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A162020 Positive k such that there is at least one partition into parts {14, 52, 78, 133, 248} (terms of A113907).

%I A162020 #20 May 23 2021 06:25:01
%S A162020 14,28,42,52,56,66,70,78,80,84,92,94,98,104,106,108,112,118,120,122,
%T A162020 126,130,132,133,134,136,140,144,146,147,148,150,154,156,158,160,161,
%U A162020 162,164,168,170,172,174,175,176,178,182,184,185,186,188,189,190,192,196,198,199,200,202,203,204,206,208,210,211,212
%N A162020 Positive k such that there is at least one partition into parts {14, 52, 78, 133, 248} (terms of A113907).
%C A162020 All integers greater than 327 are also in the sequence. There are 163 values less than 327, and 164 in the complement of this sequence. The first value arising in two different ways is 156 = 52 + 52 + 52 = 78 + 78. The first triples of three consecutive values: (132,133,134), (160,161,162). "Use the Frobenius problem. Since gcd(52,133)=1 the maximum value not in the sequence is 52*133-52-133=6731 even using only 52 and 133, by a simple dp code you can check all values up to this limit and in fact n=327 is the maximum not in the original sequence." -- _Robert Gerbicz_.
%H A162020 Wikipedia, <a href="http://en.wikipedia.org/wiki/Coin_problem">Coin problem</a>.
%p A162020 L := [14,52,78,133,248] ; # check whether n is a linear combination of op(..,L) with # some nonnegative a,b,c,d,e frob := proc(n,L) local a,b,c,d,e,nres ; for a from 0 do if a*op(1,L) > n then break; fi; for b from 0 do if a*op(1,L)+b*op(2,L) > n then break; fi; for c from 0 do if a*op(1,L)+b*op(2,L)+c*op(3,L) > n then break; fi; for d from 0 do if a*op(1,L)+b*op(2,L)+c*op(3,L)+d*op(4,L) > n then break; fi; nres := n-a*op(1,L)-b*op(2,L)-c*op(3,L)-d*op(4,L) ; if nres < 0 then break; fi; if nres = 0 then RETURN([a,b,c,d,0]) ; fi; if ( nres mod op(5,L) ) = 0 then RETURN([a,b,c,d,nres/op(5,L)]) ; fi; od: od; od: od: RETURN([]) ; end: for n from 14 to 200 do f := frob(n,L) ; if f <> [] then print(n,f) ; fi; od: Extended and b-file by _R. J. Mathar_.
%t A162020 ok[n_] := Reduce[a >= 0 && b >= 0 && c >= 0 && d >= 0 && e >= 0 && 14a + 52b + 78c + 133d + 248e == n, {a, b, c, d, e}, Integers] =!= False; Select[Range[196], ok] (* _Jean-François Alcover_, Sep 06 2011 *)
%Y A162020 Cf. A113907.
%K A162020 nonn,easy,less
%O A162020 1,1
%A A162020 _Jonathan Vos Post_, Jun 24 2009
%E A162020 Better name from _Joerg Arndt_, May 23 2021