A367781 Numbers with prime signature p^3 * q^2 whose difference from the next such number is a record low.

72, 1323, 1352, 49923, 27725061981125

Offset: 1

Author

Jon E. Schoenfield, Nov 29 2023

Keywords

• nonn
• fini

Comments

If it exists, a(6) > 10^20.

Do there exist two numbers with prime signature p^3 * q^2 that differ by less than 3? If not, this sequence terminates at a(5) = 27725061981125.

Examples

			A143610, the sequence of numbers with prime signature p^3 * q^2, is 72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, ...; the differences between its successive terms are 36, 92, 192, 108, 175, 293, 157, 198, 29, 20, 940, ..., and the first five record lows among those differences are 36, 29, 20, 5, 3, ...
------------------------------------------------  ----  -  --------------
  Two successive numbers of the form p^3 * q^2    diff  n            a(n)
------------------------------------------------  ----  -  --------------
2^3 *      3^2 =    72 | 3^3 *       2^2 =   108    36  1              72
3^3 *      7^2 =  1323 | 2^3 *      13^2 =  1352    29  2            1323
2^3 *     13^2 =  1352 | 7^3 *       2^2 =  1372    20  3            1352
3^3 *     43^2 = 49923 | 2^3 *      79^2 = 49928     5  4           49923
5^3 * 470957^2 =   ... | 2^3 * 1861621^2 =   ...     3  5  27725061981125
		

Crossrefs

Cf. A143610.