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

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 1296, 960, 1440, 2160

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• tabf
• look

Comments

The exponents can be read off Abramowitz and Stegun, p. 831, column labeled "pi".

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)

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)

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.

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).

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).

When viewed as a table the n-th row has partition(n) (A000041(n)) terms. - Alford Arnold, Jul 31 2003

A closely related sequence, A096443(n), gives the number of partitions of the n-th multiset. - Alford Arnold, Sep 29 2005

Examples

			1;
2;
4, 6;
8, 12, 30;
16, 24, 36, 60, 210;
32, 48, 72, 120, 180, 420, 2310;
64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030;
128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510;
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Links

• Peter Luschny, Rows n = 0..25, flattened
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
• John Baez, What happens when a particle gets created?
• OEIS Wiki, Prime signature.

Crossrefs

A025487 in a different order. Cf. A035098, A002110, A000110 and A000670.

Cf. A025487, A059901, A096443.

Programs

Maple

with(combinat):
A036035_row := proc(n) local e, w; w := proc(e) local i, p;
p := [seq(ithprime(nops(e)-i+1), i=1..nops(e))];
mul(p[i]^e[i], i=1..nops(e)) end:
seq(w(conjpart(e)), e = partition(n)) end:
seq(A036035_row(i), i=0..10);  # Peter Luschny, Aug 01 2013

Mathematica

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 *)

PARI

Row(n)={[prod(i=1, #p, prime(i)^p[#p+1-i]) | p<-partitions(n)]} \\ Andrew Howroyd, Oct 19 2020

Extensions

More terms from Alford Arnold; corrected Sep 10 2002

More terms from Ray Chandler, Jul 13 2003

Definition corrected by Daniel Forgues, Jan 16 2011