A116700 -id:A116700 - cp's OEIS frontend

A132134 Base 3 "Punctual Bird" numbers: write the natural numbers, base 3, in a string 12101112202122100101102... Sequence gives numbers which do not occur in the string ahead of their natural place.

1, 2, 3, 4, 6, 9, 11, 15, 18, 27, 29, 30, 32, 33, 35, 42, 45, 54, 60, 81, 83, 86, 87, 89, 92, 95, 96, 98, 99, 101, 104, 105, 107, 123, 126, 135, 141, 153, 162, 243, 245, 248, 249, 251, 252, 254, 257, 258, 260, 261, 263, 266, 267, 269, 275, 276, 278, 285, 287, 288

Offset: 1

Graeme McRae, Aug 11 2007

			a(5)=6 because 6 (20, base 3) is the fifth number that appears first in its "natural" place in the string of concatenated base-3 numbers.

Cf. A033307, A117804, A116700, A131881, A132131, A132132, A132133, A132135.

A141817 "Early bird" squares: write the natural numbers in a string 12345678910111213.... Sequence gives squares which occur in the string ahead of their natural place.

Original entry on oeis.org

64, 81, 121, 324, 361, 441, 484, 529, 625, 676, 729, 784, 841, 961, 1521, 1681, 2116, 2401, 2601, 2916, 3025, 3136, 3249, 3364, 3481, 3721, 4225, 4356, 4624, 4761, 5041, 5184, 5329, 5476, 5625, 5929, 6084, 6241, 6400, 6561, 6724, 7056, 7225, 7396, 7569

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 08 2008

Subset of A116700

			"64" appears in ....44454"64"74849, ahead of its position after "63", so is a member.

Cf. A116700.

A181585 "Early bird" squares: write the square numbers in a string 149162536496481100... . Sequence gives numbers k such that k^2 occurs in the string ahead of its natural place.

Original entry on oeis.org

7, 8, 21, 25, 46, 97, 129, 161, 196, 221, 245, 258, 277, 296, 350, 436, 460, 592, 661, 694, 789, 804, 875, 877, 1250, 2025, 2221, 3500, 3959, 4020, 5461, 5920, 7925, 9607, 12500, 14772, 19821, 20010, 21825, 22011, 22221, 24012, 25225, 25375, 25388, 26013, 28014

Offset: 1

Zak Seidov, Oct 31 2010

Corresponding positions of the k^2's in a string 149162536496481100... are 2, 9, 23, 5, 112, 209, 395, 336, 496, 465, 656, 935, 65, 486, 603, 75, 112, 1115, 2317, 3163, 2329, 1987, 252, 421, 4036, 4279, 7092, ... .

Michael S. Branicky, Table of n, a(n) for n = 1..145

Cf. A000290, A116700, A141817.

s="1";ss={};Do[tsn=ToString[n^2];If[ !StringFreeQ[s,tsn],AppendTo[ss,n];Print[n]];s=s<>tsn,{n,2,99999}];

def aupto(limit):
    s, alst = "", []
    for k in range(1, limit+1):
        ss = str(k*k)
        if ss in s: alst.append(k)
        s += ss
    return alst
print(aupto(28028)) # Michael S. Branicky, Jul 08 2021

a(46) and beyond from Michael S. Branicky, Jul 08 2021

A187752 Number of times the binary representation of n occurs in the concatenation of the binary representation of all smaller numbers.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 2, 0, 1, 0, 3, 2, 3, 4, 3, 0, 1, 1, 2, 1, 2, 0, 6, 2, 3, 3, 5, 5, 4, 6, 4, 0, 1, 1, 2, 0, 3, 2, 3, 1, 3, 1, 4, 1, 3, 3, 8, 2, 3, 4, 4, 3, 5, 3, 8, 5, 5, 5, 6, 8, 5, 8, 5, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 1, 5, 3, 4, 1, 3, 2, 5, 2, 4, 2, 6, 1, 4, 3, 6, 2, 6, 4, 10, 2, 3, 4, 4, 3

Offset: 0

M. F. Hasler, Jan 03 2013

nonn

Related to "early bird" (decimal: A116700, binary: A161373) and Hannah Rollman's numbers (cf. A048991, A048992 for decimal; A118248 and A118247-A118251 for binary versions). The latter would correspond to a variant of this sequence which has indices of nonzero terms omitted from the concatenation.

			a(3) = 1 since concatenation of 0,1,2 in binary yields "0110", and 3 = "11"[2] occurs once in this string.

(nMax)->my(c=[],cnt(t,s,M)=M=2^#s-1;sum(i=0,#t-#s,vecextract(t,M<

A229123 a(n) gives the number of bases, b>1, in which n is an early bird.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 2, 3, 2, 4, 3, 6, 4, 5, 3, 7, 2, 7, 5, 7, 6, 7, 4, 9, 7, 6, 5, 8, 5, 10, 4, 8, 8, 7, 5, 13, 8, 8, 6, 12, 7, 12, 7, 8, 11, 11, 5, 13, 9, 12, 9, 11, 5, 13, 11, 13, 12, 12, 5, 17, 11, 11, 8, 13, 9, 14, 9, 12, 7, 14, 8, 18, 11, 9, 11, 13, 11

Offset: 1

Paul Tek, Sep 14 2013

A number n is called an early bird in base b, if its digits in base b appear in the concatenation of the digits in base b of the numbers from 1 to n-1.

			The number 1 is never an early bird, so a(1)=0.
The number 3 is an early bird only in base 2, so a(3)=1.
The number 7 is an early bird in bases 2, 3 and 5, so a(7)=3.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, C program for this sequence
Paul Tek, Illustration of the bases in which n is an early bird, where n ranges from 1 to 1000

Cf. A116700, A161373, A135549.

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A341537 a(n) is the number of digits from the end of the concatenation of all previous terms where n last appears. If n has not previously appeared then a(n) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 2, 14, 15, 16, 17, 18, 19, 20, 15, 22, 33, 24, 25, 26, 27, 28, 29, 30, 31, 17, 20, 54, 35, 36, 37, 38, 39, 40, 53, 35, 17, 44, 75, 46, 5, 48, 49, 50, 16, 52, 23, 13, 55, 95, 57, 58, 7, 60, 87, 69, 51, 64, 26, 66, 115, 68, 15, 70, 106, 79, 71, 61, 36

Offset: 0

Scott R. Shannon, Feb 14 2021

			a(9) = 9 as the concatenation of all previous terms is "012345678" which does not include 9, so a(9) = 9.
a(12) = 13 as the concatenation of all previous terms is "01234567891011" which includes "12" as a substring, starting 13 digits from the end of the concatenation.
a(13) = 2 as the concatenation of all previous terms is "0123456789101113" which includes "13" as a substring, starting 2 digits from the end of the concatenation.

Scott R. Shannon, Image of the terms for n=0..500000.

Cf. A333921, A337227, A007908, A116700, A131881.

A141818 "Early bird" prime numbers: write the natural numbers in a string 12345678910111213.... Sequence gives prime numbers which occur in the string ahead of their natural place.

Original entry on oeis.org

23, 31, 41, 43, 53, 61, 67, 71, 73, 83, 89, 97, 101, 131, 151, 181, 191, 211, 223, 241, 251, 263, 271, 281, 311, 313, 317, 331, 353, 401, 419, 421, 431, 433, 461, 463, 491, 503

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 08 2008

Subset of A116700

			"23" appears in 1"23"4567..., ahead of its position after "22", so is a member.

Cf. A116700.

A178889 Numbers n such that R(n) (n-reversed) appears to the left of n in the string 12345678910111213....

Original entry on oeis.org

19, 21, 29, 31, 32, 39, 41, 42, 43, 49, 51, 52, 53, 54, 59, 61, 62, 63, 64, 65, 69, 71, 72, 73, 74, 75, 76, 79, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 110, 111, 119, 120, 121, 130, 131, 140, 141, 150, 151, 160, 161, 170, 171, 180, 181

Offset: 1

Paolo P. Lava & Giorgio Balzarotti, Jun 21 2010

Consider the infinite string of digits A007908(row->infinity). Whenever the digit-reversed variant of n, R(n) = A004086(n), appears ahead of n in that string, adjoin n to the current sequence.

			19 is generated where 9 meets 10, 12345678(91)01112131415161718(19)20...
21 is generated by 1 and 2: (12)34567891011121314151617181920(21)22...
29 is generated where 19 joins 20: 1234567891011121314151617181(92)02122232425262728(29)30...

Cf. A116700.

Definition shortened, examples detailed, keyword:base added by R. J. Mathar, Jul 13 2010

cp's OEIS Frontend

A132134 Base 3 "Punctual Bird" numbers: write the natural numbers, base 3, in a string 12101112202122100101102... Sequence gives numbers which do not occur in the string ahead of their natural place.

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A141817 "Early bird" squares: write the natural numbers in a string 12345678910111213.... Sequence gives squares which occur in the string ahead of their natural place.

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A181585 "Early bird" squares: write the square numbers in a string 149162536496481100... . Sequence gives numbers k such that k^2 occurs in the string ahead of its natural place.

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A187752 Number of times the binary representation of n occurs in the concatenation of the binary representation of all smaller numbers.

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PARI

A229123 a(n) gives the number of bases, b>1, in which n is an early bird.

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C

A341537 a(n) is the number of digits from the end of the concatenation of all previous terms where n last appears. If n has not previously appeared then a(n) = n.

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A141818 "Early bird" prime numbers: write the natural numbers in a string 12345678910111213.... Sequence gives prime numbers which occur in the string ahead of their natural place.

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Examples

Crossrefs

A178889 Numbers n such that R(n) (n-reversed) appears to the left of n in the string 12345678910111213....

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Extensions