A330486 Earliest start of a run of n numbers divisible by a seventh power larger than one.

128, 76544, 2372890624, 390491792890623, 2083234733888734218749, 18962123650219836035505781245

Offset: 1

Author

Jud McCranie, Dec 15 2019

Keywords

• nonn
• more
• hard

Comments

In De Koninck's book, a(4) was probable but not certain. It also gives probable terms for a(5) and a(6).

From David A. Corneth, Dec 18 2019: (Start)

a(6) <= 18962123650219836035505781245

a(7) <= 105574810222799317012520534891328125

a(8) <= 352061173826332779662070232460962002804453118

a(9) <= 74322635218313783849790472504951802188239215127109371

If any a(k) of these values are less than presented values, one of a(k) + m is divisible only by at least one p^7 for some p > 31, 0 <= m < k.

For a(6) this minimum prime is some prime p > 59. (End)

Examples

			2372890624 is divisible by 2^7, 2372890625 is divisible by 5^7, 2372890626 is divisible by 3^7.  This is the smallest number with this property, so a(3)=2372890624.
		

References

• J.-M. De Koninck, Those Fascinating Numbers, Entry 242, p. 63, Amer. Math. Soc., 2009.

Links

• David A. Corneth, PARI program for finding candidates
• Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 63.

Crossrefs

Cf. A045882, A271443, A001015.

Extensions

a(5) from Giovanni Resta, Dec 17 2019

a(6) from Giovanni Resta, Dec 19 2019