = 1 since 145 = 7^2 + phi(7^2) + phi(9^2) with 7 prime.
a(388) = 1 since 388 = 2^2 + phi(7^2) + phi(19^2) with 7 and 19 both prime.
a(649) = 1 since 649 = 11^2 + phi(7^2) + phi(27^2) with 7 prime.

A327172 If there is a divisor d of n such that phi(d)*d = n, then a(n) = d, otherwise a(n) = 0.

Original entry on oeis.org

1, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10, 0, 7, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15

Offset: 1

Antti Karttunen, Sep 28 2019

nonn

If such a divisor exists, it is necessarily unique. See Franz Vrabec's Dec 12 2012 comment in A002618.

Each natural number n > 0 occurs exactly once in this sequence, at position A002618(n).

Antti Karttunen, Table of n, a(n) for n = 1..21000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Left inverse of A002618.
Cf. A000010.
Cf. A082473 (the indices of nonzero terms), A194507 (nonzero terms in the order of appearance).

With[{s = EulerPhi /@ Range@ 120}, Table[DivisorSum[n, # &, # s[[#]] == n &], {n, Length@ s}]] (* Michael De Vlieger, Sep 29 2019 *)

A327172(n) = { fordiv(n,d,if(eulerphi(d)*d == n, return(d))); (0); };

a(A002618(n)) = n.

a(A082473(n)) = A194507(n).

A174857 The minimum distance k > 0 such that A020639(n+k) = A020639(n).

Original entry on oeis.org

2, 6, 2, 20, 2, 42, 2, 6, 2, 110, 2, 156, 2, 6, 2, 272, 2, 342, 2, 6, 2, 506, 2, 10, 2, 6, 2, 812, 2, 930, 2, 6, 2, 20, 2, 1332, 2, 6, 2, 1640, 2, 1806, 2, 6, 2, 2162, 2, 28, 2, 6, 2, 2756, 2, 10, 2, 6, 2, 3422, 2, 3660, 2, 6, 2, 20, 2, 4422, 2, 6, 2, 4970, 2, 5256, 2, 6, 2, 14, 2, 6162

Offset: 2

Vladimir Shevelev, Mar 31 2010

nonn

The sequence has the same records as A002618.

Antti Karttunen, Table of n, a(n) for n = 2..16385

Cf. A020639, A002618, A036689, A174869.

A174857 := proc(n) local k,aref ; aref := A020639(n) ; for k from 1 do if A020639(n+k) = aref then return k; end if; end do: end proc:
seq(A174857(n),n=2..80) ; # R. J. Mathar, Dec 07 2010

Block[{s = Array[FactorInteger[#][[1, 1]] &, 10^4]}, Array[If[EvenQ[#], 2, Block[{k = 1, n = s[[#]]}, While[n != s[[# + k]], k++; If[# + k > Length[s], AppendTo[s, FactorInteger[# + k][[1, 1]] ]] ]; k]] &, 78, 2]] (* Michael De Vlieger, Apr 06 2021 *)

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A174857(n) = if(isprime(n), (n-1)*n, my(spf=A020639(n)); for(k=1,oo,if(A020639(n+k)==spf,return(k)))); \\ Antti Karttunen, Apr 06 2021

If n is even, then a(n) = 2.
If n = 3k and A020639(k) >= 3, then a(n) = 6.
If n is prime, then a(n) = A036689(n).

cp's OEIS Frontend

A115910 Numbers k such that phi(k)*k is a triangular number.

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A189393 a(n) = phi(n^4).

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Magma

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A374456 The Euler phi function values of the exponentially odd numbers (A268335).

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A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

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Maple

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A127649 A127648 * A054523 as infinite lower triangular matrices.

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A194507 a(n) = y is the unique solution to y*phi(y) = A082473(n).

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A262747 Number of ordered ways to write n as x^2 + y^2 + phi(z^2) with 0 <= x <= y, z > 0, 2 | x*y*z, and phi(k^2) < n for all 0 < k < z, where phi(.) is Euler's totient function given by A000010.

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A262781 Number of ordered ways to write n as x^2 + phi(y^2) + phi(z^2) (x >= 0 and 0 < y <= z) with y or z prime, where phi(.) is Euler's totient function given by A000010.

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A262747 Number of ordered ways to write n as x^2 + y^2 + phi(z^2) with 0 <= x <= y, z > 0, 2 | xyz, and phi(k^2) < n for all 0 < k < z, where phi(.) is Euler's totient function given by A000010.