Our Beloved Prof V.K Doraswamy - A Memoir (†) - goto memoir

It is possible to express EVERY NUMBER above 23,
as Sum Of Powers !
To arrive at the least number of constituent
powers is an intellectual task.
Rudiments of this exercise are depicted below.

Evolved readers can think of devising suitable Java script
to instantly arrive at the various constituent powers
for high value numbers.

Numbers > 23	Sum Of Powers		Numbers > 23	Sum Of Powers
24	24 + 23		74	72 + 52
25	52		75	52 + 52 + 52
26	32 + 32 + 23		76	62 + 62 + 22
27	33		77	33 + 52 + 52
28	24 + 23 + 22		78	72 + 52 + 22
29	52 + 22		79	33 + 33 + 52
30	32 + 32 + 23 + 22		80	26 + 24
31	33 + 22		81	34
32	25		82	26 + 32 + 32
33	24 + 32 + 23		83	72 + 52 + 32
34	24 + 32 + 32		84	26 + 24 + 22
35	33 + 23		85	72 + 62
36	62		86	25 + 33 + 33
37	52 + 23 + 22		87	62 + 33 + 24 + 23
38	52 + 32 + 22		88	26 + 24 + 23
39	33 + 23 + 22		89	72 + 25 + 23
40	25 + 23		90	34 + 32
41	52 + 24		91	26 + 33
42	52 + 32 + 23		92	72 + 33 + 24
43	33 + 24		93	34 + 23 + 22
44	62 + 23		94	72 + 62 + 32
45	62 + 32		95	26 + 33 + 22
46	52 + 32 + 23 + 22		96	26 + 25
47	33 + 24 + 22		97	34 + 24
48	25 + 24		98	72 + 72
49	72		99	72 + 52 + 52
50	52 + 52		100	102
51	33 + 24 + 23		101	34 + 24 + 22
52	33 + 52		102	72 + 72 + 22
53	72 + 22		103	72 + 33 + 33
54	33 + 33		104	102 + 22
55	33 + 24 + 23 + 22		105	26 + 25 + 32
56	25 + 24 + 23		106	72 + 72 + 23
57	25 + 24 + 32		107	26 + 33 + 24
58	72 + 32		108	102 + 23
59	25 + 33		109	102 + 32
60	62 + 24 + 23		110	34 + 52 + 22
61	72 + 23 + 22		111	26 + 33 + 24 + 22   OR 72 + 72 + 32 + 22
62	72 + 32 + 22		112	102 + 23 + 22
63	62 + 33		113	34 + 25
64	26		114	72 + 72 + 24
65	72 + 24		115	34 + 52 + 32
66	72 + 32 + 23		116	102 + 24
67	72 + 32 + 32		117	34 + 62
68	26 + 22		118	102 + 32 + 32
69	72 + 24 + 22		119	34 + 52 + 32 + 22
70	33 + 33 + 24		120	102 + 24 + 22
71	33 + 62 + 23		121	112
72	26 + 23		122	34 + 25 + 32
73	72 + 24 + 23		123	72 + 72 + 52

From B.S. Rangaswamy [ November 19, 2011 ]

A few days back, I came across a moving vehicle bearing reg. no '9441'.
Last number 9441 cajoled me to set out in quest of the constituent
powers of this 4 digiter :

Vehicle Number	Sum Of Powers	# tiers
9441	212 + 942 + 122 + 24 + 22	5 tier
	212 + 942 + 102 + 26	4 tier
	212 + 902 + 302	3 tier
	212 + 203 + 103	3 tier
	213 + 122 + 62	3 tier
	962 + 152	2 tier
	972 + 25	2 tier

Thus it was possible to arrive also at 2 tier results. This is to illustrate
that all numbers have several sets of such constituent powers. To arrive
at the least number of constituent powers is really an intellectual task.

With due regards to Mr Albert H Beiler, I have attempted to find
typical sets of constituent powers related to each
of ninedigital & pandigital squares, depicted in your webpage
The Nine Digits Page 2 under the following headings.

A. Reverse of ninedigital squares (30 nos)
B. Reverse of pandigital squares (87 nos)
C. Palindromic twin ninedigitals (30 nos)
D. Palindromic twin pandigitals (87 nos)

No pandigital cubes exist in 10 digit numbers. Only 6 cubes with
all numerals from 0 to 9 exist in eleven digit series !
(Episode 24 'Cubes with all numerals' of my book "Wonders of Numerals" )

E. Reverse of elevendigital cubes (6 nos)
F. Palindromic twin elevendigitals (12 nos)
G. Palindromic 32 digitals (6 nos)
H. Palindromic 63 digitals (10 nos)

Enclosing detailed sums of powers for all the above combinations,
for your kind perusal and considerations for inclusion
in your renowned website World!Of Numbers.