A291167 -id:A291167 - cp's OEIS frontend

A292571 Lucas-Carmichael numbers whose Dedekind psi value is a square.

935, 31535, 76751, 1707839, 3106799, 11141999, 24685199, 43383167, 83618639, 151524071, 161841239, 189099039, 212133599, 213884999, 219155615, 233743319, 241485839, 271038599, 287432495, 338340239, 353107799, 624840479, 660423455, 945236159, 1171355471

Offset: 1

Amiram Eldar, Sep 19 2017

nonn

			psi(935) = 36^2.

Amiram Eldar, Table of n, a(n) for n = 1..241 (terms below 10^12)

Intersection of A006972 and A291167.

Cf. A001615, A272798, A292572, A292573.

psi[n_] := If[n < 1, 0, n*Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]]; s = Import["b006972.txt","Data"][[All,-1]]; Select[s, IntegerQ@Sqrt[psi@#] &]

A291549 Numbers n such that both phi(n) and psi(n) are perfect squares.

Original entry on oeis.org

1, 60, 170, 240, 315, 540, 679, 680, 960, 1500, 2142, 2160, 2720, 2835, 3840, 4250, 4365, 4860, 5770, 6000, 7875, 8568, 8640, 9154, 9809, 10880, 13500, 14322, 15360, 15435, 17000, 19278, 19440, 22413, 23080, 24000, 25515, 29682, 33271, 34272, 34560, 36616, 37114, 37500

Offset: 1

Amiram Eldar and Altug Alkan, Aug 26 2017

Intersection of A039770 and A291167.
Squarefree terms are 1, 170, 679, 5770, 9154, 9809, 14322, ...
From Robert Israel, May 16 2019: (Start)
If n is in the sequence and p is a prime factor of n then p^2*n is in the sequence.
If n and m are coprime members of the sequence, then n*m is in the sequence. (End)

			60 is a term because phi(60) = 16 and psi(60) = 144 are both perfect squares.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000010, A000290, A001615, A039770, A291167.

filter:= proc(n) local F,psi,phi,p;
   F:= numtheory:-factorset(n);
   issqr( n*mul(1-1/p, p=F)) and issqr(n*mul(1+1/p,p=F))
end proc:
select(filter, [$1..50000]); # Robert Israel, May 15 2019

Select[Range[10^5], AllTrue[{EulerPhi@ #, If[# < 1, 0, # Sum[MoebiusMu[d]^2/d, {d, Divisors@ #}]]}, IntegerQ@ Sqrt@ # &] &] (* Michael De Vlieger, Aug 26 2017, after Michael Somos at A001615 *)

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))
isok(n) = issquare(eulerphi(n)) && issquare(a001615(n)); \\ after Charles R Greathouse IV at A001615

A292064 Triangular numbers k such that psi(k) is a square, where psi(k) is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 66, 210, 276, 1128, 2346, 2556, 4278, 5778, 7140, 7750, 7875, 11781, 13041, 18336, 22578, 27966, 28920, 31878, 32131, 32640, 35511, 51681, 70125, 73536, 79800, 89676, 93096, 100128, 102378, 122760, 139128, 169653, 173755, 177906, 209628, 223446, 253116

Offset: 1

Amiram Eldar, Sep 08 2017

nonn

The indices of these triangular numbers are 1, 2, 11, 20, 23, 47, 68, 71, 92, 107, 119, 124, 125, 153, 161, 191, 212, 236, 240, ...
The indices of the square psi values are 1, 2, 12, 24, 24, 48, 72, 72, 96, 108, 144, 120, 120, 144, 144, 192, 216, 240, 264, ...
Intersection of A000217 and A291167. - Altug Alkan, Sep 08 2017

			66 is in the sequence since 66 = 11*12/2 is triangular, and psi(66) = 144 = 12^2 is square.

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

Cf. A000217, A000290, A001615, A287473, A291167.

psi[n_] := If[n<1, 0, n*Sum[MoebiusMu[d]^2/d, {d, Divisors @ n}]];
Select[Accumulate[Range[1000]], IntegerQ[Sqrt[psi[#]]]&]

A291638 Numbers n such that psi(n) and sigma(n) both are perfect squares and n is not a squarefree number.

Original entry on oeis.org

370440, 1704024, 3926664, 11039112, 13854456, 21707784, 25264008, 28375704, 40822488, 44378712, 57862728, 59196312, 63937944, 75051144, 79051896, 79940952, 103500936, 107946216, 128394504, 134766072, 162178632, 169735608, 177737112, 191517480, 193530168, 195221880, 196407288, 215077464

Offset: 1

Altug Alkan, Aug 28 2017

nonn

			370440 = 2^3*3^3*5*7^3 is a term because psi(370440) = 2^2*3^2*7^2*(1+2)*(1+3)*(1+5)*(1+7) = 2^8*3^4*7^2 and sigma(370440) = (1+2+2^2+2^3)*(1+3+3^2+3^3)*(1+5)*(1+7+7^2+7^3) = 2^8*3^2*5^4.

Cf. A000203, A001615, A006532, A291167.

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = !issquarefree(n) && issquare(a001615(n)) && issquare(sigma(n)) \\ after Charles R Greathouse IV at A001615

cp's OEIS Frontend

A292571 Lucas-Carmichael numbers whose Dedekind psi value is a square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A291549 Numbers n such that both phi(n) and psi(n) are perfect squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A292064 Triangular numbers k such that psi(k) is a square, where psi(k) is the Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A291638 Numbers n such that psi(n) and sigma(n) both are perfect squares and n is not a squarefree number.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI