A051022 -id:A051022 - cp's OEIS frontend

A092908 Primes in A051022.

2, 3, 5, 7, 101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 10007, 10009, 10103, 10301, 10303, 10501, 10601, 10607, 10709, 10903, 10909, 20101, 20107, 20201, 20407, 20507, 20509, 20707, 20807, 20809, 20903

Offset: 1

Jorge Coveiro, Apr 15 2004

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Select[Table[FromDigits[Riffle[IntegerDigits[n],0]],{n,300}],PrimeQ] (* Harvey P. Dale, Nov 17 2013 *)

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004

A092907 Duplicate of A051022.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, 201

Offset: 0

dead

A039723 Numbers in base -10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 150, 151, 152, 153, 154, 155

Offset: 0

Robert Lozyniak (11(AT)onna.com)

			Decimal 25 is "185" in base -10 because 100 - 80 + 5 = 25.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest
Eric Weisstein's World of Mathematics, Negadecimal
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Nonnegative numbers in negative bases: this sequence (b=-10), A039724 (b=-2), A073785 (b=-3), A007608 (b=-4), A073786 (b=-5), A073787 (b=-6), A073788 (b=-7), A073789 (b=-8), A073790 (b=-9).
Cf. A051022.
Cf. A305238: negative numbers in base -10.

a039723 0 = 0
a039723 n = a039723 n' * 10 + m where
   (n',m) = if r < 0 then (q + 1, r + 10) else qr where
            qr@(q, r) = quotRem n (negate 10)
-- Reinhard Zumkeller, Apr 20 2011

ToNegaBases[i_Integer, b_Integer] := FromDigits@ Rest@ Reverse@ Mod[ NestWhileList[(# - Mod[ #, b])/-b &, i, # != 0 &], b]

A039723 = base(n, b=-10) = if(n, base(n\b, b)*10 + n%b, 0) \\ M. F. Hasler, Oct 16 2018 [Corrected by Jianing Song, Oct 21 2018]

def A039723(n):
    s, q = '', n
    while q >= 10 or q < 0:
        q, r = divmod(q, -10)
        if r < 0:
            q += 1
            r += 10
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 10 2016

A338754 Duplicate each decimal digit of n, so 0 -> 00, ..., 9 -> 99.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 2200, 2211, 2222, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 3300, 3311, 3322, 3333, 3344, 3355, 3366, 3377, 3388, 3399, 4400, 4411, 4422, 4433, 4444, 4455, 4466

Offset: 0

Kevin Ryde, Nov 06 2020

This is equivalent to changing decimal digits 0,1,..,9 to base 100 digits 0,11,..,99, so the sequence is numbers which can be written in base 100 using only digits 0,11,..,99. Also, numbers whose decimal digit runs are all even lengths (including 0 as no digits at all).

This sequence first differs from A044836 (apart from term 0) at a(100) = 110000 whereas A044836(100) = 10011, because A044836 allows odd length digit runs provided there are more even than odd.

			For n=5517, digits duplicate to a(n) = 55551177.

Kevin Ryde, Table of n, a(n) for n = 0..10000

Cf. A051022 (0 above each digit), A044836.

Other bases: A001196, A338086.

PARI
```
a(n) = fromdigits(digits(n),100)*11;
```

def A338754(n): return int(''.join(d*2 for d in str(n))) # Chai Wah Wu, May 07 2022

a(n) = Sum_{i=0..k} 11*d[i]*100^i where the decimal expansion of n is n = Sum_{i=0..k} d[i]*10^i with digits 0 <= d[i] <= 9.

a(n) = A051022(n)*11 for n > 0. - Kritsada Moomuang, Oct 20 2019

A252480 Numbers whose decimal representation has at least one '0' digit in a position other than the final digit.

Original entry on oeis.org

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 600, 601, 602, 603, 604, 605, 606

Offset: 1

M. F. Hasler, Dec 28 2014

Similar but different sequences are the "Cyclops numbers" A134808 and A032945 and A051022, which are subsequences, except for the 1- and 2-digit terms.
Also, numbers whose decimal representation cannot be split up between any two digits without producing a string with a leading zero (other than "0" itself).
Also, numbers n > 9 such that floor(n/10) is in A011540, i.e., has a digit '0'.

M. F. Hasler, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Select[Range[10,700],DigitCount[Floor[#/10],10,0]>0&] (* Harvey P. Dale, May 10 2020 *)

PARI
```
is(n)=n>9 && !vecmin(digits(n\10))
```

A343550 Numbers k > 9 such that the number m formed by inserting a digit 0 between each pair of digits in k is divisible by k.

Original entry on oeis.org

10, 15, 18, 20, 30, 40, 45, 50, 60, 70, 80, 90, 100, 111, 120, 126, 150, 180, 200, 222, 240, 250, 285, 300, 333, 360, 400, 444, 450, 480, 500, 555, 600, 666, 700, 750, 777, 800, 888, 900, 999, 1000, 1041, 1110, 1185, 1200, 1260, 1395, 1443, 1500, 1554, 1665

Offset: 1

Lars Blomberg, Apr 19 2021

One-digit terms are not considered since no 0 digits can be inserted.
If k is a term then so is k*10^i, i > 0.
If k is a term then so is k*i, 2 <= i <= 9 as long as no carry occurs in the multiplication.
The number of terms with n digits is (12, 29, 51, 107, 149, 240, 308, 438, 566, 789, 1007), 2 <= n <= 12.

			18 is a term because 108/18=6, and so is 1185 because 1010805/1185=853.
10101/111=91, 1010100/1110=910, 101010000/11100=9100, ... so 111, 1110, 11100, ... are all terms.
1000401/1041=961 and 2000802/2082=961 so 1041 and 2082 are terms but 3123 is not since it does not divide 3010203.

Cf. A051022, A285176, A343551, A343552.

Cf. A062846 (binary), A062891 (ternary).

A092909 Interpolate 0's between each pair of digits of n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 101, 103, 107, 109, 203, 209, 301, 307, 401, 403, 407, 503, 509, 601, 607, 701, 703, 709, 803, 809, 907, 10001, 10003, 10007, 10009, 10103, 10207, 10301, 10307, 10309, 10409, 10501, 10507, 10603, 10607, 10703, 10709, 10801, 10901, 10903, 10907, 10909

Offset: 1

Jorge Coveiro, Apr 15 2004

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A051022, A092908.

Table[FromDigits[Riffle[IntegerDigits[p],0]],{p,Prime[Range[50]]}] (* Harvey P. Dale, Aug 15 2021 *)

a(n)={fromdigits(concat([[0,d] | d<-digits(prime(n))]))} \\ Andrew Howroyd, Feb 12 2020

a(n) = A051022(A000040(n)). - Andrew Howroyd, Feb 12 2020

Name clarified and terms a(31) and beyond from Andrew Howroyd, Feb 12 2020

A343551 Numbers m/k where the number m is formed by inserting a digit 0 between each pair of digits in k, and m is divisible by k.

Original entry on oeis.org

10, 7, 6, 10, 10, 10, 9, 10, 10, 10, 10, 10, 100, 91, 85, 81, 70, 60, 100, 91, 85, 82, 73, 100, 91, 85, 100, 91, 90, 85, 100, 91, 100, 91, 100, 94, 91, 100, 91, 100, 91, 1000, 961, 910, 853, 850, 810, 739, 721, 700, 676, 637, 600, 571, 546, 1000, 961, 91

Offset: 1

Lars Blomberg, Apr 19 2021

This sequence is parallel to A343550 and A343552, therefore some values are repeated.

			6 is a term because 108/18=6, and so is 853 because 1010805/1185=853.
10101/111=91, 1010100/1110=910, 101010000/11100=9100, etc. are all terms.
1000401/1041=961 and 2000802/2082=961 are terms but not 3123.

Cf. A051022, A285176, A343550, A343552.

A343552 Numbers m such that for some number k dividing n, m is formed by inserting a digit 0 between each pair of digits of k.

Original entry on oeis.org

100, 105, 108, 200, 300, 400, 405, 500, 600, 700, 800, 900, 10000, 10101, 10200, 10206, 10500, 10800, 20000, 20202, 20400, 20500, 20805, 30000, 30303, 30600, 40000, 40404, 40500, 40800, 50000, 50505, 60000, 60606, 70000, 70500, 70707, 80000, 80808, 90000

Offset: 1

Lars Blomberg, Apr 19 2021

If m is a term then so is m*10^(2i), i > 0.

If m is a term then so is m*i, 2 <= i <= 9 as long as no carry occurs in the multiplication.

			108 is a term because 108/18=6, and so is 1010805 because 1010805/1185=853.

Cf. A051022, A285176, A343550, A343551.

cp's OEIS Frontend

A092908 Primes in A051022.

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A092907 Duplicate of A051022.

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A039723 Numbers in base -10.

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A338754 Duplicate each decimal digit of n, so 0 -> 00, ..., 9 -> 99.

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A252480 Numbers whose decimal representation has at least one '0' digit in a position other than the final digit.

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A343550 Numbers k > 9 such that the number m formed by inserting a digit 0 between each pair of digits in k is divisible by k.

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A092909 Interpolate 0's between each pair of digits of n-th prime.

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A343551 Numbers m/k where the number m is formed by inserting a digit 0 between each pair of digits in k, and m is divisible by k.

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A343552 Numbers m such that for some number k dividing n, m is formed by inserting a digit 0 between each pair of digits of k.

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