A380953 Numbers m such that the sum of its distinct prime factors and the sum of its nonprime divisors are both squares.

1, 323, 3887, 5183, 149903, 311790, 777923, 1327103, 6718463, 12446783, 14605487, 16402499, 20373435, 28128270, 30856494, 33144430, 37058230, 37380745, 68661901, 86755609, 139557721, 159954570, 221294682, 222538813, 229159043, 269108440, 360590058, 412621345

Offset: 1

Author

Michel Lagneau, Feb 09 2025

Keywords

• nonn

Comments

Or numbers m such that A008472(m) and (A000203(m) - A008472(m)) are both squares.

Examples

			s1 is the sum of the prime factors, s2 is the sum of the nonprime divisors.
+----------------------------+-------------------------+-----+-------+
|    m   |  prime factors    |   nonprimedivisors      |  s1 |  s2   |
+----------------------------+-------------------------+-----+-------+
|    323 | {17, 19}          | {1, 323}                | 6^2 |  18^2 |
+----------------------------+-------------------------+-----+-------+
|   3887 | {13, 23}          | {1, 169, 299, 3887}     | 6^2 |  66^2 |
+----------------------------+-------------------------+-----+-------+
|   5183 | {71, 73}          | {1, 5183}               |12^2 |  72^2 |
+----------------------------+-------------------------+-----+-------+
| 149903 | {13, 887}         | {1, 169, 11531, 149903} |30^2 | 402^2 |
+----------------------------+-------------------------+-----+-------+
		

Crossrefs

Cf. A000203, A008472, A164722.

Programs

Maple

with(numtheory):nn:=10^8:print(1):
for m from 2 to nn do:
 d:=factorset(m):n0:=nops(d):s:=sum('d[i]', 'i'=1..n0):
    if issqr(s) and issqr(sigma(m)-s) then print(m):
     else
    fi:
 od:

PARI

isok(m) = my(f=factor(m), s=sum(k=1, #f~, f[k,1])); issquare(s) && issquare(sigma(f)-s); \\ Michel Marcus, Feb 09 2025

Extensions

More terms from Jinyuan Wang, Feb 11 2025