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A036035 Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents. 21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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.

EXAMPLE

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

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-Fran├žois Alcover, Nov 16 2011 *)

CROSSREFS

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

Cf. A025487, A059901, A096443.

Sequence in context: A173941 A194406 A087443 * A063008 A059901 A136101

Adjacent sequences:  A036032 A036033 A036034 * A036036 A036037 A036038

KEYWORD

nonn,easy,nice,tabf,look

AUTHOR

N. J. A. Sloane.

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

STATUS

approved

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Last modified February 19 16:28 EST 2018. Contains 299356 sequences. (Running on oeis4.)