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A036035 Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.

%I A036035 #53 Mar 31 2025 11:56:06
%S A036035 1,2,4,6,8,12,30,16,24,36,60,210,32,48,72,120,180,420,2310,64,96,144,
%T A036035 216,240,360,900,840,1260,4620,30030,128,192,288,432,480,720,1080,
%U A036035 1800,1680,2520,6300,9240,13860,60060,510510,256,384,576,864,1296,960,1440,2160
%N A036035 Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.
%C A036035 The exponents can be read off Abramowitz and Stegun, p. 831, column labeled "pi".
%C A036035 Here are the partitions in the order used by Abramowitz and Stegun (graded reflected colexicographic order): 0; 1; 2, 1+1; 3, 1+2, 1+1+1; 4, 1+3, 2+2, 1+1+2, 1+1+1+1; 5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1; ... (Cf. A036036)
%C A036035 Here are the partitions in graded colexicographic order: 0; 1; 2, 1+1; 3, 2+1, 1+1+1; 4, 3+1, 2+2, 2+1+1, 1+1+1+1; 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1; ... (Cf. A036037)
%C A036035 Since the prime signature is a partition of Omega(n), so to speak, the internal order is only a matter of convention and has no effect on the least integer with a given prime signature.
%C A036035 The graded colexicographic order has the advantage that the exponents are in the same order as the least integer with a given prime signature (also used on the wiki page, see links).
%C A036035 Embedded values include the primorial numbers 1, 2, 6, 30, 210, 2310, 30030 ... (A002110) with unordered factorizations counted by A000110 (Bell numbers) and ordered factorizations by A000670 (ordered Bell numbers).
%C A036035 When viewed as a table the n-th row has partition(n) (A000041(n)) terms. - _Alford Arnold_, Jul 31 2003
%C A036035 A closely related sequence, A096443(n), gives the number of partitions of the n-th multiset. - _Alford Arnold_, Sep 29 2005
%D A036035 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).
%H A036035 Peter Luschny, <a href="/A036035/b036035.txt">Rows n = 0..25, flattened</a>
%H A036035 M. Abramowitz and I. A. Stegun, eds., <a href="https://personal.math.ubc.ca/~cbm/aands/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
%H A036035 John Baez, <a href="https://plus.google.com/u/0/117663015413546257905/posts/XAokddKJEBt">What happens when a particle gets created?</a>
%H A036035 OEIS Wiki, <a href="http://oeis.org/wiki/Prime_signature">Prime signature</a>.
%e A036035 1;
%e A036035 2;
%e A036035 4, 6;
%e A036035 8, 12, 30;
%e A036035 16, 24, 36, 60, 210;
%e A036035 32, 48, 72, 120, 180, 420, 2310;
%e A036035 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030;
%e A036035 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510;
%p A036035 with(combinat):
%p A036035 A036035_row := proc(n) local e, w; w := proc(e) local i, p;
%p A036035 p := [seq(ithprime(nops(e)-i+1), i=1..nops(e))];
%p A036035 mul(p[i]^e[i], i=1..nops(e)) end:
%p A036035 seq(w(conjpart(e)), e = partition(n)) end:
%p A036035 seq(A036035_row(i), i=0..10);  # _Peter Luschny_, Aug 01 2013
%t A036035 nmax = 52; primeSignature[n_] := Sort[ FactorInteger[n], #1[[2]] > #2[[2]] & ][[All, 2]]; ip[n_] := Reverse[ Sort[#]] & /@ Split[ Sort[ IntegerPartitions[n], Length[#1] < Length[#2] & ], Length[#1] == Length[#2] & ]; tip = Flatten[ Table[ip[n], {n, 0, 8}], 2]; a[n_] := (sig = tip[[n+1]]; k = 1; While[sig =!= primeSignature[k++]]; k-1); a[0] = 1; a[1] = 2; Table[an = a[n]; Print[an]; an, {n, 0, nmax}](* _Jean-François Alcover_, Nov 16 2011 *)
%o A036035 (PARI) Row(n)={[prod(i=1, #p, prime(i)^p[#p+1-i]) | p<-partitions(n)]} \\ _Andrew Howroyd_, Oct 19 2020
%Y A036035 A025487 in a different order. Cf. A035098, A002110, A000110 and A000670.
%Y A036035 Cf. A025487, A059901, A096443.
%K A036035 nonn,easy,nice,tabf,look
%O A036035 0,2
%A A036035 _N. J. A. Sloane_
%E A036035 More terms from _Alford Arnold_; corrected Sep 10 2002
%E A036035 More terms from _Ray Chandler_, Jul 13 2003
%E A036035 Definition corrected by _Daniel Forgues_, Jan 16 2011