A001704 -id:A001704 - cp's OEIS frontend

A087737 Value of (n,n+1) concatenated in binary representation.

6, 11, 28, 37, 46, 55, 120, 137, 154, 171, 188, 205, 222, 239, 496, 529, 562, 595, 628, 661, 694, 727, 760, 793, 826, 859, 892, 925, 958, 991, 2016, 2081, 2146, 2211, 2276, 2341, 2406, 2471, 2536, 2601, 2666, 2731, 2796, 2861, 2926, 2991, 3056, 3121

Offset: 1

Reinhard Zumkeller, Oct 01 2003

a(n) = n * 2^A070939(n+1) + (n+1).

			n=12: 12->'1100', 12+1->'1101', a(12)='1100''1101'='11001101'->205.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A007088, A001704.

a:= n-> n*(2^(ilog2(n+1)+1)+1)+1:
seq(a(n), n=1..50);  # Alois P. Heinz, May 20 2017

Table[FromDigits[Join[IntegerDigits[n,2],IntegerDigits[n+1,2]],2],{n,50}] (* Harvey P. Dale, Jan 14 2013 *)
Table[n 2^IntegerLength[#, 2] + # &[n + 1], {n, 48}] (* Michael De Vlieger, May 20 2017 *)

a(n) = fromdigits(Vec(concat(binary(n), binary(n+1))), 2); \\ Michel Marcus, May 20 2017

A309809 a(n) is the concatenation of n and 2n+1.

Original entry on oeis.org

13, 25, 37, 49, 511, 613, 715, 817, 919, 1021, 1123, 1225, 1327, 1429, 1531, 1633, 1735, 1837, 1939, 2041, 2143, 2245, 2347, 2449, 2551, 2653, 2755, 2857, 2959, 3061, 3163, 3265, 3367, 3469, 3571, 3673, 3775, 3877, 3979, 4081, 4183, 4285, 4387, 4489, 4591, 4693, 4795, 4897

Offset: 1

Robert Israel, Aug 17 2019

			a(3)=37 is the concatenation of 3 and 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001704, A309808.

seq(n*10^(ceil(log[10](2*n+1)))+2*n+1, n=1..100);

Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[2n+1]]],{n,50}] (* Harvey P. Dale, Jan 30 2024 *)

a(n) = n*10^(ceiling(log_10(2*n+1))) + 2*n + 1.

A032607 Concatenation of n and n + 2 or {n,n+2}.

Original entry on oeis.org

13, 24, 35, 46, 57, 68, 79, 810, 911, 1012, 1113, 1214, 1315, 1416, 1517, 1618, 1719, 1820, 1921, 2022, 2123, 2224, 2325, 2426, 2527, 2628, 2729, 2830, 2931, 3032, 3133, 3234, 3335, 3436, 3537, 3638, 3739, 3840, 3941, 4042, 4143, 4244

Offset: 1

Patrick De Geest, May 15 1998

Cf. A001704, A020338.

Table[FromDigits[Flatten[IntegerDigits/@{n,n+2}]],{n,50}] (* Harvey P. Dale, Dec 18 2013 *)

A032608 Concatenation of n and n + 3.

Original entry on oeis.org

14, 25, 36, 47, 58, 69, 710, 811, 912, 1013, 1114, 1215, 1316, 1417, 1518, 1619, 1720, 1821, 1922, 2023, 2124, 2225, 2326, 2427, 2528, 2629, 2730, 2831, 2932, 3033, 3134, 3235, 3336, 3437, 3538, 3639, 3740, 3841, 3942, 4043, 4144, 4245

Offset: 1

Patrick De Geest, May 15 1998

Cf. A001704, A020338.

Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[n+3]]],{n,50}] (* Harvey P. Dale, Jul 11 2011 *)
#[[1]]*10^IntegerLength[#[[2]]]+#[[2]]&/@Table[{n,n+3},{n,50}] (* Harvey P. Dale, May 21 2018 *)

A032609 Concatenation of n and n + 4 or {n,n+4}.

Original entry on oeis.org

15, 26, 37, 48, 59, 610, 711, 812, 913, 1014, 1115, 1216, 1317, 1418, 1519, 1620, 1721, 1822, 1923, 2024, 2125, 2226, 2327, 2428, 2529, 2630, 2731, 2832, 2933, 3034, 3135, 3236, 3337, 3438, 3539, 3640, 3741, 3842, 3943, 4044, 4145, 4246

Offset: 1

Patrick De Geest, May 15 1998

Cf. A001704, A020338.

A032611 Concatenation of n and n + 6 or {n,n+6}.

Original entry on oeis.org

17, 28, 39, 410, 511, 612, 713, 814, 915, 1016, 1117, 1218, 1319, 1420, 1521, 1622, 1723, 1824, 1925, 2026, 2127, 2228, 2329, 2430, 2531, 2632, 2733, 2834, 2935, 3036, 3137, 3238, 3339, 3440, 3541, 3642, 3743, 3844, 3945, 4046, 4147, 4248

Offset: 1

Patrick De Geest, May 15 1998

Cf. A001704, A020338.

A032613 Concatenation of n and n + 8 or {n,n+8}.

Original entry on oeis.org

19, 210, 311, 412, 513, 614, 715, 816, 917, 1018, 1119, 1220, 1321, 1422, 1523, 1624, 1725, 1826, 1927, 2028, 2129, 2230, 2331, 2432, 2533, 2634, 2735, 2836, 2937, 3038, 3139, 3240, 3341, 3442, 3543, 3644, 3745, 3846, 3947, 4048, 4149

Offset: 1

Patrick De Geest, May 15 1998

Cf. A001704, A020338.

A032614 Concatenation of n and n + 9 or {n,n+9}.

Original entry on oeis.org

110, 211, 312, 413, 514, 615, 716, 817, 918, 1019, 1120, 1221, 1322, 1423, 1524, 1625, 1726, 1827, 1928, 2029, 2130, 2231, 2332, 2433, 2534, 2635, 2736, 2837, 2938, 3039, 3140, 3241, 3342, 3443, 3544, 3645, 3746, 3847, 3948, 4049, 4150, 4251, 4352, 4453, 4554, 4655

Offset: 1

Patrick De Geest, May 15 1998

Tanya Khovanova, Non Recursions

Cf. A001704, A020338.

a[n_]:=n*10^Floor[Log10[n+9]+1]+n+9; Array[a,46] (* Stefano Spezia, Sep 04 2023 *)

a(42)-a(46) from Stefano Spezia, Sep 04 2023

A253253 a(n) = smallest divisor of the concatenation of n and n+1 that did not occur earlier.

Original entry on oeis.org

1, 23, 2, 3, 4, 67, 6, 89, 5, 337, 8, 1213, 9, 283, 379, 7, 859, 17, 10, 43, 1061, 13, 14, 25, 421, 37, 11, 41, 293, 433, 12, 53, 1667, 15, 16, 3637, 21, 349, 20, 449, 19, 4243, 24, 35, 2273, 1549, 1187, 373, 18, 5051, 28, 51, 2677, 1091, 463, 5657, 2879, 27

Offset: 1

Eric Angelini and Reinhard Zumkeller, Jun 05 2015

Is this a permutation of the integers > 0?
Comment from N. J. A. Sloane, May 19 2017: (Start)
It should not be difficult to prove that every positive integer appears.
If not, let m be the smallest missing number. There is an n_0 such that for all n >= n_0, a(n) > m. The theorem will follow if we can find an N > n_0 such that
m divides the concatenation of N and N+1.
Let N have k digits and suppose that
10^(k-1) <= N <= 10^k - 2.
The concatenation of N and N+1 is N*(10^k+1)+1, so we want to find numbers k and N such that
N*(10^k+1) == -1 mod m.
Case (i). If gcd(m,10)=1, then by Euler's theorem, 10^phi(m) == 1 mod m, so we can take k to be a sufficiently large multiple of phi(m), and then take N to be a number of the form r*m-1 in the range 10^(k-1) <= N <= 10^k - 2.
Case (ii). If m = 2^r or 5^r, then for large k, 10^k+1 == 1 mod m, and we take N to be of the form m*s-1 in the range 10^(k-1) <= N <= 10^k - 2.
The other cases are left to the reader. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Éric Angelini, Divisors of the concatenation of n and n+1, SeqFan list, Jun 03 2015.

Cf. A027750, A001704, A256285.

import Data.List (insert); import Data.List.Ordered (minus)
a253253 n = a253253_list !! (n-1)
a253253_list = f a001704_list [] where
   f (x:xs) ds = y : f xs (insert y ds) where
                 y = head (a027750_row' x `minus` ds)

A085776 Numbers k such that k concatenated with k+1 is a semiprime.

Original entry on oeis.org

3, 10, 14, 17, 18, 20, 21, 26, 30, 32, 33, 38, 45, 46, 48, 53, 54, 57, 66, 70, 72, 77, 81, 84, 88, 98, 101, 105, 110, 116, 118, 122, 125, 128, 132, 140, 141, 142, 146, 152, 158, 162, 164, 170, 173, 176, 177, 178, 185, 190, 194, 198, 204, 206, 208, 210, 212, 218, 222

Offset: 1

Jason Earls, Jul 23 2003

			10 is a term because 1011 = 3*337.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001358, A001704.

Select[Range[300], PrimeOmega[FromDigits[IntegerString[#] <> IntegerString[#+1]]] == 2 &] (* Paolo Xausa, Apr 17 2024 *)

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A087737 Value of (n,n+1) concatenated in binary representation.

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A309809 a(n) is the concatenation of n and 2n+1.

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A032607 Concatenation of n and n + 2 or {n,n+2}.

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A032608 Concatenation of n and n + 3.

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A032609 Concatenation of n and n + 4 or {n,n+4}.

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A032611 Concatenation of n and n + 6 or {n,n+6}.

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A032613 Concatenation of n and n + 8 or {n,n+8}.

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A032614 Concatenation of n and n + 9 or {n,n+9}.

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A253253 a(n) = smallest divisor of the concatenation of n and n+1 that did not occur earlier.

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A085776 Numbers k such that k concatenated with k+1 is a semiprime.

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