A075349 -id:A075349 - cp's OEIS frontend

A121997 Count up to n, n times.

1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7

Offset: 1

Franklin T. Adams-Watters, Sep 11 2006

The n-th block consists of n subblocks, each of which counts from 1 to n.
This a fractal sequence: removing the first instance of each value leaves the original sequence.
The first comment implies that this gives the column index of the n-th element of a sequence whose terms are coefficients, read by rows, of a sequence of matrices of size 1 X 1, 2 X 2, 3 X 3, etc.; cf. example. The row index is given by A238013(n), and the size of the matrix by A074279(n). - M. F. Hasler, Feb 16 2014

			Sequence begins:
  1;
  1,2;
  1,2;
  1,2,3;
  1,2,3;
  1,2,3;
  ...
The blocks of n subblocks of n terms (n=1,2,3,...) can be cast into a square matrices of order n; then the terms are equal to the index of the column they fall into.

Cf. A081489 (locations of new values), A075349 (locations of 1's).
Cf. A000290 (row lengths), A002411 (row sums), A036740 (row products).
Cf. A002024 and references there, esp. in PROG section.
Cf. A238013.

A121997(N=9)=concat(vector(N,i,concat(vector(i,j,vector(i,k,k))))) \\ Note: this creates a vector; use A121997()[n] to get the n-th term. - M. F. Hasler, Feb 16 2014

from sympy import integer_nthroot
def A121997(n): return 1+(n-(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k-1)*((k<<1)-1)//6-1)%k # Chai Wah Wu, Nov 04 2024

A100143 Unique sequence with a(1)=1 where each a(n) occurs in the same order a(n) times consecutively in its sequence of first differences which contains no other terms.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 18, 22, 28, 34, 40, 46, 52, 58, 68, 78, 88, 98, 108, 118, 128, 138, 148, 158, 172, 186, 200, 214, 228, 242, 256, 270, 284, 298, 312, 326, 340, 354, 372, 390, 408, 426, 444, 462, 480, 498, 516, 534, 552, 570, 588, 606, 624, 642, 660, 678, 700

Offset: 1

Rick L. Shepherd, Nov 09 2004

nonn

By definition, for each n the sequence contains exactly one arithmetic progression (beginning at the A100250(n)-th term) of length a(n)+1 with common difference a(n). Contrast with A075349 and its sequence of first differences, A002024, which includes each A075349(n) term A075349(n) times but also includes terms that are not found in A075349.

Cf. A100144 (first differences), A001462 (Golomb's sequence), A075349, A002024.

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

A121997 Count up to n, n times.

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A100143 Unique sequence with a(1)=1 where each a(n) occurs in the same order a(n) times consecutively in its sequence of first differences which contains no other terms.

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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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