A046756 Numbers k such that d(k)^4 divides k.

1, 625, 6561, 117649, 4100625, 31250000, 37515625, 73530625, 771895089, 1000000000, 2147483648, 6442450944, 10737418240, 15032385536, 23622320128, 25937424601, 27917287424, 32212254720, 33059881728, 36507222016, 40802189312, 45097156608, 49392123904

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Proper subset of A033950, A046754 and A046755. Relatively prime terms are in the sequence together with their products like 73530625 or 771895089000000000.

2^31 is a term, as is every integer of the form 2^31*p, 2^31*p^3, and 2^31*p*q, where p and q are distinct odd primes; each of these has 32, 64, or 128 divisors. Of the first 10000 terms, 9609 are of one of those forms. Of the remaining 391 terms, 316 are of the form 2^8 * 3^17 * m, where m is 1, a prime > 3, or 5^4. - Jon E. Schoenfield, Aug 13 2022

Examples

			If k=625, d(k) = sigma(0,k) = 5. Its 4th power is 625, which divides k.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A033950, A046754, A046755.

Programs

Mathematica

Select[ Range[ 1, 14500000 ], IntegerQ[ #/(DivisorSigma[ 0, # ])^2 ]& ]

Extensions

a(6)-a(20) from Donovan Johnson, Nov 30 2008

a(21)-a(23) from Donovan Johnson, Jun 08 2011