A001750 -id:A001750 - cp's OEIS frontend

A036468 Number of ways to represent 2n+1 as a+b with 0 < a < b and a^2 + b^2 prime.

1, 2, 2, 2, 3, 3, 4, 4, 4, 3, 4, 8, 4, 6, 5, 4, 9, 8, 6, 9, 7, 7, 7, 5, 7, 9, 14, 8, 9, 11, 7, 17, 11, 10, 9, 11, 9, 8, 13, 9, 15, 20, 11, 14, 13, 8, 18, 14, 10, 18, 16, 10, 17, 16, 13, 20, 20, 13, 14, 17, 12, 23, 18, 14, 22, 15, 17, 18, 21, 12, 19, 29, 16, 23, 21, 14, 27, 24

Offset: 1

R. K. Guy

nonn

Zhang Ming-Zhi (zamiz(AT)mail.sc.cninfo.net) asks if a(m) is always > 0.
I have confirmed that a(n) > 0 for 0 < n < 10^7. - T. D. Noe, Oct 17 2004
This open problem is mentioned by Guy at the end of section C1. - T. D. Noe, Apr 22 2009
a(n) <= phi(2n+1)/2, where phi(m) = A000010(m), while a(n) = phi(2n+1)/2 only for n = 1, 2, and 7. - Thomas Ordowski, Jan 25 2014
Records in a(n) are for 2n+1 = 3, 5, 11, 15, 25, 35, 55, 65, 85, 125, 145, 185, 205, 215, 235, 265, 295, 325, 365, 415, ... cf. A001750. - Thomas Ordowski, Mar 02 2017
a(n) tends to be larger for n == 2 (mod 5): see plot in Links. - Robert Israel, Mar 02 2017
Number of primes p = ((2n+1)^2 + x^2)/2 for positive integers x < 2n+1. - Thomas Ordowski, Mar 06 2017

R. K. Guy, Unsolved Problems in Theory of Numbers, Section C1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Gordon Hamilton, Unsolved K-12: Grade 7, 2014. (video)
Robert Israel, a(5k+j) for j=0,1,2,3,4

Cf. A082519, A099468, A281543.

a:= n-> add(`if`(isprime(i^2+(2*n+1-i)^2), 1, 0), i=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Jul 09 2016

Table[cnt=0; m=2n+1; Do[If[PrimeQ[k^2+(m-k)^2], cnt++ ], {k, n}]; cnt, {n, 100}]

a(n)=sum(k=1,n,isprime(k^2+(2*n-k+1)^2)) \\ Charles R Greathouse IV, Jan 09 2014

a(n) = O(n/log(n)). - Thomas Ordowski, Feb 11 2013

More terms from David W. Wilson and Michael Kleber

A339116 Triangle of all squarefree semiprimes grouped by greater prime factor, read by rows.

Original entry on oeis.org

6, 10, 15, 14, 21, 35, 22, 33, 55, 77, 26, 39, 65, 91, 143, 34, 51, 85, 119, 187, 221, 38, 57, 95, 133, 209, 247, 323, 46, 69, 115, 161, 253, 299, 391, 437, 58, 87, 145, 203, 319, 377, 493, 551, 667, 62, 93, 155, 217, 341, 403, 527, 589, 713, 899

Offset: 2

Gus Wiseman, Dec 01 2020

A squarefree semiprime is a product of any two distinct prime numbers.

			Triangle begins:
   6
  10  15
  14  21  35
  22  33  55  77
  26  39  65  91 143
  34  51  85 119 187 221
  38  57  95 133 209 247 323
  46  69 115 161 253 299 391 437
  58  87 145 203 319 377 493 551 667
  62  93 155 217 341 403 527 589 713 899

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)

A339194 gives row sums.
A100484 is column k = 1.
A001748 is column k = 2.
A001750 is column k = 3.
A006094 is column k = n - 1.
A090076 is column k = n - 2.
A319613 is the central column k = 2*n.
A087112 is the not necessarily squarefree version.
A338905 is a different triangle of squarefree semiprimes.
A339195 is the generalization to all squarefree numbers, row sums A339360.
A001358 lists semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd terms A046388.
A024697 is the sum of semiprimes of weight n.
A025129 is the sum of squarefree semiprimes of weight n.
A332765 gives the greatest squarefree semiprime of weight n.
A338898/A338912/A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
Cf. A000040, A001221, A014342, A098350, A112798, A168472, A320656, A338901, A339003, A339114/A339115.
Subsequence of A019565.

Table[Prime[i]*Prime[j],{i,2,10},{j,i-1}]

row(n) = {prime(n)*primes(n-1)}
{ for(n=2, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

T(n,k) = prime(n) * prime(k) for k < n.

Offset corrected by Andrew Howroyd, Jan 19 2023

A138636 a(n) = 6 * prime(n).

Original entry on oeis.org

12, 18, 30, 42, 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 14 2008

Column 5 of A272214. - Omar E. Pol, Apr 29 2016

			2*6=12, 3*6=18, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A001748, A001749, A001750, A100484.

[6*p: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

Mathematica
```
6*Prime[Range[100]]
```

vector(50, n, 6*prime(n)) \\ G. C. Greubel, Feb 02 2019

[6*nth_prime(n) for n in (1..50)] # G. C. Greubel, Feb 02 2019

A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.

Original entry on oeis.org

21, 27, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 125, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 230, 235, 237, 249, 250, 253, 259, 265, 267, 287, 290, 291, 295, 301, 303, 305, 309, 310, 319, 321, 327, 329, 335

Offset: 1

Hartmut F. W. Hoft, Nov 03 2015

The areas of the first two regions are (2^(m+1) - 1) * (p * q + 1) / 2 and (2^(m+1) - 1) * (p + q) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (q + 1).

For a proof of the formula for this sequence see the link.

			65 = 5*13 is in the sequence since m = 0 and 2 < 5 < 10 < 13. The first two regions in the symmetric representation of sigma(65) = 84 start with legs 1 and 5 of the Dyck path and have areas 33 and 9, respectively.
406 = 2*7*29 is in the sequence since m=1 and 4 < 7 < 28 < 29. The first two regions in the symmetric representation of sigma(406) = 720 start with legs 1 and 7 and have areas 306 and 54, respectively. Note also that 406 is a triangular number and the middle two regions meet at the center of the Dyck path.
One case in the formula for the sequence is the 3-parameter expression n = 2^m * p * q with p and q distinct primes satisfying the stated conditions. That subsequence can be visualized as a skew tetrahedron since the start of each "line" on an irregular "triangular" side of the "tetrahedron" is determined by a different prime number and each layer is determined by a different power of two. Below are the first three layers with primes p designating columns and primes q rows.
m=0| 3    5    7    11   13
-----------------------------
7  | 21
11 | 33   55
13 | 39   65
17 | 51   85   119
19 | 57   95   133
23 | 69   115  161  253
29 | 87   145  203  319  377
31 | 93   155  217  341  403
37 | 111  185  259  407  481
41 | 123  205  287  451  533
...
89 | 267  445  623  979  1157
...
Column 1 is A001748 except for the first three terms and column 2 is A001750 except for the first four terms in the two resepctive sequences.
m=1| 3    5    7    11   13
-------------------------------
23 |     230
29 |     290  406
31 |     310  434
37 |     370  518
41 |     410  574
43 |     430  602
47 |     470  658  1034
53 |     530  742  1166  1378
...
89 |     890  1246 1958  2314
...
m=2| 3    5    7    11   13
-------------------------------
89 |               3916
97 |               4268
101|               4444
103|               4532
107|               4708  5564
109|               4796  5668
...
The fourth layer for m = 3 starts with number 37672 in column p = 17 and row q = 277.
The subsequence of the 2-parameter case n = 2^m * p^3 with 2^(m+1) < p gives rise to the following irregular triangle:
p\m| 0      1       2       3
----------------------------------
3  | 27
5  | 125    250
7  | 343    686
11 | 1331   2662    5324
13 | 2197   4394    8788
17 | 4913   9826    19652   39304
19 | 6859   13718   27436   54872
23 | 12167  24334   48668   97336
29 | 24389  48778   97556   195112
...
The first column in this triangle is A030078 except for the first term and the second column is A172190 except for the first two terms respectively in the two sequences.

Hartmut F. W. Hoft, Diagram of symmetric representations of sigma
Hartmut F. W. Hoft, Proof of formula for 4 regions of width 1

Cf. A001748, A001750, A030078, A172190.
For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.
Subsequence of A280107.

mpStalk[m_, p_, bound_] := Module[{q=NextPrime[2^(m+1)*p], list={}}, While[2^m*p*q<=bound, AppendTo[list, 2^m*p*q]; q=NextPrime[q]]; If[2^m*p^3<=bound, AppendTo[list, 2^m*p^3]]; list]
mTriangle[m_, bound_] := Module[{p=NextPrime[2^(m+1)], list={}}, While[2^m*p*NextPrime[2^(m+1)*p]<=bound, list=Union[list, mpStalk[m, p, bound]]; p=NextPrime[p]]; list]
(* 2^(4m+3)<=bound is a simpler test, but computes some empty stalks *)
a264102[bound_] := Module[{m=0, list={}}, While[2^m*NextPrime[2^(m+1)]*NextPrime[2^(m+1)*NextPrime[2^(m+1)]]<=bound, list=Union[list, mTriangle[m, bound]]; m++]; list]
a264102[335] (* data *)

n = 2^m * p * q where m >= 0, p > 2 is prime, 2^(m+1) < p < 2^(m+1) * p < q, and either q is prime or q = p^2.

A272470 7 times the primes.

Original entry on oeis.org

14, 21, 35, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 371, 413, 427, 469, 497, 511, 553, 581, 623, 679, 707, 721, 749, 763, 791, 889, 917, 959, 973, 1043, 1057, 1099, 1141, 1169, 1211, 1253, 1267, 1337, 1351, 1379, 1393, 1477, 1561, 1589, 1603, 1631, 1673, 1687, 1757, 1799, 1841, 1883, 1897

Offset: 1

Omar E. Pol, Apr 30 2016

Column 4 of A272214.

k times the primes (k=1..6): A000040, A100484, A001748, A001749, A001750, A138636.

7 Prime@ Range@ 58 (* Michael De Vlieger, May 01 2016 *)

a(n) = 7*prime(n); \\ Michel Marcus, May 01 2016

from sympy import prime
for n in range(1,1000):print(7*prime(n),end=", ") # Soumil Mandal, May 08 2016

a(n) = 7*prime(n) = 7*A000040(n).

A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes.

Original entry on oeis.org

1, 0, 2, 2, 3, 2, 5, 2, 5, 4, 7, 2, 11, 2, 9, 8, 9, 2, 17, 2, 17, 10, 13, 2, 23, 6, 15, 10, 23, 2, 29, 2, 17, 14, 19, 12, 35, 2, 21, 16, 37, 2, 41, 2, 35, 38, 25, 2, 47, 8, 47, 20, 41, 2, 53, 16, 51, 22, 31, 2, 59, 2, 33, 52, 33, 18, 65, 2, 53, 26, 67, 2, 71, 2, 39, 68, 59, 18, 77, 2, 77, 28, 43, 2, 83, 22, 45, 32, 79

Offset: 0

Alois P. Heinz, Dec 04 2018

Numbers where a(n) + A000010(n) != n + 1: A102467. - Robert G. Wilson v, Aug 23 2021

			a(6) = |{0,2,3,4,6}| = 5.
a(9) = |{0,3,6,9}| = 4.
a(10) = |{0,2,4,5,6,8,10}| = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Matt Baker, The Balanced Centrifuge Problem, Math Blog, 2018.
Holly Krieger and Brady Haran, The Centrifuge Problem, Numberphile video (2018)
T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, arXiv:math/9511209 [math.NT], 1995.
Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.
Robert G. Wilson v, Graph of the first 100001 terms.

Cf. A000040, A001248, A001748, A001750, A004280, A008588, A008864, A100484, A103306, A103314, A163300, A182986, A272470, A306275.

a:= proc(n) option remember; local f, b; f, b:=
       map(i-> i[1], ifactors(n)[2]),
       proc(m, i) option remember; m=0 or i>0 and
        (b(m, i-1) or f[i]<=m and b(m-f[i], i))
       end; forget(b); (t-> add(
      `if`(b(j, t) and b(n-j, t), 1, 0), j=0..n))(nops(f))
    end:
seq(a(n), n=0..100);

$RecursionLimit = 4096;
a[1] = 0;
a[n_] := a[n] = Module[{f, b}, f = FactorInteger[n][[All, 1]];
     b[m_, i_] := b[m, i] = m == 0 || i > 0 &&
     (b[m, i - 1] || f[[i]] <= m && b[m - f[[i]], i]);
     With[{t = Length[f]}, Sum[
     If[b[j, t] && b[n - j, t], 1, 0], {j, 0, n}]]];
Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Dec 13 2018, after Alois P. Heinz, corrected and updated Aug 07 2021 *)
f[n_] := Block[{c = 2, k = 2, p = First@# & /@ FactorInteger@ n}, While[k < n, If[ IntegerPartitions[k, All, p, 1] != {} && IntegerPartitions[n - k, All, p, 1] != {}, c++]; k++]; c]; f[0] = 1; f[1] = 0; Array[f, 75] (* Robert G. Wilson v, Aug 22 2021 *)

a(n) = |{ k : k and n-k can be written as a sum of prime factors of n }|.
a(n) = 2 <=> n is prime (A000040).
a(n) >= n-1 <=> n in {1,2,3,4} union { A008588 }.
a(n) = (n+4)/2 <=> n in { A100484 } minus { 4 }.
a(n) = (n+9)/3 <=> n in { A001748 } minus { 9 }.
a(n) = (n+25)/5 <=> n in { A001750 } minus { 25 }.
a(n) = (n+49)/7 <=> n in { A272470 } minus { 49 }.
a(n^2) = n+1 <=> n = 0 or n is prime <=> n in { A182986 }.
a(A001248(n)) = A008864(n).
a(n) is odd <=> n in { A163300 }.
a(n) is even <=> n in { A004280 }.

A281543 Number of partitions n = x + y with y >= x > 0 such that x^2 + y^2 or (x^2 + y^2)/2 is prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 1, 4, 3, 4, 1, 4, 4, 3, 2, 4, 1, 8, 4, 4, 3, 6, 3, 5, 3, 4, 4, 9, 3, 8, 4, 6, 6, 9, 2, 7, 4, 7, 5, 7, 3, 5, 7, 7, 6, 9, 4, 14, 4, 8, 4, 9, 4, 11, 7, 7, 6, 17, 5, 11, 6, 10, 8, 9, 5, 11, 6, 9, 7, 8, 3, 13, 9, 9, 5, 15, 5, 20, 8, 11, 8, 14, 7, 13, 9, 8, 6, 18, 7, 14, 10, 10, 8

Offset: 1

Thomas Ordowski and Altug Alkan, Mar 01 2017

nonn

Conjecture: a(n) > 0 for all n > 1.
We have a(n) <= phi(n)/2 for n <> 2, because must be gcd(x,y) = 1.
Numbers n such that a(n) = phi(n)/2 are 3, 4, 5, 6, 10, 12, 15, and 20.
Record values of a(n) are for n = 1, 2, 5, 11, 15, 25, 35, 55, 65, 85, 125, 145, 185, 205, 215, 235, 265, 295, 325, 365, 415, ... cf. A001750.

			a(5) = 2 because 5 = 1 + 4 and 5 = 2 + 3 are only options; 1^2 + 4^2 = 17 and 2^2 + 3^2 = 13 are primes.
a(6) = 1 because 6 = 1 + 5 is only option; (1^2 + 5^2)/2 = 13 is prime.
a(7) = 2 because 7 = 1 + 6, 7 = 2 + 5 and 7 = 3 + 4, but 3^2 + 4^2 = 5^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative Scatterplot of A281543

Cf. A000010, A002313, A036468, A069004.

a(n) = if(n==2, 1, if(n%2==0, sum(k=1, n/2-1, isprime(n^2/4+k^2)), sum(k=1, (n-1)/2, isprime(k^2+(n-k)^2))));

a(2m+1) = A036468(m) for m > 0.
a(2m) = A069004(m) for m > 1.
a(n) = O(n/log(n)).

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31

Offset: 1

Stefano Spezia, Jul 14 2019

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Wikipedia, Toeplitz matrix

Cf. A025581, A318173, A325516.

Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

[[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output

a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);

Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]

T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output

[[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

T(n, k) = A025581(n, k)*A000040(1 + k).

A068547 Numbers m such that mtau(m)>5prime(m).

Original entry on oeis.org

2520, 3360, 3780, 3960, 4200, 4320, 5040, 6300, 6720, 7200, 7560, 7920, 8400, 8640, 8820, 9240, 9360, 9900, 10080, 10560, 10800, 10920, 11088, 11340, 11520, 11700, 11760, 11880, 12096, 12240, 12480, 12600, 12960, 13104, 13200, 13440, 13680

Offset: 1

Benoit Cloitre, Mar 22 2002

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

Cf. A001750, A038040.

Cf. A068544, A068545, A068546.

R:= NULL: count:= 0: p:= 0:
for n from 1 while count < 100 do
  p:= nextprime(p);
  if n * numtheory:-tau(n) > 5 * p then R:= R,n; count:= count+1;  fi;
od:
R; # Robert Israel, May 13 2025

Select[Range[15000],# DivisorSigma[0,#]>5 Prime[#]&] (* Harvey P. Dale, Jul 17 2023 *)

A133321 Inserting any (identical) digit between adjacent digits of an odd semiprime k never yields a prime.

Original entry on oeis.org

15, 25, 35, 55, 65, 85, 95, 115, 121, 143, 145, 155, 185, 187, 205, 215, 235, 253, 265, 295, 299, 305, 335, 341, 355, 365, 393, 395, 411, 415, 437, 445, 451, 473, 485, 505, 515, 535, 545, 565, 583, 635, 655, 671, 679, 685, 695, 717, 745, 755, 781, 785, 815

Offset: 1

Jonathan Vos Post, Oct 18 2007

Odd semiprime analog of A050805. Trivially true for any digit if we substitute "even semiprime" for "odd semiprime." Trivially true for any semiprime which is a multiple of 5 (A001750). The nonmultiples of 5 in this sequence begin 121, 143, 187, 253, 299, 341.

			121 is in the sequence because 10201, 11211, 12221, 13231, 14241, 15251, 16261, 17271, 18281, 19291 are all composite.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A001750, A050674-A050719, A050805, A050806.

Select[Select[Range[11,900,2],PrimeOmega[#]==2&],AllTrue[Table[ FromDigits[ Riffle[ IntegerDigits[#],n]],{n,0,9}],CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 07 2018 *)

More terms from R. J. Mathar, Oct 22 2007

cp's OEIS Frontend

A036468 Number of ways to represent 2n+1 as a+b with 0 < a < b and a^2 + b^2 prime.

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A339116 Triangle of all squarefree semiprimes grouped by greater prime factor, read by rows.

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A138636 a(n) = 6 * prime(n).

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A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.

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A272470 7 times the primes.

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A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes.

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A281543 Number of partitions n = x + y with y >= x > 0 such that x^2 + y^2 or (x^2 + y^2)/2 is prime.

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A068547 Numbers m such that mtau(m)>5prime(m).