A266890 -id:A266890 - cp's OEIS frontend

A348425 Squares whose second arithmetic derivative is a square.

0, 1, 4, 49, 529, 2209, 6241, 27889, 28561, 35344, 49729, 128881, 192721, 250000, 431649, 528529, 703921, 1181569, 1495729, 1610361, 1868689, 3411409, 4870849, 5755201, 9138529, 11390625, 12250000, 13830961, 13845841, 15737089, 22648081, 25391521, 31618129

Offset: 1

Marius A. Burtea, Oct 18 2021

nonn

For prime numbers of the form p = k^2 - 2 (A028871) the number m = p^2 is a term because m'' = (p^2)'' = (2*p*p')' = (2*p)'= p + 2*p' = p + 2 = k^2.
If m is a term in A028873 then p = m^2 - 3 is prime and k = p^4 is a term. Indeed: k' = 4*p^3 and k'' = 4*p^3 + 12*p^2 = 4*p^2*(p + 3) = 4*p^2*m^2.
If m is a term in A201787  then p = 5*m^2 - 6 is prime and k = p^6 is a term. Indeed: k' = 6*p^5 and k'' = 5*p^5 + 30*p^4 = 5*p^4*(p + 6) = 25*p^4*m^2.

			4'' = 4' = 4 so 4 is a term.
49'' = 14' = 9 so 49 is a term.

Cf. A000290, A003415, A028871, A028873, A068346, A201787, A266890, A192192, A328244.

f:=func; [n:n in [s*s:s in [0.. 5623]] | IsSquare(Floor(f(Floor(f(n)))))];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 6000]^2, IntegerQ @ Sqrt[d[d[#]]] &] (* Amiram Eldar, Oct 18 2021 *)

ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k, 2]/f[k, 1])); \\ A003415
lista(nn) = {for (n=0, nn, if (issquare(ad(ad(n^2))), print1(n^2, ", ")); ); } \\ Michel Marcus, Oct 30 2021

A352330 Squares whose arithmetic derivative (A003415) is a cube.

Original entry on oeis.org

0, 1, 11664, 20736, 2313441, 2985984, 9150625, 28005264, 236421376, 655360000, 1871773696, 3340840000, 4294967296, 10435031104, 10485760000, 11716114081, 33556377856, 50054665441, 80706559921, 156531800881, 203928109056, 258439040161, 282429536481, 414998793616

Offset: 1

Marius A. Burtea, Mar 13 2022

nonn

For p prime number of the form p = 4*m^3 - 1 (A199367) the number k = 2^8*p^4 is a term. Indeed, k' = (2^8*p^4)' = 8*2^7*p^4 + 2^8*4*p^3 = 2^9*p^3*(2*(p + 1)) = 2^9*p^3*(2*(p + 1)) = 2^9*p^3*2*4*m^3 = (2^3*p*8*m)^3 so k is a term.
The sequence is infinite because numbers of the form m = 2^(2^(6*k + 5)), k >= 0, are terms. Indeed: m' = 2^(6*k + 5)*2^(2^(6*k + 5) - 1) = 2^(6*k + 4 + 2^(6*k + 5)) = 2^(6*k + 3 + 2^(6*k + 5) + 1), and the exponent 6*k + 3 + 2^(6*k + 5) + 1 is divisible by 3.
If p is a prime number then the numbers of the form m = p^(64^k), k >= 1 are terms.

			11664 = 108^2 and 11664' = 46656 = 36^3 so 11664 is a term.
20736 = 144^2 and 20376' = 110592 = 48^3 so 20736 is a term.

Cf. A003415, A199367, A266890.

f:=func; [p:p in [s*s:s in [0.. 450000]]| IsPower(Floor(f(p)),3)];

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 6.5*10^5]^2, IntegerQ@Surd[d[#], 3] &] (* Amiram Eldar, Mar 13 2022 *)

cp's OEIS Frontend

A348425 Squares whose second arithmetic derivative is a square.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI