A221284 -id:A221284 - cp's OEIS frontend

A221285 Square values taken by totient function phi(m) = A000010(m).

1, 4, 16, 36, 64, 100, 144, 196, 256, 324, 400, 576, 676, 784, 900, 1024, 1296, 1600, 1764, 1936, 2304, 2500, 2704, 2916, 3136, 3600, 4096, 4356, 4624, 4900, 5184, 5476, 6400, 7056, 7744, 8100, 8836, 9216, 10000, 10816, 11664, 12100, 12544, 12996, 13456, 14400, 15376, 15876

Offset: 1

Charles R Greathouse IV, Feb 05 2013

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
W. D. Banks, J. B. Friedlander, C. Pomerance, and I. E. Shparlinski, Multiplicative structure of values of the Euler function, in High primes and misdemeanours: Lectures in honour of the sixtieth birthday of Hugh Cowie Williams, Fields Inst. Comm. 41 (2004), pp. 29-47.
Tristan Freiberg, Carl Pomerance, A note on square totients, arXiv:1410.8109 [math.NT], 2014.
Paul Pollack and Carl Pomerance, Square values of Euler's function, Bulletin of the London Mathematical Society 46:2 (April 2014), pp. 403-414.

Cf. A002202, A221284.

inversePhiSingle[(m_)?EvenQ] := Module[{p, nmax, n}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; While[n <= nmax, If[EulerPhi[n] == m, Return[n]]; n++]; 0];
Reap[For[k = 1, k <= 200, k = k + If[k==1, 1, 2], If[inversePhiSingle[k^2] > 0, Print[k^2]; Sow[k^2]]]][[2, 1]] (* Jean-François Alcover, Dec 11 2018 *)

PARI
```
is(n)=issquare(n) && istotient(n)
```

A002202 INTERSECTION A000290.
a(n) = A221284(n)^2.
Pollack & Pomerance show that n^2 log^.0126 n << a(n) << n^2 log^6 n.

A228061 Numbers k such that k^2 = sigma(m) for some m.

Original entry on oeis.org

1, 2, 6, 11, 12, 16, 18, 20, 24, 28, 30, 31, 32, 36, 40, 42, 44, 48, 52, 54, 56, 60, 62, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 108, 112, 114, 120, 124, 126, 128, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 172, 174

Offset: 1

T. D. Noe, Sep 04 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).
Paul Pollack and Carl Pomerance, Square values of Euler's function, Bull. London Math. Soc., Vol. 46, No. 2 (2014), pp. 403-414; alternative link. See section 5.

Cf. A000203, A038688 (squares of these numbers).

Cf. A221284 (similar numbers for the phi function).

nn = 40000; t = Select[Union[DivisorSigma[1, Range[nn]]], IntegerQ[Sqrt[#]] &]; t = Sqrt[t]; t = Select[t, # < Sqrt[nn] &]
With[{nn=50000},Union[Select[Sqrt[DivisorSigma[1,Range[nn]]],IntegerQ[ #] && #<=Sqrt[nn]&]]] (* Harvey P. Dale, Jul 12 2021 *)

lista(kmax) = for(k = 1, kmax, if(invsigmaNum(k^2) > 0, print1(k, ", "))); \\ Amiram Eldar, Aug 12 2024, using Max Alekseyev's invphi.gp

a(n) = sqrt(A038688(n)).

A306722 Number of pairs of primes (p,q), p < q, which are a solution of the Diophantine equation (p-1)*(q-1) = (2n)^2.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 06 2019

nonn

a(n) is also the number of semiprimes p*q whose totient is a square (A247129) and equal to (2*n)^2.
From Robert G. Wilson v, Mar 30 2019, Mar 30 2019: (Start)
First occurrence of k=1,2,3,...: 1, 3, 10, 27, 60, 72, 120, 180, 270, 480, 252, 1155, 720, 792, 1260, 630, ..., . = A307245.
Start of table:
a(k_i) = n:
\i  1   2   3   4   5   6    7    8    9   10   11   12   13   14   15  ...
n\
0  11  17  19  23  29  31   34   38   39   41   43   46   49   51   53  ...
1   1   2   4   5   7   8    9   13   14   15   16   21   22   25   26  ...
2   3   6  54  56  58  87  100  115  116  123  138  148  160  170  176  ...
3  10  12  18  20  24  28   30   36   40   42   48   84   88   99  144  ...
4  27  33  45  63  66  70   75   80  112  126  135  153  156  162  165  ...
5  60  78  90 102 140 168  200  260  264  285  288  315  378  408  432  ...
6  72 105 108 130 150 306  348  357  450  495  528  560  672  696  708  ...
7 120 132 240 297 312 330  390  588  750  882  980 1140 1176 1190 1215  ...
8 180 198 210 280 396 468  540  612  648  700  810  910  945  960 1020  ...
9 270 420 660 858 918 990 1248 1620 1782 1920 2088 2184 2352 2376 2688  ...
... (End).
If n is a prime <> 3, then a(n) = 1 if n is in A052291 and 0 otherwise, and a(n^2) = 1 if 2*n+1 and 2*n^3+1 are primes and 0 otherwise. - Robert Israel, Apr 04 2019

			a(2) = 1 because (2*2)^2 = (2-1) * (17-1), also, phi(2*17) = 4^2.
a(3) = 2 because (2*3)^2 = (2-1) * (37-1) = (3-1) * (19-1), also, phi(2*37) = phi(3*19) = 6^2.
a(11) = 0  because (2*11)^2 can't be written as (p-1)*(q-1) with p < q.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A039770, A052291, A062732, A221284, A221285, A247129, A306440, A307245.

f:= proc(n) local w;
  w:= (2*n)^2;
  nops(select(t -> t < 2*n and isprime(t+1) and isprime(w/t + 1),  numtheory:-divisors(w)))
end proc:
map(f, [$1..100]); # Robert Israel, Apr 04 2019

f[n_] := Length@ Select[ Divisors[ 4n^2], # < 2n && PrimeQ[# + 1] && PrimeQ[ 4n^2/# + 1] &]; Array[f, 81] (* Robert G. Wilson v, Mar 30 2019 *)

a(n) = {my(nb = 0, nn = 4*n^2); fordiv(nn, d, if (d == 2*n, break); if (isprime(d+1) && isprime(nn/d+1), nb++);); nb;} \\ Michel Marcus, Mar 06 2019

A306882 Even numbers k such that phi(m) = k^2 has no solution.

Original entry on oeis.org

22, 34, 38, 46, 58, 62, 76, 78, 82, 86, 92, 98, 102, 106, 118, 122, 138, 142, 152, 154, 158, 164, 166, 172, 178, 182, 190, 194, 202, 212, 214, 218, 226, 238, 244, 254, 258, 262, 266, 274, 278, 282, 298, 302, 304, 310, 316, 318, 322, 328, 332, 334, 338, 344, 346, 356, 358, 362

Offset: 1

Bernard Schott, Mar 15 2019

nonn

In the link, P. Pollack and C. Pomerance "show that almost all squares are missing from the range of Euler's phi-function".
Except for m=1 and m=2, phi(m) is always even, so, the odd numbers >= 3 are not included in the data for clarity.
Includes 2*p if p is a prime not in A052291. - Robert Israel, Apr 10 2019

			phi(489) = 18^2, phi(401) = 20^2, phi(577) = 24^2, phi(677) = 26^2, but there is no integer m such that phi(m) = 22^2 = 484.

Robert Israel, Table of n, a(n) for n = 1..10000
P. Pollack and C. Pomerance, Square values of Euler's function, preprint (2013); Bulletin of the London Mathematical Society, Volume 46, Issue 2, 1 April 2014, Pages 403-414.

Cf. A000010, A002202, A052291, A058277, A062732, A221284, A221285.

select(t -> numtheory:-invphi(t^2)=[], [seq(i,i=2..400,2)]);  # Robert Israel, Apr 10 2019

isok(n) = !(n%2) && !istotient(n^2); \\ Michel Marcus, Mar 15 2019

cp's OEIS Frontend

A221285 Square values taken by totient function phi(m) = A000010(m).

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A228061 Numbers k such that k^2 = sigma(m) for some m.

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A306722 Number of pairs of primes (p,q), p < q, which are a solution of the Diophantine equation (p-1)*(q-1) = (2n)^2.

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A306882 Even numbers k such that phi(m) = k^2 has no solution.

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