A080883 -id:A080883 - cp's OEIS frontend

A080439 a(1) = 11, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).

11, 101, 10061, 100000651, 10000000000060571, 100000000000000000000000600052761, 10000000000000000000000000000000000000000000000060000000502271641

Offset: 1

Amarnath Murthy, Feb 22 2003

Conjecture: Only one digit needs to be inserted between each pair of digits of a(n-1) to get a(n); i.e., a(n) contains exactly 2n-1 digits for n > 1.

The conjecture above is false: a(5)=10000000000060571 has 17 digits instead of 2*5-1=9. A refined conjecture is: a(n) contains exactly 2^(n-1) + 1 digits for all n>0. This follows trivially by induction from the initial conjecture (above) of only one digit needed between each pair, and the fact that we start with 11, a 2-digit number, and holds true at least till a(12). - Julio Cesar Hernandez-Castro, Jul 05 2011

			a(2) = 101 and a(3) is obtained by inserting a '0' and a '6' in the two inner spaces of 101: (1,-,0,-,1).

Julio Cesar Hernandez-Castro, Table of n, a(n) for n = 1..12

Cf. A080440, A080441, A080442, A080883 - A080914.

a[n_] := Block[{d = IntegerDigits[n]}, k = Length[d]; While[k > 1, d = Insert[d, 0, k]; k-- ]; d = FromDigits[d]; e = d; k = 0; While[ !PrimeQ[e], k++; e = d + 10FromDigits[ IntegerDigits[k], 100]]; e]; NestList[a, 11, 6]

Edited, corrected and extended by Robert G. Wilson v, Feb 22 2003

A080440 a(1) = 13, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).

Original entry on oeis.org

13, 103, 10093, 100000963, 10000000000092653, 100000000000000000000000902060523, 10000000000000000000000000000000000000000000000090002000600051233

Offset: 1

Amarnath Murthy, Feb 22 2003

Conjecture: Only one digit needs to be inserted between each pair of digits of a(n-1) to get a(n); i.e., a(n) contains exactly 2n-1 digits for n > 1.

The conjecture above is false: a(4) = 100000963 has 9 digits instead of 2*4 - 1 = 7. A refined conjecture is: a(n) contains exactly 2^(n-1) + 1 digits for all n > 0. This follows trivially by induction from the initial above conjecture of only one digit needed between each pair, and the fact that we start with 13, a 2-digit number, and holds true at least till a(12). - Julio Cesar Hernandez-Castro, Jul 06 2011

Julio Cesar Hernandez-Castro, Table of n, a(n) for n = 1..12

Cf. A080439, A080441, A080442, A080883 - A080914.

a[n_] := Block[{d = IntegerDigits[n]}, k = Length[d]; While[k > 1, d = Insert[d, 0, k]; k-- ]; d = FromDigits[d]; e = d; k = 0; While[ !PrimeQ[e], k++; e = d + 10FromDigits[ IntegerDigits[k], 100]]; e]; NestList[a, 13, 6]

Edited, corrected and extended by Robert G. Wilson v, Feb 22 2003

A080441 a(1) = 17, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).

Original entry on oeis.org

17, 107, 10007, 100000007, 10000000000003037, 100000000000000000000000003000307, 10000000000000000000000000000000000000000000000000003000000030057

Offset: 1

Amarnath Murthy, Feb 22 2003

Conjecture: Only one digit needs to be inserted between each pair of digits of a(n-1) to get a(n); i.e. a(n) contains exactly 2n-1 digits for n > 1.

The conjecture above is false: a(5) = 10000000000003037 has 17 digits instead of 2*5 - 1 = 9. A refined conjecture is: a(n) contains exactly 2^(n-1) + 1 digits for all n > 0. This follows trivially by induction from the initial conjecture above of only one digit needed between each pair, and the fact that we start with 17, a 2-digit number, and holds true at least till a(12).

Julio Cesar Hernandez-Castro, Table of n, a(n) for n = 1..12

Cf. A080439, A080440, A080442, A080883 - A080914.

a[n_] := Block[{d = IntegerDigits[n]}, k = Length[d]; While[k > 1, d = Insert[d, 0, k]; k-- ]; d = FromDigits[d]; e = d; k = 0; While[ !PrimeQ[e], k++; e = d + 10FromDigits[ IntegerDigits[k], 100]]; e]; NestList[a, 17, 6]

Edited and extended by Robert G. Wilson v, Feb 22 2003

A080442 a(1) = 19, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).

Original entry on oeis.org

19, 109, 10009, 100000039, 10000000000002359, 100000000000000000000000002031519, 10000000000000000000000000000000000000000000000000002000301050179

Offset: 1

Amarnath Murthy, Feb 22 2003

Conjecture: Only one digit needs to be inserted between each pair of digit of a(n-1) to get a(n); i.e., a(n) contains exactly 2n-1 digits for n > 1.

The conjecture above is false: a(5) = 10000000000002359 has 17 digits instead of 2*5 - 1 = 9. A refined conjecture is: a(n) contains exactly 2^(n-1) + 1 digits for all n > 0. This follows trivially from the initial conjecture of only one digit needed between each pair, and the fact that we start with 19, a 2-digit number, and holds true at least till a(12). - Julio Cesar Hernandez-Castro, Jul 07 2011

Julio Cesar Hernandez-Castro, Table of n, a(n) for n = 1..12

Cf. A080439, A080440, A080441, A080883 - A080914.

a[n_] := Block[{d = IntegerDigits[n]}, k = Length[d]; While[k > 1, d = Insert[d, 0, k]; k-- ]; d = FromDigits[d]; e = d; k = 0; While[ !PrimeQ[e], k++; e = d + 10FromDigits[ IntegerDigits[k], 100]]; e]; NestList[a, 19, 6]

Edited and extended by Robert G. Wilson v, Feb 22 2003

A256243 Smallest positive integer m such that n + 2m is a square.

Original entry on oeis.org

4, 1, 3, 6, 2, 5, 1, 4, 8, 3, 7, 2, 6, 1, 5, 10, 4, 9, 3, 8, 2, 7, 1, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 14, 6, 13, 5, 12, 4, 11, 3, 10, 2, 9, 1, 8, 16, 7, 15, 6, 14, 5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 18, 8, 17, 7, 16, 6, 15, 5, 14, 4, 13, 3, 12, 2, 11, 1, 10, 20, 9, 19, 8, 18, 7, 17, 6, 16, 5, 15, 4, 14, 3, 13, 2, 12, 1, 11, 22

Offset: 1

Zak Seidov, Mar 20 2015

			1 + 2*4 = 9 = 3^2 so a(1) = 4.
2 + 2*1 = 4 = 2^2, so a(2) = 1.
3 + 2*3 = 9 = 3^2, so a(3) = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A256244, A080883.

Table[m = 1; While[! IntegerQ[Sqrt[n + 2*m]], m++]; m, {n, 100}] (* Michael De Vlieger, Mar 20 2015 *)

a(n)=m=1;while(!issquare(n+2*m),m++);m
vector(100,n,a(n)) \\ Derek Orr, Mar 22 2015

a(n)=my(s=sqrtint(n)); (if((s-n)%2,s+1,s+2)^2-n)/2 \\ Charles R Greathouse IV, Mar 23 2015

1 <= a(n) <= 2*sqrt(n) + 2. - Charles R Greathouse IV, Mar 23 2015

a(n) = (1/4)*(6*floor(sqrt(n)) + 2*floor(sqrt(n))^2 + (2*floor(sqrt(n)) + 3)*(-1)^(n - floor(sqrt(n))) - 2*n + 5). - Ridouane Oudra, Oct 09 2020

A373016 a(n) is the least positive integer k such that 3n^2 + 2n + k is a square.

Original entry on oeis.org

4, 9, 3, 8, 15, 1, 8, 17, 28, 4, 15, 28, 43, 9, 24, 41, 60, 16, 35, 56, 4, 25, 48, 73, 11, 36, 63, 92, 20, 49, 80, 113, 31, 64, 99, 9, 44, 81, 120, 20, 59, 100, 143, 33, 76, 121, 3, 48, 95, 144, 16, 65, 116, 169, 31, 84, 139, 196, 48, 105, 164, 8, 67, 128, 191, 25, 88, 153, 220, 44, 111, 180

Offset: 1

Claude H. R. Dequatre, May 20 2024

The scatterplot shows an interesting crosshatch structure where all terms are at the intersection of ascending and descending hatches.
Terms on each hatch are quite well fitted by a polynomial of degree 2.
For terms on ascending hatches, the parity of the term indices does not change on a given hatch but alternates from one hatch to the next and on the same hatch, the parity of two consecutive terms alternates.
For terms on descending hatches, the parity of the indices of two consecutive terms alternates on the same hatch and that of terms does not change on the same hatch but alternates from one hatch to the next.
All squares exclusively are in ascending order on the same ascending hatch at n = 6, 10, 14, 18, 22, ... but some squares can be also found at the intersection of other hatches.
The first differences of the indices of the terms located on ascending and descending hatches are respectively equal to 4 and 3. For terms that are on the ascending and descending hatches, the differences of order 2 quickly become constant and equal to 2 and 4, respectively.
The fixed points begin 3, 48, 675, 9408, etc. They are all divisible by 3 and their parity seems to alternate. It appears that they are the positive terms of A007654.

			a(1) = 4 because 3*1^2 + 2*1 = 5 and 5 + 1, 5 + 2, 5 + 3 are not squares, but 5 + 4 is. So, 4 is a term.
a(2) = 9 because 3*2^2 + 2*2 = 16 and 16 + 1, 16 + 2, 16 + 3, 16 + 4, 16 + 5, 16 + 6, 16 + 7, 16 + 8 are not squares, but 16 + 9 is. So, 9 is a term.

Cf. A000005, A000290, A005408.
Cf. A045944, A080883.
Cf. A373026, A373028.
Sequences with similar scatterplot and pin plot graphs: A141130, A141131, A141134, A141135.

a(n)=my(t=3*n^2+2*n); (sqrtint(t)+1)^2-t \\ Charles R Greathouse IV, May 21 2024

a(n) is the smallest square greater than 3*n^2 + 2*n, minus 3*n^2 + 2*n. - Charles R Greathouse IV, May 21 2024
1 <= a(n) <= floor(sqrt(12)*n) + 3. I believe both bounds are tight infinitely often. - Charles R Greathouse IV, May 21 2024
a(n) = A080883(A045944(n)). - Michel Marcus, May 22 2024

A121490 Rectangular array T by antidiagonals: T(n,k) = [y+1]^2-y^2, where y=n*sqrt(k) and [ ] denotes the floor function.

Original entry on oeis.org

3, 2, 5, 1, 1, 7, 5, 4, 7, 9, 4, 9, 9, 4, 11, 3, 5, 13, 1, 14, 13, 2, 1, 4, 17, 6, 9, 15, 1, 8, 10, 1, 21, 13, 2, 17, 7, 4, 1, 4, 19, 25, 22, 16, 19, 6, 13, 9, 9, 19, 16, 29, 4, 7, 21, 5, 9, 19, 16, 21, 9, 11, 33, 13, 25, 23, 4, 5, 10, 25, 25, 4, 30, 4, 37, 24, 14, 25, 3, 1, 1, 9, 31, 1, 18, 16

Offset: 1

Clark Kimberling, Aug 02 2006

(Row 1)=A080883; (Column 1)=A005408 except for initial term; (Column 2)=A087059; (Column 3)=A120865; (Column 4)=A016813 except for initial term; (Column 5)=A120867; Columns 1,4,9,16,25,... are arithmetic sequences; (Main diagonal)=A121489.

			Northwest corner:
3 2 1 5 4
5 1 4 9 5
7 7 9 13 4
9 4 1 17 1

T(n,k) = [y+1]^2-y^2

A333884 Difference between smallest cube > n and n.

Original entry on oeis.org

1, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45

Offset: 0

Ilya Gutkovskiy, Apr 08 2020

nonn

a(n) is the smallest positive number k such that n + k is a cube.

Michael De Vlieger, Table of n, a(n) for n = 0..9999

Cf. A000578, A048763, A055400, A068527, A080883, A154840.

Table[Floor[n^(1/3) + 1]^3 - n, {n, 0, 80}]

a(n) = floor(n^(1/3) + 1)^3 - n.

A106033 a(n) is the least number k such that n*prime(n)+k is a perfect square.

Original entry on oeis.org

2, 3, 1, 8, 9, 3, 2, 17, 18, 34, 20, 40, 43, 23, 24, 52, 21, 58, 23, 24, 67, 26, 27, 73, 75, 78, 28, 29, 88, 91, 32, 33, 103, 35, 114, 40, 120, 47, 48, 136, 57, 142, 68, 157, 160, 62, 83, 112, 113, 214, 217, 116, 223, 135, 26, 156, 43, 158, 41, 40, 161, 59, 259, 260, 104, 103

Offset: 1

Zak Seidov, May 05 2005

			a(10)=34 because 10*prime(10)+34 = 10*29+34 = 324 = 18^2.

Cf. A033286 (n*prime(n)), A080883 (distance of n to next square).

a[n_]:=(Floor[Sqrt[n*Prime[n]]]+1)^2-n*Prime[n]
lnk[n_]:=With[{c=n Prime[n]},(Floor[Sqrt[c]]+1)^2-c]; Array[lnk,70] (* Harvey P. Dale, Feb 17 2024 *)

a(n) = (floor(sqrt(n*prime(n)))+1)^2 - n*prime(n).

a(n) = A080883(A033286(n)). - Michel Marcus, Mar 29 2020

cp's OEIS Frontend

A080439 a(1) = 11, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).

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A080440 a(1) = 13, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).

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A080441 a(1) = 17, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).

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A080442 a(1) = 19, a(n) is the smallest prime obtained by inserting digits between every pair of digits of a(n-1).

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A256243 Smallest positive integer m such that n + 2m is a square.

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A373016 a(n) is the least positive integer k such that 3*n^2 + 2*n + k is a square.

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A121490 Rectangular array T by antidiagonals: T(n,k) = [y+1]^2-y^2, where y=n*sqrt(k) and [ ] denotes the floor function.

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A333884 Difference between smallest cube > n and n.

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A373016 a(n) is the least positive integer k such that 3n^2 + 2n + k is a square.