A385238 -id:A385238 - cp's OEIS frontend

A386623 Numbers k such that k - A224787(k) is a square.

1, 64, 630, 1225, 1296, 1750, 1925, 2079, 3125, 3402, 3888, 7150, 11495, 13000, 16445, 16464, 17160, 17500, 25578, 25935, 26082, 27508, 36975, 39083, 42688, 47125, 55955, 57188, 61740, 66671, 85085, 88451, 99372, 104544, 111375, 120736, 122452, 128898, 137547, 141427, 145509, 146927, 152592

Offset: 1

Will Gosnell and Robert Israel, Jul 27 2025

nonn

Numbers k such that k is the sum of a square and the cubes of the prime factors of k, counted with multiplicity.
Except for a(1) = 1, all terms are the product of at least 4 (not necessarily distinct) primes.
a(1) = 1 and a(7) = 1925 have k - A224787(k) = 1.  Are there any others? - Will Gosnell and Robert Israel, Aug 01 2025

			a(3) = 630 = 2 * 3^2 * 5 * 7 is a term because 630 - 2^3 - 2 * 3^3 - 5^3 - 7^3 = 100 is a square.

Robert Israel, Table of n, a(n) for n = 1..2511

Cf. A224787, A385238, A386245, A386304, A386640.

filter:= proc(n) local t;
   issqr(n - add(t[1]^3*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^6]);

A386640 Numbers k such that k + A224787(k) is a square.

Original entry on oeis.org

1, 225, 270, 1900, 4988, 5656, 6120, 8704, 11180, 16588, 17710, 19228, 24475, 28449, 29458, 32330, 34606, 38088, 39292, 40221, 41181, 42476, 48545, 48640, 53795, 56832, 57288, 64975, 78793, 84925, 86242, 117116, 124135, 128478, 129673, 134044, 136224, 136896, 147149, 150528, 168055, 183141

Offset: 1

Will Gosnell and Robert Israel, Jul 27 2025

nonn

Numbers k such that the sum of k and the cubes of the prime factors of k, counted with multiplicity, is a square.

			a(3) = 270 = 2 * 3^3 * 5 is a term because 270 + 2^3 + 3 * 3^3 + 5^3 =  484 = 22^2 is a square.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A224787, A385238, A386246, A386257, A386623.

filter:= proc(n) local t;
   issqr(n + add(t[1]^3*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^6]);

lim=184000;f[{p_,e_}]:=e*p^3;a224787[k_]:=If[k==1,0,Total[f/@FactorInteger[k]]];q[k_]:=IntegerQ[Sqrt[k+a224787[k]]];Select[Range[lim],q[#]&] (* James C. McMahon, Jul 30 2025 *)

cp's OEIS Frontend

A386623 Numbers k such that k - A224787(k) is a square.

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