A007952 -id:A007952 - cp's OEIS frontend

A048859 A sieve: keep the first 2 numbers, delete the next 3 numbers; keep the next 3 numbers, delete the next 4 numbers; keep the next 4 numbers, delete the next 5 numbers; and so on. In other words, keep the next k numbers and delete the next k+1 numbers, for k = 2, 3, ...

1, 2, 6, 7, 8, 13, 14, 15, 16, 22, 23, 24, 25, 26, 33, 34, 35, 36, 37, 38, 46, 47, 48, 49, 50, 51, 52, 61, 62, 63, 64, 65, 66, 67, 68, 78, 79, 80, 81, 82, 83, 84, 85, 86, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com)

nonn

			List the natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
Keep the first two numbers 1, 2 and delete the next three numbers 3, 4, 5.
Keep the next three numbers 6, 7, 8 and delete the next four numbers 9, 10, 11, 12. And so on.

C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.
M. Le, On the Smarandache n-ary Sieve, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 146-147.
F. Smarandache, Properties of Numbers, 1972.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
Index entries for sequences generated by sieves

Cf. A007950, A007951, A007952.

Cf. A064801.

a048859 n = a048859_list !! (n-1)
a048859_list = f 2 [1..] where
   f k xs = us ++ f (k + 1) (drop (k + 1) vs)
            where (us, vs) = splitAt k xs
-- Reinhard Zumkeller, May 16 2014

ss[n_]:=Module[{c=n^2+4n+1},Range[c,c+n+1]]; Flatten[Array[ss,10,0]] (* Harvey P. Dale, Sep 10 2014 *)

Corrected and revised by the author, Mar 24 2004
More terms from Bernardo Boncompagni Jul 27 2004
Offset changed by Reinhard Zumkeller, May 16 2014

A073047 Least k such that x(k)=0 where x(1)=n and x(k)=k*floor(x(k-1)/k).

Original entry on oeis.org

2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16

Offset: 1

Benoit Cloitre, Aug 31 2002; revised May 03 2003

Length of n-th run of consecutive identical terms is given by A028913 - Ralf Stephan.

			If x(1)=4, x(2)= 2*floor(4/2)=4, x(3)=3*floor(4/3)=3; x(4)=4*floor(3/4)=0 hence a(4)=4.

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

Cf. A002491, A007952, A082527, A082528.

f:= proc(n,k) option remember;
  if n = 0 then return k-1 fi;
  procname(k*floor(n/k),k+1)
end proc:
map(f, [$1..100], 1); # Robert Israel, Jul 25 2019

a[n_] := Module[{x}, x[1] = n; x[k_] := x[k] = k Floor[x[k-1]/k]; For[k = 1, True, k++, If[x[k] == 0, Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Jun 07 2020 *)

a(n)=if(n<0,0,s=n; c=1; while(s-s%c>0,s=s-s%c; c++); c)

Presumably a(n) = sqrt(Pi*n)+O(1).

A108696 Generated by a sieve: see comments.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 19, 23, 31, 35, 43, 49, 59, 61, 79, 83, 103, 109, 119, 133, 151, 155, 175, 193, 211, 215, 241, 259, 275, 283, 323, 331, 361, 373, 403, 419, 443, 455, 499, 511, 541, 571, 613, 623, 649, 673, 719, 733, 781, 803, 841, 871, 919

Offset: 1

David Applegate, Oct 11 2007

nonn

Start with the natural numbers:
1 2 3 4 5 6 7 8 9 10 11 ...
Accept the 2nd number, 2 and erase every 2nd number after it, giving:
1 2 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 ...
Accept the 3rd number, 3 and erase every 3rd number after it, giving:
1 2 3 5 7 11 13 17 19 23 25 29 31 35 ...
Accept the 4th number, 5 and erase every 4th number after it, giving:
1 2 3 5 7 11 13 19 23 25 31 35 ...
Repeat!

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Equals A007952 + 2 or equally A002491(n) + 1.

a108696 n = a108696_list !! (n-1)
a108696_list = 1 : sieve' 2 [2..] where
   sieve' n (x:xs) = x : (sieve' (n+1) $ sieving xs) where
      sieving xs = (take (n-1) xs) ++ (sieving $ drop n xs)
-- Reinhard Zumkeller, Jul 04 2011

source /tclutils/utils.tcl
set l [range 0 10000]; set z z
for {set i 2} {$i*2 <= [llength $l]} {incr i} {
set k [expr {[llength $l]-1}]
set k [expr {$k - ($k % $i)}]
while {$k > $i} {
set l [lreplace $l $k $k]
incr k -$i
}
puts "after $i: length [llength $l], prefix [join [lrange $l 0 10] { }]"
}

A140060 Array of quotients.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 03 2008

1. k divides Q(n,k) for each k.

2. The numbers in row n are distinct if and only if n is a term of the Sieve of Tchoukaillon (or Mancala, or Kalahari), A007952.

			First 8 rows:
1
2 2
3 2
4 4 3
5 4 3
6 6 6 4
7 6 6 4
8 8 6 4
For row 5: Q(5,1)=5, Q(5,2)=2*[5/2]=4, Q(5,3)=3*[4/3]=3.

Cf. A140061.

For n>=1, for k=1,2,...,A082447(n), Q(n,1)=n, Q(n,k)=k*Floor(Q(n,k-1)/k).

A204539 a(n) is the number of integers N=4k whose "basin" sequence (cf. comment) ends in n^2.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 2, 4, 3, 5, 1, 9, 2, 10, 3, 5, 7, 9, 2, 10, 9, 9, 2, 13, 9, 8, 4, 20, 4, 15, 6, 15, 8, 12, 6, 22, 6, 15, 15, 21, 5, 13, 12, 23, 7, 24, 11, 19, 15, 24, 6, 30, 6, 26, 7, 27, 26, 13, 6, 33, 27, 30, 5, 13, 30, 30, 5, 37, 15, 26, 28, 32, 7, 17, 25, 54, 9, 30, 21, 41, 25

Offset: 2

Colm Fagan, Jan 16 2012

nonn

The "basin" (analogous to river basins, for reasons set out below) is the number of positive integers N=4k which end in the "sea" at n^2. The "sea" of N is found as follows:
Starting out with N, in step i=1,2,3,..., stop if you have reached N=(i+1)^2 (the "sea" of N), otherwise set N to the next higher, odd or even (according to the parity of i), multiple of i+2, and go to step i+1.
Partial sums of this sequence appear to be A104738 (with a shift in offset). This has been confirmed for at least the first 4000 terms, but it is not at all clear why this is the case. - Ray Chandler, Jan 20 2012
After the first term, this sequence agrees with A028914 except for offset. Therefore this sequence is related to A028913, A007952, A002491 and A108696 dealing with the sieve of Tchoukaillon (or Mancala, or Kalahari). - Ray Chandler, Jan 20 2012

			For integers N=4,8,12,16,... we have the following sequences:
  {4}
  {8, 9}   (8 -> the next higher odd multiple of 3, which is 9 -> STOP)
  {12, 15, 16}  (12 -> 3*5=15 -> 4*4=16 -> STOP)
  {16, 21, 24, 25}
  {20, 21, 24, 25}
  {24, 27, 32, 35, 36}
  {28, 33, 40, 45, 48, 49}
  {32, 33, 40, 45, 48, 49}
  {36, 39, 40, 45, 48, 49}
  ...
Thus there is 1 integer N=4k ending in the sea at 2^2, whence basin a(2)=1, and idem for 3 and 4.
The two integers 16 and 20 end at 5^2, so the basin of 5 is a(5)=2.
There is again a(6)=1 integer ending in 6^2, while the basin of 7 are the 3 integers 28, 32, and 36, which all merge into the "river" that enters the "sea" in 7^2=49.
Thus the first 6 terms in the sequence are 1, 1, 1, 2, 1, 3.
Take N=100 as an example: the next integer on the same line is the next higher odd multiple of 3, i.e., smallest 3*(2m+1) > 100, which is 105. The next number is the least even multiple of 4, 4*(2m) = 112, etc., leading to 115 = 5*(2m+1), followed by 120 = 6*(2m), 133 = 7*(2m+1), 144 = 8*2m (where we have a square, but not the square of 8), 153 =9*(2m+1), 160 = 10*2m, 165 = 11*(2m+1), 168 = 12*(2m) and finally 169 = 13*13.

Ray Chandler, Table of n, a(n) for n = 2..10001
Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, Journal of Integer Sequences, Vol. 24 (2021), Article 21.7.1.
Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, arXiv:2202.02381 [math.NT], 2022.

Cf. A002491, A007952, A028913, A104738, A108696, A204540.

Essentially the same as A028914.

cumul[n_Integer] := Module[{den1 = n, num = n^2, den2}, While[num > 4 && den1 != 2, num = num - 1; den1 = den1 - 1; den2 = Floor[num/den1]; If[Not[EvenQ[den1 + den2]], den2 = den2 - 1]; num = den1 den2]; Return[num/4]]; basin[2] := 1; basin[n_Integer] := cumul[n] - cumul[n - 1]; Table[basin[n], {n, 2, 75}] (* Alonso del Arte, Jan 19 2012 *)

bs(n,s,m=2)={while(n>m^2,n=(n\m+++2-bittest(n\m-m,0))*m; s & print1(n","));n}
n=4; for(c=2,50, for(k=1,9e9, bs(n+=4)==c^2 || print1(k",")||break)) \\ M. F. Hasler, Jan 20 2012

A140061 Triangle of quotients.

Original entry on oeis.org

1, 3, 2, 5, 4, 3, 9, 8, 6, 4, 11, 10, 9, 8, 5, 17, 16, 15, 12, 10, 6, 21, 20, 18, 16, 15, 12, 7, 29, 28, 27, 24, 20, 18, 14, 8, 33, 32, 30, 28, 25, 24, 21, 16, 9, 41, 40, 39, 36, 35, 30, 28, 24, 18, 10, 47, 46, 45, 44, 40, 36, 35, 32, 27, 20, 11, 57, 56, 54, 52, 50, 48, 42, 40, 36

Offset: 1

Clark Kimberling, May 03 2008

Column 1 is essentially A007952. - Clark Kimberling, Aug 27 2008

			First 6 rows:
1
3 2
5 4 3
9 8 6 4
11 10 9 8 5
17 16 15 12 10 6

Cf. A140060.

Cf. A007952.

Flatten@Table[Reverse@FoldList[#2*Floor[#1/#2+1]&,i,Reverse@Range[i-1]],{i,10}] (* Birkas Gyorgy, Feb 26 2011 *)

The triangular subarray of A140060 consisting of rows whose terms are distinct. If n is the first term in a row (i.e., if n is a term of A082447), then Q(n,1)=n, Q(n,k)=k*Floor(Q(n,k-1)/k).

A357431 Triangle read by rows where each term in row n is the next greater multiple of n..1.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 4, 6, 8, 9, 5, 8, 9, 10, 11, 6, 10, 12, 15, 16, 17, 7, 12, 15, 16, 18, 20, 21, 8, 14, 18, 20, 24, 27, 28, 29, 9, 16, 21, 24, 25, 28, 30, 32, 33, 10, 18, 24, 28, 30, 35, 36, 39, 40, 41, 11, 20, 27, 32, 35, 36, 40, 44, 45, 46, 47

Offset: 1

Tamas Sandor Nagy, Sep 28 2022

Row n has length n and columns are numbered k = 1..n.
Row n begins with n which is trivially divisible by n. This is followed by the least number greater than n that is divisible by n-1. Next comes the least number that is greater than this preceding one and is divisible by n-2. Then it continues the same way until the last one is reached, which is trivially divisible by 1.
The end-most terms of the rows are A007952.

			Triangle begins:
  n/k|  1   2   3   4   5   6   7
  --------------------------------
  1  |  1;
  2  |  2,  3;
  3  |  3,  4,  5;
  4  |  4,  6,  8,  9;
  5  |  5,  8,  9, 10, 11;
  6  |  6, 10, 12, 15, 16, 17;
  7  |  7, 12, 15, 16, 18, 20, 21;
  ...
For row n=6, the numbers of the chain, and below them their divisors are:
  6 10 12 15 16 17
  6  5  4  3  2  1

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..9870 (first 140 rows flattened)

Cf. A357417 (row sums), A357498, A007952 (right diagonal).

row[n_] := Module[{k = n, s = Table[0, n], r}, s[[1]] = n;Do[k++; k += If[(r = Mod[k, i]) == 0, 0, i - Mod[k, i]]; s[[n+1-i]] = k, {i, n - 1, 1, -1}]; s]; Array[row, 11] // Flatten (* Amiram Eldar, Sep 28 2022 *)

row(n) = my(v=vector(n)); v[1] = n; for (k=2, n, v[k] = v[k-1] + (n-k+1) - (v[k-1] % (n-k+1));); v; \\ Michel Marcus, Nov 16 2022

T(n,1) = n.
T(n,k) = T(n,k-1) + (n-k+1) - (T(n,k-1) mod (n-k+1)), for k >= 2.
T(n,n) = A007952(n).

A357498 Triangle read by rows where each term in row n is the next greater multiple of n..1 divided by n..1.

Original entry on oeis.org

1, 1, 3, 1, 2, 5, 1, 2, 4, 9, 1, 2, 3, 5, 11, 1, 2, 3, 5, 8, 17, 1, 2, 3, 4, 6, 10, 21, 1, 2, 3, 4, 6, 9, 14, 29, 1, 2, 3, 4, 5, 7, 10, 16, 33, 1, 2, 3, 4, 5, 7, 9, 13, 20, 41, 1, 2, 3, 4, 5, 6, 8, 11, 15, 23, 47, 1, 2, 3, 4, 5, 6, 8, 10, 13, 18, 28, 57

Offset: 1

Tamas Sandor Nagy, Oct 01 2022

This triangle is related to A357431. Terms there are divisible by n..1 and here that division is performed, leaving the respective multiple of each.
Row n has length n and columns are numbered k = 1..n corresponding to multiples n..1.
Row n begins with n/n = 1. The end-most terms of the rows are A007952.

			Triangle begins:
  n/k|  1   2   3   4   5   6   7
  --------------------------------
  1  |  1;
  2  |  1,  3;
  3  |  1,  2,  5;
  4  |  1,  2,  4,  9;
  5  |  1,  2,  3,  5, 11;
  6  |  1,  2,  3,  5,  8, 17;
  7  |  1,  2,  3,  4,  6, 10, 21;
  ...
For row n=6, we have:
  A357431 row  6 10 12 15 16 17
  divided by   6  5  4  3  2  1
  results in   1  2  3  5  8 17

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10011 (141 rows, flattened)

Cf. A358435 (row sums), A357431, A007952 (right diagonal).

row[n_] := Module[{k = n, s = Table[0, n], r}, s[[1]] = 1; Do[k++; k += If[(r = Mod[k, i]) == 0, 0, i - Mod[k, i]]; s[[n + 1 - i]] = k/i, {i, n - 1, 1, -1}]; s]; Array[row, 12] // Flatten (* Amiram Eldar, Oct 01 2022 *)

row(n) = my(v=vector(n)); v[1] = n; for (k=2, n, v[k] = v[k-1] + (n-k+1) - (v[k-1] % (n-k+1));); vector(n, k, v[k]/(n-k+1)); \\ Michel Marcus, Nov 16 2022

T(n,k) = A357431(n,k) / (n-k+1).
T(n,1) = 1.
T(n,k) = (T(n,k-1)*(n-k+2) + (n-k+1) - (T(n,k-1)*(n-k+2)) mod (n-k+1))/(n-k+1), for k >= 2.
T(n,n) = A007952(n).

A386520 Column sums of the triangle in A386755.

Original entry on oeis.org

1, 5, 13, 13, 31, 35, 57, 61, 85, 85, 111, 99, 235, 89, 353, 173, 171, 341, 343, 229, 489, 423, 415, 435, 661, 525, 535, 559, 1161, 427, 931, 653, 1201, 787, 941, 885, 1629, 537, 1443, 1839, 1723, 931, 1119, 1525, 2415, 741, 2257, 2327, 1947, 2005, 2767, 1131, 3181, 1055, 3131, 2147

Offset: 1

Tamas Sandor Nagy, Jul 24 2025

nonn

It appears that A007952(n) is the index of the row where n first appears.

It appears that A007952(n)-1 is the index of the row where the last nonzero term of the n-th column is seen. - Michel Marcus, Aug 02 2025

			Triangle whose columns are summed.
  m/n| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
  ----------------------------------------------------------------
   1 | 1
   2 |    1
   3 |    2  1
   4 |    2     1
   5 |       3  2  1
   6 |       3  2     1
   7 |       3        2  1
   8 |       3        2     1
   9 |          4     3     2  1
  10 |          4     3     2     1
  11 |             5        4  3  2  1
  12 |             5        4  3  2     1
  13 |             5        4  3        2  1
  14 |             5        4  3        2     1
  15 |             5        4           3     2  1
  16 |             5        4           3     2     1
  17 |                6           5     4        3  2  1
  18 |                6           5     4        3  2     1
  19 |                6           5     4        3        2  1
  20 |                6           5     4        3        2     1
  ...
The completed column for n=5 is definitely fully visible here because in column 6 for n=6 the divisor k=6 already appeared. That means that column 5 cannot have more divisors in it under the last k=5 in row 17 because in that row only k=7 may follow k=6 in theory, but 7 does not divide 5. So, all similarly proven, definitely fully visible completed columns in this sample array are readily summable by sight. E.g. column 5: a(5) = 1 + 5 + 5 + 5 + 5 + 5 + 5 = 31.

Michel Marcus, Table of n, a(n) for n = 1..500

Cf. A386755.

Cf. A007952 (row number where k=n first appears).

\\ uses row(n) from A386755
a(n) = my(ok=1, k=1, last=-1, s=0, r); while(ok, r=row(k); if (#r >= n, s+=r[n]); k++; if (#r>=n, if ((last==n) && (r[n]==0), ok = 0, last = r[n]))); s; \\ Michel Marcus, Aug 02 2025

\\ uses row(n) from A386755
lista(nn) = my(ok=1, k=1, vlast=vector(nn,i,-1), vs=vector(nn)); while(ok, my(r=row(k)); for (i=1, nn, if (#r>=i, vs[i]+=r[i])); k++; my(nbok=0); for (i=1, nn, if (#r>=i, if ((vlast[i]==i) && (r[i]==0), nbok++, vlast[i] = r[i]))); if (nbok == nn, ok = 0);); vs; \\ Michel Marcus, Aug 02 2025

A141262 Mancala numbers that are prime numbers.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 47, 59, 101, 107, 131, 149, 173, 191, 239, 257, 281, 359, 401, 509, 569, 647, 839, 929, 1277, 1427, 1487, 1847, 1931, 2039, 2339, 2579, 2939, 4451, 4457, 4799, 4931, 5231, 5381, 5717, 5741, 6029, 6317, 6833, 7451, 7547, 7901, 9011, 9437

Offset: 1

Reinhard Zumkeller, Jun 21 2008

nonn

Intersection of A007952 and A000040.

R. Zumkeller, Table of n, a(n) for n = 1..250

Cf. A000040, A007952.

f[n_] := Fold[#2*Floor[#1/#2 + 1] &, n, Reverse@ Range[n - 1]]; a007952=Array[f, 170] ;Select[a007952,PrimeQ] (* James C. McMahon, Jul 19 2025 *)

cp's OEIS Frontend

A048859 A sieve: keep the first 2 numbers, delete the next 3 numbers; keep the next 3 numbers, delete the next 4 numbers; keep the next 4 numbers, delete the next 5 numbers; and so on. In other words, keep the next k numbers and delete the next k+1 numbers, for k = 2, 3, ...

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A073047 Least k such that x(k)=0 where x(1)=n and x(k)=k*floor(x(k-1)/k).

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A108696 Generated by a sieve: see comments.

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A140060 Array of quotients.

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A204539 a(n) is the number of integers N=4k whose "basin" sequence (cf. comment) ends in n^2.

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Mathematica

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A140061 Triangle of quotients.

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Mathematica

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A357431 Triangle read by rows where each term in row n is the next greater multiple of n..1.

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Mathematica

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A357498 Triangle read by rows where each term in row n is the next greater multiple of n..1 divided by n..1.

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