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%I #25 Mar 24 2020 03:25:02
%S 21,33,39,65,85,95,115,133,145,155,161,185,203,205,215,217,235,259,
%T 261,265,279,287,295,301,305,329,335,341,355,365,371,395,407,413,415,
%U 427,445,451,469,473,481,485,497
%N Integers which are prime power-like but not prime powers.
%C For an integer n>0 and not the unity we define DTD(n) to be the difference table of the divisors of n. We say that DTD(n) is positive if all entries in the table are positive and we call DTD(n) monotone if all rows and all columns of the table are nondecreasing (reading from left to right and from top to bottom).
%C We define an integer n to be prime power-like if and only if DTD(n) is positive and monotone. All prime powers (in the sense of A246655 (but not in the sense of A000961)) are prime power-like integers. Sequence A273200 provides the prime power-like integers. This sequence (A273201) lists the integers which are prime power-like but not prime powers.
%C Thus we have the inclusions A000040 < A246655 < A273200 and the union A273200 = A273201 U A246655. Integers which have a positive but not monotone DTD are listed in A273199. Integers with a positive DTD are listed in A273130.
%e 95 is in this sequence because the DTD of 95 has positive entries and nondecreasing rows and columns:
%e [ 1 5 19 95]
%e [ 4 14 76]
%e [10 62]
%e [52]
%t pplikeQ[n_] := Module[{T, DTD, DTD2}, If[n == 1 || PrimePowerQ[n], Return[False]]; T = Divisors[n]; DTD = Table[Differences[T, k], {k, 0, Length[T]-1}]; If[AnyTrue[Flatten[DTD], NonPositive], Return[False]]; DTD2 = Transpose[PadRight[#, Length[T], Infinity]& /@ DTD]; AllTrue[DTD, OrderedQ] && AllTrue[DTD2, OrderedQ]];
%t Select[Range[500], pplikeQ] (* _Jean-François Alcover_, Jun 28 2019 *)
%o (Sage) # uses[is_prime_power_like from A273200]
%o # Compare script in A273200.
%o def is_A273201(n):
%o return not is_prime_power(n) and is_prime_power_like(n)
%o print(list(filter(is_A273201, range(1, 500))))
%Y Cf. A000040, A246655, A273102, A273130, A273199, A273200.
%K nonn
%O 1,1
%A _Peter Luschny_, May 17 2016
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