The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A293627 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A293627

Number of knapsack factorizations whose factors sum to n.
(history; published version)

#19 by Peter Luschny at Sun Oct 29 21:30:13 EDT 2017

STATUS

reviewed

approved

#18 by Andrey Zabolotskiy at Sat Oct 28 15:19:16 EDT 2017

STATUS

proposed

reviewed

#17 by Gus Wiseman at Tue Oct 24 03:41:05 EDT 2017

STATUS

editing

proposed

#16 by Gus Wiseman at Tue Oct 24 03:40:50 EDT 2017

CROSSREFS

Cf. A000041, A001055, A002033, A002865, A108917, A126796, A275972, A281116, A292886, Q02A294150.

#15 by Gus Wiseman at Mon Oct 23 21:52:45 EDT 2017

NAME

allocated for Gus WisemanNumber of knapsack factorizations whose factors sum to n.

DATA

1, 1, 1, 2, 2, 4, 4, 6, 8, 11, 12, 19, 21, 27, 34, 45, 51, 69, 77, 100, 117, 146

OFFSET

1,4

COMMENTS

A knapsack factorization is a finite multiset of positive integers greater than one such that every distinct submultiset has a different product.

EXAMPLE

The a(12) = 19 partitions are:

(12),

(10 2), (9 3), (8 4), (7 5), (6 6),

(8 2 2), (7 3 2), (6 4 2), (6 3 3), (5 5 2), (5 4 3), (4 4 4),

(6 2 2 2), (5 3 2 2), (4 3 3 2), (3 3 3 3),

(3 3 2 2 2),

(2 2 2 2 2 2).

MATHEMATICA

nn=22;

apsQ[y_]:=UnsameQ@@Times@@@Union[Rest@Subsets[y]];

Table[Length@Select[IntegerPartitions[n], apsQ], {n, nn}]

CROSSREFS

Cf. A000041, A001055, A002033, A002865, A108917, A126796, A275972, A281116, A292886, Q02.

KEYWORD

allocated

nonn,more

AUTHOR

Gus Wiseman, Oct 23 2017

STATUS

approved

editing

#14 by Gus Wiseman at Mon Oct 23 21:52:45 EDT 2017

NAME

allocated for Gus Wiseman

KEYWORD

recycled

allocated

#13 by N. J. A. Sloane at Mon Oct 23 20:04:33 EDT 2017

STATUS

proposed

approved

#12 by Andrey Zabolotskiy at Tue Oct 17 17:17:41 EDT 2017

STATUS

editing

proposed

#11 by Andrey Zabolotskiy at Tue Oct 17 17:17:14 EDT 2017

NAME

Take an integer k and sum the numbers obtained deleting only one of the digit of k, repeating the process for any digit of k. Sequence lists numbers such that the sum is equal to the reverse of k.

DATA

10, 20, 30, 40, 50, 60, 70, 80, 90, 531, 27681, 67743, 70692, 4812363, 5741514, 34999992, 36600123, 63690684, 65523315, 94444650, 261400662, 315000072, 385601193, 906400737

OFFSET

1,1

EXAMPLE

With 70692 we have 692 + 7692 + 7092 + 7062 + 7069 = 29607 that is the reverse of 70692;

With 36600123 we have 6600123 + 3600123 + 3600123 + 3660123 + 3660123 + 3660023 + 3660013 + 3660012 = 32100663 that is the reverse of 36600123.

MAPLE

T:=proc(w) local x, y, z; x:=0; y:=w; for z from 1 to ilog10(w)+1 do

x:=10*x+(y mod 10); y:=trunc(y/10); od; x; end:

P:=proc(q) local c, d, k, n; for n from 1 to q do d:=ilog10(n)+1; c:=0;

for k from 0 to d-1 do c:=n mod 10^k+trunc(n/10^(k+1))*10^k+c; od;

if c=T(n) then print(n); fi; od; end: P(10^9);

CROSSREFS

Cf. A004086, A131639.

KEYWORD

nonn,base,more,changed

recycled

AUTHOR

Paolo P. Lava, Oct 13 2017

STATUS

proposed

editing

Discussion

Tue Oct 17

17:17

Andrey Zabolotskiy: Since you don't mind, as well as anybody else, I'll do it. Also, let me explain why I think so: it is the case of the 3rd main bullet of the 3rd section of the essay https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Is_this_sequence_interesting : the composition of 3 unrelated concepts, namely: take A004086(n), take A093882(n), compare them.

#10 by Paolo P. Lava at Tue Oct 17 05:14:13 EDT 2017

STATUS

editing

proposed

Discussion

Tue Oct 17

05:24

Andrey Zabolotskiy: Seems a bit contrived to me...

05:34

Paolo P. Lava: It is similar to A131639. Here I used a verbose definition that could be relaxed. Anyway, if the problem is the "reverse" then recycle