A340303
Numbers k such that k and the next two numbers after k with the same prime signature as k also have the same set of distinct prime divisors as k.
Original entry on oeis.org
10800, 12600, 54000, 75600, 148176, 162000, 264600, 324000, 609840, 453600, 529200, 793800, 444528, 648000, 1058400, 1323000, 1764000, 1620000, 2116800, 2646000, 2721600, 3704400, 3969000, 6403320, 4233600, 5292000, 4860000, 6223392, 7938000, 7408800, 8467200
Offset: 1
10800 = 2^4 * 3^3 * 5^2 is a term since 10800 and the next two numbers with the same prime signature as 10800, 16200 = 2^3 * 3^4 * 5^2 and 18000 = 2^4 * 3^2 * 5^3, have the same set of distinct prime divisors, {2, 3, 5}.
A340304
Numbers k such that k and the next three numbers after k with the same prime signature as k also have the same set of distinct prime divisors as k.
Original entry on oeis.org
529200, 1058400, 2646000, 3704400, 5292000, 9720000, 15876000, 22226400, 18522000, 31752000
Offset: 1
529200 = 2^4 * 3^3 * 5^2 * 7^2 is a term since 529200 and the next three numbers with the same prime signature as 529200, 793800 = 2^3 * 3^4 * 5^2 * 7^2, 882000 = 2^4 * 3^2 * 5^3 * 7^2, and 1234800 = 2^4 * 3^2 * 5^2 * 7^3, have the same set of distinct prime divisors, {2, 3, 5, 7}.
Original entry on oeis.org
12, 420, 540, 2268, 7020, 10692, 11340, 17640, 24948, 42750, 56700, 87120, 152460, 409500, 413100, 609840, 996072, 2478600, 3822000, 5287500, 9189180, 9447840, 14871600, 20241900, 20567520, 23510592, 23832800, 27766152, 28552500, 39358800, 41135040, 44783648, 49985100
Offset: 1
12 is a term since the A065642(12) = A081761(12) = 18, i.e., 18 = 2 * 3^2 is the least number with the same set of prime divisors, {2, 3}, and the same prime signature as 12 = 2^2 * 3.
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rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; next[n_] := Module[{r = rad[n]}, SelectFirst[Range[n + 1, n^2], rad[#] == r &]]; sig[n_] := Sort@FactorInteger[n][[;; , 2]]; nextsig[n_] := Module[{sign = sig[n], k = n + 1}, While[sig[k] != sign, k++]; k]; Select[Range[2, 600], sig[#] == sig[next[#]] && rad[#] == rad[nextsig[#]] &]
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