Andrew S. Plewe - cp's OEIS frontend

A132435 Composite integers n with two prime factors nearly equidistant from the integer part of the square root of n.

4, 6, 9, 10, 14, 22, 25, 35, 49, 55, 65, 77, 85, 91, 119, 121, 143, 169, 187, 209, 221, 247, 253, 289, 299, 319, 323, 361, 377, 391, 407, 437, 493, 527, 529, 551, 589, 629, 667, 697, 703, 713, 841, 851, 899, 943, 961, 989, 1073, 1081, 1147, 1189

Offset: 1

Andrew S. Plewe, Nov 13 2007

nonn

An integer n is included if, for some value y >= 0: n = A007918(A000196(n) + y) * A007918(A000196(n) - y) Or: n = nextprime(sqrtint(n) + y) * nextprime(sqrtint(n) - y) Where "nextprime(x)" is the smallest prime number >= to x and "sqrtint(z)" is the integer part of the square root of z.

Has many terms in common with A078972. - Bill McEachen, Dec 24 2020

			25 = nextprime(5 + 0) * nextprime(5 - 0) = 5 * 5 = 25
35 = nextprime(5 + 1) * nextprime(5 - 1) = 7 * 5 = 35
119 = nextprime(10 + 4) * nextprime(10 - 4) = 17 * 7 = 119

Michel Marcus, Table of n, a(n) for n = 1..3406

Cf. A007918, A000196, A078972.

bal(x,y) = nextprime(sqrtint(x)+y) * nextprime(sqrtint(x)-y);
findbal(x) = local(z,y); z=sqrtint(x); while( 0<=z, y=bal(x,z); if(y==x, print1(x", ");break;); z--;);
for (n=1,1200, findbal(n));

A129861 Smallest square s such that A024352(n) + s is square.

Original entry on oeis.org

1, 4, 9, 1, 16, 25, 4, 36, 1, 9, 64, 81, 16, 4, 121, 1, 144, 9, 36, 196, 225, 4, 16, 1, 64, 324, 25, 9, 400, 441, 100, 4, 529, 1, 576, 49, 144, 676, 9, 25, 64, 841, 4, 900, 1, 36, 16, 1089, 256, 100, 1225, 49, 1296, 25, 324, 4, 1521, 81, 144, 1681, 16, 36, 169, 81, 1936, 9, 484

Offset: 1

Andrew S. Plewe, May 23 2007

nonn

			5(5+6) = 55, smallest square = (6/2)^2 = 9
4(4+10) = 56, smallest square = (10/2)^2 = 25
3(3+16) = 57, smallest square = (16/2)^2 = 64
1(1+58) = 59, smallest square = (58/2)^2 = 841
6(6+4) = 60, smallest square = (4/2)^2 = 4
1(1+60) = 61, smallest square = (60/2)^2 = 900
7(7+2) = 63, smallest square = (2/2)^2 = 1
etc.

Cf. A024352.

y(y+e) = A024352(n), where e is the smallest even number that satisfies this equation, A(n) = (e/2)^2.

A125585 Array of constant-spaced integers read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 1, 4, 5, 4, 2, 4, 6, 7, 5, 3, 5, 7, 8, 9, 6, 1, 6, 8, 10, 10, 11, 7, 2, 5, 9, 11, 13, 12, 13, 8, 3, 6, 9, 12, 14, 16, 14, 15, 9, 4, 7, 10, 13, 15, 17, 19, 16, 17, 10, 1, 8, 11, 14, 17, 18, 20, 22, 18, 19, 11, 2, 6, 12, 15, 18, 21, 21, 23, 25, 20, 21, 12

Offset: 1

Andrew S. Plewe, Jan 04 2007

Iteratively taking sums of the values in each row starting with 1 produces the "figurate" numbers. For instance: 1, 1 + 2 = 3, 1 + 2 + 3 = 6 (the triangular numbers -- A000217) 1, 1 + 3 = 4, 1 + 3 + 5 = 9 (the square numbers -- A000290) 1, 1 + 4 = 5, 1 + 4 + 7 = 10 (the pentagonal numbers -- A000326) etc.
Iterative sums of the rows in between produce sequences related to the figurate numbers: 2, 2+4=6, 2+4+6=10 (oblong, or pronic, or heteromecic numbers -- A002378) 2, 2+5=7, 2+5+8=15 (second pentagonal numbers -- A005449) 3, 3+6=9, 3+6+9=18 (triangular matchstick numbers -- A045943) etc.
Iterative products produce the n-factorial numbers: 1, 1*3=3, 1*3*5=15 (double factorial numbers: (2*n-1)!! -- A001147) 2, 2*4=8, 2*4*6=48 (double factorial numbers: (2*n)!! -- A000165) 1, 1*4=4, 1*4*7=28, (triple factorial numbers (3*n-2)!!! -- A007559) etc.

			The array begins:
  1, 2, 3,  4,  5,  6, ...
  1, 3, 5,  7,  9, 11, ...
  2, 4, 6,  8, 10, 12, ...
  1, 4, 7, 10, 13, 16, ...
  2, 5, 8, 11, 14, 17, ...
  3, 6, 9, 12, 15, 18, ...

Alois P. Heinz, Antidiagonals n = 1..141

Cf. A000217, A000290, A000326, A002378, A005449, A045943, A001147, A000165, A007559.
Columns k=1-2 give A002260, A108872.
Main diagonal gives A380548.

A:= proc(n, k) local m;
      m:= floor((sqrt(8*n-7)-1)/2);
      n + (m+1)*(k-1-m/2)
    end:
seq(seq(A(1+d-k, k), k=1..d), d=1..12); # Alois P. Heinz, Jul 16 2012

imax = 5;
A = Table[k, {i, 1, imax}, {j, 1, i}, {k, j, j + i*imax*(imax+1)/2 - 1, i} ] // Flatten[#, 1]&;
Table[A[[n-k+1, k]], {n, 1, Length[A]}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 23 2016 *)

A117048 Prime numbers that are expressible as the sum of two positive triangular numbers.

Original entry on oeis.org

2, 7, 11, 13, 29, 31, 37, 43, 61, 67, 73, 79, 83, 97, 101, 127, 137, 139, 151, 157, 163, 181, 191, 193, 199, 211, 227, 241, 263, 277, 281, 307, 331, 353, 367, 373, 379, 389, 409, 421, 433, 443, 461, 463, 487, 499, 541, 571, 577, 587, 601, 619, 631, 659, 661

Offset: 1

Andrew S. Plewe, Apr 15 2006

If the triangular number 0 is allowed, only one additional prime occurs: 3. In that case, the sequence becomes A117112.

A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

			2 = 1 + 1
7 = 1 + 6
11 = 1 + 10
13 = 10 + 3, etc.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217, A002243, A002244, A020756, A051533, A053614, A060773, A002636.

tri = Table[n (n + 1)/2, {n, 40}]; Select[Union[Flatten[Outer[Plus, tri, tri]]], # <= tri[[-1]]+1 && PrimeQ[#] &] (* T. D. Noe, Apr 07 2011 *)

is(n)=for(k=sqrtint(4*n+1)\2+1,(sqrtint(8*n+1)-1)\2, if(ispolygonal(n-k*(k+1)/2,3), return(n>3 && isprime(n)))); n==2 \\ Charles R Greathouse IV, Nov 07 2014

A110301 Integers written in base "triangle".

Original entry on oeis.org

0, 1, 2, 21, 22, 32, 321, 322, 332, 432, 4321, 4322, 4332, 4432, 5432, 54321, 54322, 54332, 54432, 55432, 65432, 654321, 654322, 654332, 654432, 655432, 665432, 765432, 7654321, 7654322, 7654332, 7654432, 7655432, 7665432, 7765432, 8765432

Offset: 0

Andrew S. Plewe, Sep 07 2005

To convert an integer to base triangle:
Write the integer as a group of ones (6 is shown below).
1 1 1 1 1 1
Iteratively redistribute the leftmost value, adding 1 to each value to its right.
2 1 1 1 1
2 2 1 1
3 2 1
Stop when the leftmost value is greater than the number of values to its right.

A108872 Sums of ordinal references for a triangular table read by columns, top to bottom.

Original entry on oeis.org

2, 3, 4, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 16, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22

Offset: 1

Andrew S. Plewe, Jul 13 2005

The ordinal references (i,j) for a triangular table are arranged as follows:
  (1,1) (2,1) (3,1)
  ..... (2,2) (3,2)
  ........... (3,3)
The sequence comprises the sum of each reference in each column, read top to bottom. A similar sequence is A003057, which consists of the sums of the ordinal references for an array read by antidiagonals.
Subtriangle of triangle in A051162. - Philippe Deléham, Mar 26 2013
First 9 rows coincide with triangle A248110; T(n,k) = A002260(n,k) + n; T(2*n-1,n) = A016789(n-1). - Reinhard Zumkeller, Oct 01 2014

			a(1) = (1,1) = 1 + 1 = 2
a(2) = (2,1) = 2 + 1 = 3
a(3) = (2,2) = 2 + 2 = 4
a(4) = (3,1) = 3 + 1 = 4, etc.
Triangle begins:
  2
  3, 4
  4, 5, 6
  5, 6, 7, 8
  6, 7, 8, 9, 10
  7, 8, 9, 10, 11, 12
  8, 9, 10, 11, 12, 13, 14
  9, 10, 11, 12, 13, 14, 15, 16
  ... - _Philippe Deléham_, Mar 26 2013

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
NRICH discussion thread, Floor Function Identity. [Via Wayback Machine]

Cf. A003057.
Cf. A002024, A002260.
Cf. A005408, A005843.
Cf. A016789 (central terms), A248110.

a108872 n k = a108872_tabl !! (n-1) !! (k-1)
a108872_row n = a108872_tabl !! (n-1)
a108872_tabl = map (\x -> [x + 1 .. 2 * x]) [1..]
-- Reinhard Zumkeller, Oct 01 2014

Flatten[ Table[i + j, {j, 1, 12}, {i, 1, j}]] (* Jean-François Alcover, Oct 07 2011 *)

from math import isqrt
def A108872(n): return n+((r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-r)>>1) # Chai Wah Wu, Nov 08 2024

a(n) = a(i, j) = i + j
a(n) = A002024(n) + A002260(n) = floor(1/2 + sqrt(2n)) + n - (m(m+1)/2) + 1, where m = floor((sqrt(8n+1) - 1) / 2 ). The floor function may be computed directly by using the expression floor(x) = x + (arctan(cot(Pi*x)) / Pi) - 1/2 (equation from nrich.maths.org, see links).
Sum_{k=0..n} T(n,k) = A005449(n+1). - Philippe Deléham, Mar 26 2013

Offset changed by Reinhard Zumkeller, Oct 01 2014

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121

Offset: 0

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n > 0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007
A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007
Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m) = m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008

			Array begins:
  1  4  9 16 25 36  49  64  81 100 ...
  3  8 15 24 35 48  63  80  99 120 ...
  5 12 21 32 45 60  77  96 117 140 ...
  7 16 27 40 55 72  91 112 135 160 ...
  9 20 33 48 65 84 105 128 153 180 ...
  ...
Triangle begins:
   1;
   3,  4;
   5,  8,  9;
   7, 12, 15, 16;
   9, 16, 21, 24, 25;
  11, 20, 27, 32, 35, 36;
  13, 24, 33, 40, 45, 48, 49;
  15, 28, 39, 48, 55, 60, 63, 64;
  17, 32, 45, 56, 65, 72, 77, 80, 81;
  19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.
Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.
Diagonals give A033428, A045944, A067725.
Cf. A000384, A002412, A024352, A105020, A128076.

[(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023

t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]]
(* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *)
Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)

def A105020(n,k): return (k+1)*(2*n-k+1)
flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023

a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021
a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022
From G. C. Greubel, Mar 15 2023: (Start)
Sum_{k=0..n} T(n, k) = A002412(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)

More terms from Robert G. Wilson v, Jul 11 2005

A105047 Form an addition table of the primes; a(n) is the number of even numbers that appear for the first time in column n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 5, 4, 5, 3, 4, 3, 3, 7, 4, 5, 4, 3, 5, 5, 6, 5, 5, 4, 4, 3, 7, 5, 5, 7, 5, 5, 6, 4, 6, 5, 5, 7, 4, 6, 5, 4, 6, 5, 8, 5, 7, 4, 5, 6, 5, 3, 3, 8, 8, 5, 4, 5, 8, 8, 5, 5, 9, 4, 8, 7, 7, 6, 6, 5, 5, 7, 5, 7, 7, 6, 6, 6, 6, 5, 7, 7, 6, 6, 5, 6, 5, 5, 7, 4, 8, 4, 8, 5, 8, 7, 8, 9, 7, 5, 9

Offset: 1

Andrew S. Plewe, Apr 06 2005

nonn

For n > 2: a(n) = A102696(n-1) - A102696(n-2); a(n+1) = length of n-th row in the triangle A260580. - Reinhard Zumkeller, Aug 11 2015

			The addition table is as follows:
   + | 2  3  5  7 11
   --+--------------
   2 | 4  5  7  9 13
   3 |    6  8 10 14
   5 |      10 12 16
   7 |         14 18
  11 |            22

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A102696, A260580.

a105047 1 = 1
a105047 n = length $ a260580_row (n - 1)
-- Reinhard Zumkeller, Aug 11 2015

lista(n) = {maxp = prime(n); v = vector(maxp); forprime (p=1, maxp, nb = 0; forprime (q=1, p, s = p+q; if (! (s % 2), if (!v[s/2], nb++); v[s/2] = 1;);); print1(nb, ", "););}  \\ Michel Marcus, Apr 18 2013

More terms from Reinhard Zumkeller, Apr 19 2005

A098383 Define a function f on the positive integers by: if n is 1 or composite, stop; but if n = prime(k) then f(n) = k; a(n) = sum of terms in trajectory of n under repeated application of f.

Original entry on oeis.org

1, 3, 6, 4, 11, 6, 11, 8, 9, 10, 22, 12, 19, 14, 15, 16, 28, 18, 27, 20, 21, 22, 32, 24, 25, 26, 27, 28, 39, 30, 53, 32, 33, 34, 35, 36, 49, 38, 39, 40, 60, 42, 57, 44, 45, 46, 62, 48, 49, 50, 51, 52, 69, 54, 55, 56, 57, 58, 87, 60, 79, 62, 63, 64, 65, 66, 94, 68, 69, 70, 91, 72

Offset: 1

Andrew S. Plewe, Oct 26 2004

Sum of the terms in the prime index chain for n (cf. A049076).

			a(2) = 3 because 2 is the first prime, therefore 2 + 1 = 3. a(3) = 6 because 3 is the second prime and two is the first prime, therefore 3 + 2 + 1 = 6. a(4) = 4 because 4 is composite. a(5) = 11 because five is the third prime, three is the second prime and two is the first prime, which gives us 5 + 3 + 2 + 1 = 11 and so on.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A007097, A057450, A057451, A049076.

a:= n-> n + `if`(isprime(n), a(numtheory[pi](n)), 0):
seq (a(n), n=1..80);  # Alois P. Heinz, Jul 16 2012

Table[s=n; p=n; While[PrimeQ[p], p=PrimePi[p]; s=s+p]; s, {n, 1000}] (T. D. Noe)

More terms from Ray Chandler, Nov 04 2004

A094812 Number of odd composites between 2^n and 2^(n + 1).

Original entry on oeis.org

0, 0, 0, 2, 3, 9, 19, 41, 85, 181, 375, 769, 1584, 3224, 6580, 13354, 27059, 54521, 110682, 223509, 450702, 908240, 1828936, 3680596, 7402790, 14883096, 29908688, 60081574, 120655821, 242228178, 486173375, 975559168, 1957148063, 3925643991

Offset: 0

Andrew S. Plewe, Jun 11 2004

This sequence may be related to n-ary rooted trees of a fixed height. For instance, the first few terms of A036616 are:
1, 1, 1, 2, 4, 9, 19, 41, 86, 182, 376, 776, 1579, ...
and in A036622:
1, 1, 1, 2, 4, 9, 19, 42, 88, 188, 393, 821, 1692, ...
whereas in the present sequence we have:
0, 0, 0, 2, 3, 9, 19, 41, 85, 181, 375, 769, 1584, ...

			a(3) = 2 because in the interval 2^3..2^4 = [8..16] there are two odd composites: 9 = 3^2, 15 = 3 * 5.

Cf. A036616, A036622, A071904.

f[n_] := (2^(n - 1) - PrimePi[2^(n + 1)] + PrimePi[2^n]); Table[ f[n], {n, 32}] (* Robert G. Wilson v, Jun 15 2004 *)

Members of A071904 that lie between 2^n and 2^(n + 1).

More terms from Robert G. Wilson v, Jun 15 2004

cp's OEIS Frontend

User: Andrew S. Plewe

A132435 Composite integers n with two prime factors nearly equidistant from the integer part of the square root of n.

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A129861 Smallest square s such that A024352(n) + s is square.

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A125585 Array of constant-spaced integers read by antidiagonals.

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A117048 Prime numbers that are expressible as the sum of two positive triangular numbers.

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A110301 Integers written in base "triangle".

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A108872 Sums of ordinal references for a triangular table read by columns, top to bottom.

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A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

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A105047 Form an addition table of the primes; a(n) is the number of even numbers that appear for the first time in column n.

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