Cumulative sums (and products)
Antoni Tapiès, Linia discontinua, 1967
From Wolfram MathWorld: « A cumulative sum is a sequence of partial sums of a given sequence. For example, the cumulative sums of the sequence {a, b, c, ...} are a, a+b, a+b+c, ... »
We want to build here the lexicographically earliest permutation P of the integers > 0 such that the cumultive sums Q reproduce P digit by digit. Are both P and Q correct?
The first 50 terms of P (computed by hand) seem to be:
P = 91, 10, 1, 102, 20, 4, 2, 24, 22, 8, 230, 25, 42, 7, 6, 28, 45, 14, 5, 3, 9, 58, 15, 88, 59, 46, 226, 66, 76, 81, 68, 689, 69, 87, 56, 77, 18, 599, 189, 64, 11, 90, 12, 561, 33, 21, 41, 13, 148, 2170, ...
P will be submitted soon to the OEIS.
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The variant with the cumulative products M (instead of the cumulative sums P) is fun to explore:
The first 79 terms of M (also computed by hand) seem to be:
M
= 1, 11, 2, 25, 50,
27, 500, 7, 4, 2500, 3, 71, 250000, 259, 8, 750000, 10, 39, 5000000, 2598, 7500000000,
77, 9, 6, 2500000000, 5, 53, 533, 75000000001, 38, 383, 43, 75000000000000, 35,
84, 13, 103, 12, 5000000000000, 28, 67, 30, 48, 25000000000000000, 21, 504, 78,
61, 87, 50000000000000000000, 2150, 47, 86, 18, 7500000000000000000000, 83, 868,
66, 613, 12500000000000000000000, 41, 93, 433, 306, 56, 2500000000000000000000000000,
108, 94, 539, 730, 44, 937, 5000000000000000000000000000, 81, 70, 90, 479, 783,
703, 12500000000000000000000000000000000000, ...
Arman, Accumulation de crayons de couleur, 1998
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