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A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.

%I A325241 #16 Jan 30 2023 02:39:58
%S A325241 12,18,20,28,44,45,50,52,60,63,68,72,75,76,84,90,92,98,99,108,116,117,
%T A325241 124,126,132,140,147,148,150,153,156,164,171,172,175,180,188,198,200,
%U A325241 204,207,212,220,228,234,236,242,244,245,252,260,261,268,275,276,279
%N A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.
%C A325241 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum multiplicity is one greater than their minimum (counted by A325279).
%C A325241 The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... . - _Amiram Eldar_, Jan 30 2023
%H A325241 Michael S. Branicky, <a href="/A325241/b325241.txt">Table of n, a(n) for n = 1..10000</a>
%F A325241 A051903(a(n)) - A051904(a(n)) = 1.
%e A325241 The sequence of terms together with their prime indices begins:
%e A325241   12: {1,1,2}
%e A325241   18: {1,2,2}
%e A325241   20: {1,1,3}
%e A325241   28: {1,1,4}
%e A325241   44: {1,1,5}
%e A325241   45: {2,2,3}
%e A325241   50: {1,3,3}
%e A325241   52: {1,1,6}
%e A325241   60: {1,1,2,3}
%e A325241   63: {2,2,4}
%e A325241   68: {1,1,7}
%e A325241   72: {1,1,1,2,2}
%e A325241   75: {2,3,3}
%e A325241   76: {1,1,8}
%e A325241   84: {1,1,2,4}
%e A325241   90: {1,2,2,3}
%e A325241   92: {1,1,9}
%e A325241   98: {1,4,4}
%e A325241   99: {2,2,5}
%t A325241 Select[Range[100],Max@@FactorInteger[#][[All,2]]-Min@@FactorInteger[#][[All,2]]==1&]
%t A325241 Select[Range[300],  Min[e = FactorInteger[#][[;; , 2]]] +1 == Max[e] &] (* _Amiram Eldar_, Jan 30 2023 *)
%o A325241 (Python)
%o A325241 from sympy import factorint
%o A325241 def ok(n):
%o A325241     e = sorted(factorint(n).values())
%o A325241     return n > 1 and max(e) == 1 + min(e)
%o A325241 print([k for k in range(280) if ok(k)]) # _Michael S. Branicky_, Dec 18 2021
%o A325241 (PARI) is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) + 1 == vecmax(e); } \\ _Amiram Eldar_, Jan 30 2023
%Y A325241 Positions of 1's in A062977. Supersequence of A054753, A096156.
%Y A325241 Cf. A001221, A001222, A001694, A051903, A051904, A052485, A056239, A112798, A118914, A325240, A325259, A325270, A325279.
%Y A325241 Cf. A059956, A088453.
%K A325241 nonn
%O A325241 1,1
%A A325241 _Gus Wiseman_, Apr 15 2019