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A225580 The sum of all substrings of n (including n). 4
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 66, 68, 70, 72, 74, 76, 78, 80 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This sequence differs from A071980 beginning with n = 1010, and differs formulaically beginning with n = 1000 (the first four digit number). Where A071980 is calculated as a + ab + abc + abcd + bcd + cd + d for four digit numbers abcd, this sequence also includes the term bc in the sum.
Limits: n <= a(n) < 1.73*n. Proof: a(n)/n will be maximized when substrings are as large as possible while n is as small as possible, or for numbers of the form 199999999... The sum of substrings of this number is < 222222... + < 1234567... or < 3456790123.../2000000000... or < 1.728396.
The number 111 is the smallest term that occurs twice in the sequence, when n = {96, 100}. The number 2254 is the smallest term that occurs three times in the sequence, when n = {1476, 1510, 2008}.
LINKS
Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Ulam spiral of all values of a(n)<10000, color-coded by the number of times they occur.
FORMULA
a(n) = A138953(n) + n. (Note the offset in A138953 is zero. - Zak Seidov, May 16 2013)
EXAMPLE
For n=1980, a(n) = 1 + 9 + 8 + 0 + 19 + 98 + 80 + 198 + 980 + 1980 = 3373. Note that A071980(1980) = 3258, because it does not include 9, 8, 98 in the sum.
MATHEMATICA
Table[s = IntegerDigits[n]; Total[Flatten[Table[FromDigits /@ Partition[s, i, 1], {i, Length[s]}]]], {n, 100}] (* T. D. Noe, May 13 2013 *)
PROG
(R) sapply(1:100, function(n) {tot=0; s=as.character(n); len=nchar(s); for(i in 1:len) for(j in i:len) tot=tot+as.numeric(substr(s, i, j)); tot})
CROSSREFS
Sequence in context: A317621 A298297 A331009 * A071980 A058183 A322341
KEYWORD
nonn,base
AUTHOR
EXTENSIONS
Example corrected by Zak Seidov, May 16 2013
STATUS
approved

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Last modified June 16 08:45 EDT 2024. Contains 373424 sequences. (Running on oeis4.)