A080914 -id:A080914 - cp's OEIS frontend

A080900 a(1)=1; for n>1, a(n)=a(n-1)-2 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

1, 6, 11, 16, 21, 19, 24, 29, 34, 39, 37, 42, 47, 52, 57, 55, 60, 65, 63, 68, 66, 71, 76, 74, 79, 84, 89, 94, 92, 97, 102, 107, 112, 110, 115, 120, 118, 123, 121, 126, 131, 129, 134, 139, 144, 149, 147, 152, 157, 162, 167, 165, 170, 175, 173, 178, 176

Offset: 1

N. J. A. Sloane and Benoit Cloitre, Apr 01 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000 (terms 1..1000 from Ivan Neretin)
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 2.

Cf. A080912, A080913, A080904, A080914, A080578, A080919, A080922, A080927.

Cf. A080901 (starting value = 2), A080905 (run lengths of first differences).

Fold[Append[#1, #1[[-1]] + If[MemberQ[#1, #2], -2, 5]] &, {1}, Range[2, 57]] (* Ivan Neretin, Mar 03 2016 *)

up_to = 1001;
A080900list(up_to_n) = { my(xs=Map(), v=vector(up_to_n)); mapput(xs,1,1); v[1] = 1; for(n=2,up_to_n, v[n] = v[n-1]+if(mapisdefined(xs,n), -2, +5); mapput(xs,v[n],n)); (v); };
v080900 = A080900list(up_to);
A080900(n) = v080900[n]; \\ Antti Karttunen, Jan 22 2020

Perhaps this is asymptotic to c_0*n*(1 + c_1/log n + ...), with c_0 near 2 ?

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

Original entry on oeis.org

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

A080919 Numbers that do not appear in A080900.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 38, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 56, 58, 59, 61, 62, 64, 67, 69, 70, 72, 73, 75, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 93, 95, 96, 98, 99, 100, 101, 103, 104

Offset: 1

N. J. A. Sloane, Apr 04 2003

nonn

Complement of A080914.

cp's OEIS Frontend

A080900 a(1)=1; for n>1, a(n)=a(n-1)-2 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

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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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A080919 Numbers that do not appear in A080900.

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