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%I #29 Mar 01 2026 17:19:17
%S 1,2,3,4,5,8,9,15,24,80,2025,15552,32768,274658203125,7625597484987,
%T 9007199254740992,450283905890997363,381469726562500000000000000000,
%U 17763086495282268024161967871623168,2922977339492680612451840826835216578535400390625
%N 5-smooth numbers k = s(i) such that s(i)/s(i+1) sets a record high, where s = A051037.
%C No larger terms for n <= s(78738133) = 2^1200.
%H Michael De Vlieger, <a href="/A392740/b392740.txt">Table of n, a(n) for n = 1..22</a>
%e Let facs(x) represent the standard form prime decomposition of x.
%e Table of n, a(n) = s(i) for n = 1..22:
%e n i a(n) facs(s(i)) facs(s(i+1))
%e -------------------------------------------------
%e 1 1 1 1 2
%e 2 2 2 2 3
%e 3 3 3 3 2^2
%e 4 4 4 2^2 5
%e 5 5 5 5 2 * 3
%e 6 7 8 2^3 3^2
%e 7 8 9 3^2 2 * 5
%e 8 11 15 3 * 5 2^4
%e 9 15 24 2^3 * 3 5^2
%e 10 30 80 2^4 * 5 3^4
%e 11 109 2025 3^4 * 5^2 2^11
%e 12 197 15552 2^6 * 3^5 5^6
%e 13 240 32768 2^15 3^8 * 5
%e 14 2993 3^2 * 5^15 2^38
%e 15 4190 3^27 2 * 5^18
%e 16 7716 2^53 3^10 * 5^16
%e 17 10322 3^37 2^54 * 5^2
%e 18 46264 2^17 * 5^35 3^62
%e 19 71093 2^90 * 3^15 5^49
%e 20 197754 3^84 * 5^12 2^161
%e 21 5105847 5^207 2^21 * 3^290
%e 22 8629955 2^573 3^237 * 5^85
%t s = Block[{n = 2^60}, Union@ Flatten@ Table[2^i * 3^j * 5^k, {i, 0, Log[2, n]}, {j, 0, Log[3, n/2^i]}, {k, 0, Log[5, n/(2^i * 3^j)]} ] ]; Numerator@ Union@ FoldList[Max, Divide @@@ Partition[s, 2, 1] ]
%Y Cf. A051037, A085152, A392635, A392739.
%K nonn
%O 1,2
%A _Zhicheng Wei_, Feb 25 2026