A060432 -id:A060432 - cp's OEIS frontend

A361172 a(n) is the smallest positive number not among the terms between a(n-1) and the previous most recent occurrence of a(n-1) inclusive; if a(n-1) is a first occurrence, set a(n)=1; a(1)=1.

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

Offset: 1

Neal Gersh Tolunsky, Mar 02 2023

nonn

The terms between two adjacent 1s must be strictly increasing.

The index of first occurrences appears to be A060432 (partial sums of A002024).

			a(11)=5 because between a(10)=3 and the previous 3 (3, 1, 2, 4, 1, 3), the smallest missing number is 5, so a(11)=5.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Neal Gersh Tolunsky, Scatterplot of the first 13158 terms

Cf. A060432, A358921, A361101.

{ pos = [0]; v = 1; for (n = 1, #a = vector(86), print1 (a[n] = v", "); v = 1; if (a[n] <= #pos && pos[a[n]], r = Set(a[pos[a[n]]..n]); while (setsearch(r, v), v++)); while (#pos < a[n], pos = concat(pos, vector(#pos));); pos[a[n]] = n;); } \\ Rémy Sigrist, Mar 04 2023

A214651 Count down from n to 1, n times.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5

Offset: 1

Arkadiusz Wesolowski, Jul 24 2012

This sequence contains every positive integer infinitely often.

This is a fractal sequence. Striking out the first instance of every term produces 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, ..., which is the same as the original sequence, as far as it goes.

			1;
2, 1, 2, 1;
3, 2, 1, 3, 2, 1, 3, 2, 1;
...

Arkadiusz Wesolowski, Rows n = 1..14 of triangle, flattened
Eric Weisstein's World of Mathematics, Positive Integer
Wikipedia, Fractal sequence

Cf. A056520 (locations of new values), A060432 (locations of 1's).

Cf. A000290 (row lengths), A002411 (row sums), A036740 (row products).

f[n_] := Table[Range[n, 1, -1], {n}]; Flatten@Array[f, 6] (* Wesolowski *)
Flatten[Table[Table[Range[n, 1, -1], {n}], {n, 6}]] (* Alonso del Arte, Jul 24 2012 *)

A329598 Partial sums of the nontriangular numbers (A014132).

Original entry on oeis.org

2, 6, 11, 18, 26, 35, 46, 58, 71, 85, 101, 118, 136, 155, 175, 197, 220, 244, 269, 295, 322, 351, 381, 412, 444, 477, 511, 546, 583, 621, 660, 700, 741, 783, 826, 870, 916, 963, 1011, 1060, 1110, 1161, 1213, 1266, 1320, 1376, 1433, 1491, 1550, 1610, 1671, 1733

Offset: 1

Metin Sariyar and Robert G. Wilson v, Nov 17 2019

nonn

Terms which are triangular: 6, 136, 351, 741, 2415, 3916, 5995, 12561, 17391, 23436, ..., .

			The nontriangular numbers begin 2, 4, 5, 7, ..., so their partial sums begin 2, 6, 11, 18, etc.

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.

Cf. A329472, A014132, A086849.

Cf. A000217, A060432.

triQ[n_] := IntegerQ @ Sqrt[8n + 1]; Accumulate@ Select[ Range@ 70, !triQ@# &]

from math import isqrt
def A329598(n): return (k:=(r:=isqrt(m:=n+1<<1))+int((m<<2)>(r<<2)*(r+1)+1)-1)*(k*(-k - 3) + 6*n - 2)//6 + (n*(n+3)>>1) # Chai Wah Wu, Jun 18 2024

a(n) = Sum_{i=1..n} A014132(i).

a(n) = A000217(n) + A060432(n). [corrected by Gerald Hillier, Jul 31 2022]

A256619 Numbers n such that there are no primes in the interval [b(n), b(n+1) - 1], where b(n) = 1 + Sum_{k=1..n} floor(1/2 + sqrt(2*k - 2)).

Original entry on oeis.org

1, 26, 29, 38, 47, 97, 114, 127, 216, 276, 433, 1094, 1284

Offset: 1

Arkadiusz Wesolowski, Apr 05 2015

nonn

Numbers n such that there are no primes in the interval [A075349(n), A060432(n)].

Conjecture: the sequence is finite and complete.

			1st row:  {1}              - no prime!
2nd row:  {2, 3}           - two primes (2 and 3).
3rd row:  {4, 5}           - one prime (5).
4th row:  {6, 7, 8}        - one prime (7).
5th row:  {9, 10, 11}      - one prime (11).
6th row:  {12, 13, 14}     - one prime (13).
7th row:  {15, 16, 17, 18} - one prime (17).
8th row:  {19, 20, 21, 22} - one prime (19).
9th row:  {23, 24, 25, 26} - one prime (23).
10th row: {27, 28, 29, 30} - one prime (29).
...
26th row: {120, 121, 122, 123, 124, 125, 126} - no primes!
...
29th row: {141, 142, 143, 144, 145, 146, 147, 148} - no primes!
...

lst:=[]; k:=1284; b:=1; e:=0; for n in [1..k] do b:=b+Floor(1/2+Sqrt(2*n-2)); e:=e+Floor(1/2+Sqrt(2*n)); if IsZero(#[m: m in [b..e] | IsPrime(m)]) then Append(~lst, n); end if; end for; lst;

cp's OEIS Frontend

A361172 a(n) is the smallest positive number not among the terms between a(n-1) and the previous most recent occurrence of a(n-1) inclusive; if a(n-1) is a first occurrence, set a(n)=1; a(1)=1.

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A214651 Count down from n to 1, n times.

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Mathematica

A329598 Partial sums of the nontriangular numbers (A014132).

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A256619 Numbers n such that there are no primes in the interval [b(n), b(n+1) - 1], where b(n) = 1 + Sum_{k=1..n} floor(1/2 + sqrt(2*k - 2)).

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