A138482
Mutually-praising pairs excluding autobiographical numbers A138480. Version 2: numbers may have more than 10 digits.
Original entry on oeis.org
130, 230, 430, 530, 630, 730, 830, 930, 1101, 2210, 10110, 11200, 23100, 43100, 53100, 63100, 73100, 83100, 93100, 211100, 230100, 311100, 411100, 430100, 511100, 530100, 611100, 630100, 711100, 730100, 811100, 830100, 911100, 930100
Offset: 1
130 and 1101 are a mutually-praising pair: 130 specifies a number with 1 digit equal to "0", 3 digit "1" and nothing else. 1101 specifies 1 digit "0", 1 digit "1", no digits "2" and 1 digit "3". 73100 and 21010001000 is the first pair with a number of more than 10 digits.
A046043
Autobiographical numbers (or curious numbers): list of numbers m = x_0 x_1 x_2 ... x_{b-1} (written in base b) such that x_i is the number of "digits" in m that are equal to i, for all i=0,...,b-1.
Original entry on oeis.org
1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000
Offset: 1
Robert Leduc (leduc(AT)macalester.edu)
m = 1210 is written in base 4 (since it has 4 digits), and has one 0, two 1's, one 2 and zero 3's and m = "one two one zero".
- E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
- M. Gardner, Mathematical Circus, pp. 128; 135 Prob. 7 Alfred A. Knopf NY 1979.
- Tanya Khovanova, A Story of Storytelling Numbers, Math. Horizons, Sep 2009, 14-17.
- Fred Gavin, Letters to the editor, Math. Mag 66 (4) (1993) p 276.
- Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
- Amy Harmon, Beyond 'Hidden Figures': Nurturing New Black and Latino Math Whizzes, New York Times, Feb 17, 2017.
- Tanya Khovanova, Autobiographical Numbers, arXiv:0803.0270 [math.CO], 2008.
- Chris Smith, #MegaFavNumbers Self-Descriptive Numbers (the beauty 6210001000), video (2020)
- K. Uhland, The Ten-Digit Number [Broken link?]
Compare with the "Look-and-Say" version
A047841.
-
isSelfDescribing[n_Integer] := (RotateRight[DigitCount[n]] == PadRight[IntegerDigits[n], 10]); Select[Range[10^10 - 1], isSelfDescribing] (* Martin Ettl, Oct 09 2012 *) (* Warning: This program causes Mathematica to crash! - David Callan, Feb 17 2017 *)
A359049
Autobiographical numbers k whose decimal digits are a concatenation count(0), count(1), ..., count(m) for some m, where count(j) is the number of (possibly overlapping) occurrences of j within the digits of k itself.
Original entry on oeis.org
1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 53110100002, 62200010001, 541011000021, 6401101000310, 74011001003100, 840110001031000, 1040110000031000, 9321000001201000, 94201000012110000, 1160010100041000010, 11611001000320000100, 13313000000001200000, 13313000000100200000
Offset: 1
1040110000031000 is a term: we have ten 0's, four 1's, zero 2's, one 3, one 4, three 10's and one 11 as integers in the term, visualized as follows:
Integers(j): 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
term: 10 4 0 1 1 0 0 0 0 0 3 1 0 0 0
Notice that overlapping integers are counted so 110 is one 11, one 10 (or 111 would be two 11's).
A234512
Numbers n = d(0)d(1)d(2)...d(r) such that d(i) is the number of differences |d(i)-d(i-1)| equal to i in n, i = 1,2,...,r.
Original entry on oeis.org
110, 311000, 2301000, 3003000, 3120000, 42100000, 410300000, 430100000
Offset: 1
311000 is in the sequence because the differential digits are:
|1-3| = 2;
|1-1| = 0;
|0-1| = 1;
|0-0| = 0;
|0-0| = 0, and
0 appears three times => 3;
1 appears one time => 1;
2 appears one time => 1;
3 appears zero time => 0;
4 appears zero time => 0;
5 appears zero time => 0, hence a(2) = 311000.
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with(numtheory):for n from 10 to 10^10 do:T:=array(0..9):for k from 0 to 9 do:T[k]:=0:od:x:=convert(n,base,10):n1:=nops(x):for i from 1 to n1-1 do:a:=abs(x[i]-x[i+1]):T[a]:=T[a]+1:od:s:=sum('T[i]*10^(10-i-1)','i'=0..9): for u from 9 by -1 to 1 do:if T[0]<>0 and irem(s,10^u)=0 and s/10^u = n then print(n):else fi:od:od:
A358538
Autobiographical numbers: the first digit of the term counts how many 0's the term in total has, the second how many 1's etc. up to the last digit but no more than b-1, where b is the base of the sequence. This sequence is in base b=10.
Original entry on oeis.org
1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 52200100019, 52200100108, 52200101007, 52200110006, 52201100004, 52210100003, 53010100019, 53010100108, 53010101007, 53010110006, 53011100004, 53110100002, 61200020006, 62200010001, 63010010001, 70200002007, 72100001000, 431110000299
Offset: 1
63010010001 is a term: we have six 0's, three 1's, one 3 and one 6 as digits in the term, visualized as follows:
Digits: 0123456789
term: 63010010001.
Note that this example also shows, starting from the 11th digit, there is no more representation of the frequency of that digit, because only the first b digits of its base-b expansion count the occurrences of the corresponding digit. In this case, the last digit, 1, is the 11th.
A358711
Autobiographical numbers: let the k-th digit count the k-th nonnegative integer (A001477(k)) (possibly overlapping) occurrences in the term.
Original entry on oeis.org
1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 53110100002, 62200010001, 541011000021, 6401101000310, 74011001003100, 840110001031000, 9321000001201000, 94201000012110000
Offset: 1
6401101000310 is a term: we have six 0's, four 1's, zero 2's, one 3, one 4, one 6, three 10's and one 11 as integers in the term, visualized as follows:
Digits(k): 0 1 2 3 4 5 6 7 8 9 10 11 12 (also the Integers(k))
term: 6 4 0 1 1 0 1 0 0 0 3 1 0
Note that overlapping integers are counted as well: e.g., 110 is one 11, one 10. 111 is two 11's.
A260387
Numbers n = d_0d_1...d_n (n < 10) such that d_i is the number of digits equal to i in n (base b), where b is less than 10.
Original entry on oeis.org
12, 13, 320, 3201, 72200, 89000, 132110, 345000, 643000, 2320200, 3121300, 10103111, 11300130, 42430000, 51340000, 64030000, 72300000, 86300000, 125102000, 130213000, 211220001, 220101111, 323111000, 431130000, 614110000, 667000000, 2153100000, 2521002000, 3021211100
Offset: 1
12 = 110_3, which has 1 zero and 2 ones.
13 = 1101_2, which has 1 zero and 3 ones.
320 = 11000_4, which has 3 zeros, 2 ones and 0 twos.
3201 = 100301_5, which has 3 zeros, 2 ones, 0 twos and 1 three.
72200 = 10200001002_3
89000 = 10101101110101000_2
132110 = 13211420_5
345000 = 122112020210_3
643000 1012200000211_3
42430000 = 2201312320300_4
51340000 = 3003312023200_4
64030000 = 3310100110300_4
72300000 = 122002100000_5
86300000 = 20000101111100022_3
431130000 = 110440340120_6
614110000 = 2224203010000_5
667000000 = 1201111002002222201_3
2153100000 = 104233022322_7
a(10)-a(13), a(19)-a(23), a(28)-a(29) added by
Giovanni Resta, Jul 26 2015
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