Ilya Lopatin - cp's OEIS frontend

A242392 Irregular table which shows in row n the list of distinct integer arithmetic means of distinct divisors of n, or zero if n <= 2.

0, 0, 2, 3, 3, 2, 3, 4, 4, 3, 5, 6, 2, 5, 6, 3, 6, 6, 2, 3, 4, 5, 6, 7, 8, 7, 4, 8, 2, 3, 4, 5, 5, 6, 7, 8, 7, 4, 8, 2, 3, 4, 6, 8, 9, 10, 3, 5, 6, 7, 8, 9, 9, 2, 4, 5, 6, 7, 8, 9, 10, 12, 10, 3, 5, 6, 7, 9, 10, 11, 12, 15, 2, 4, 5, 8, 12, 14, 6, 9, 12, 12

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, May 12 2014

			Table starts:
0,
0,
2,
3,
3,
2, 3, 4,
4,
3, 5, 6,
2, 5, 6,
3, 6,
6,
2, 3, 4, 5, 6, 7, 8,
7,
4, 8,
2, 3, 4, 6, 8, 9, 10,
3, 5, 6, 7, 8, 9,
9,
2, 4, 5, 6, 7, 8, 9, 10, 12,
10,
3, 5, 6, 7, 9, 10, 11, 12, 15,
2, 4, 5, 8, 12, 14,
6, 9, 12,
12.
2, 3, 4, are in 6th row because:
1) (n/6 + n/2)/2 = (6/6 + 6/2)/2 = (1 + 3)/2 = 2,
2) (n/6 + n/3 + n/2 + n/1)/4 = (6/6 + 6/3 + 6/2 + 6/1)/4 = (1 + 2 + 3 + 6)/4 = 3,
3) (n/3 + n/1)/2 = (6/3 + 6/1)/2 = (2 + 6)/2 = 4.

Cf. A027750.

A242310 Largest 1 < k <= A000005(n) such that the sum of k distinct divisors of n is equal to k*m for some m, or 1 if no such k exists.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 2, 2, 2, 2, 2, 5, 2, 2, 4, 3, 2, 5, 2, 6, 4, 4, 2, 7, 2, 2, 4, 5, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 7, 2, 8, 2, 6, 6, 4, 2, 11, 3, 4, 4, 5, 2, 8, 4, 8, 4, 2, 2, 12, 2, 4, 5, 6, 4, 8, 2, 6, 4, 8, 2, 11, 2, 2, 4, 4, 2, 8, 4, 2, 2, 11, 4, 4, 4

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, May 10 2014

nonn

k is the number of divisors of arithmetic numbers A003601.

			a(12) = 5 because 1 + 2 + 4 + 6 + 12 = 25 = 5*5 for m = 5.

Cf. A000203, A003601, A049642.

A241884 Numbers n such that n - n^2/k^2 is not prime for all k < n.

Original entry on oeis.org

1, 2, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 100

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 30 2014

nonn

Conjecture: numbers n such that n-1 is not a prime number. - Vincenzo Librandi, Jul 16 2016

Counterexamples to conjecture: 56, 306, 552, 870, ... are not in the sequence. These are p^2+p where p is prime but p^2+p-1 is not prime. - Robert Israel, Jul 04 2017

			56 is not in the sequence because 56 - 56^2/8^2 = 7 is prime for k = 8.

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

Cf. A008864, A036690, A138666 (numbers n such that 2n - n/k is not prime for all k), A242221.

filter:= n -> andmap(t -> not isprime(n - t^2), numtheory:-divisors(n)):
select(filter, [$1..1000]); # Robert Israel, Jul 04 2017

isOK(n) = fordiv(n, k, if(isprime(n-(n/k)^2), return(0))); 1
s=[]; for(n=1, 100, if(isOK(n), s=concat(s, n))); s \\ Colin Barker, May 16 2014

A241534 Number of integer arithmetic means of 2 distinct divisors of n.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 3, 3, 2, 1, 7, 1, 2, 6, 6, 1, 6, 1, 7, 6, 2, 1, 16, 3, 2, 6, 7, 1, 12, 1, 10, 6, 2, 6, 18, 1, 2, 6, 16, 1, 12, 1, 7, 15, 2, 1, 29, 3, 6, 6, 7, 1, 12, 6, 16, 6, 2, 1, 34, 1, 2, 15, 15, 6, 12, 1, 7, 6, 12, 1, 39, 1, 2, 15, 7, 6, 12, 1, 29

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, May 08 2014

nonn

			Triangle T(n, k) starts for n > 2:
2,
3,
3,
2, 4,
4,
3, 5, 6,
2, 5, 6;
where T(n, k) = the values of k such that 2k = q + g; q, g are distinct divisors of n.
a(20) = 7 because (1,5), (2,4), (2,10), (2,20), (4,10), (4,20) and (10,20) are the 7 values of (g,q) such that (g+q)/2 is an integer. - _Colin Barker_, May 10 2014

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A027750.

Table[Sum[Sum[(1 - Ceiling[(i + k)/2] + Floor[(i + k)/2]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Oct 06 2020 *)

a(n) = c=0; fordiv(n, g, fordiv(n, q, if(gColin Barker, May 10 2014

a(n) = Sum_{d1|n, d2|n, d1 < d2} (1 - ceiling((d1+d2)/2) + floor((d1+d2)/2)). - Wesley Ivan Hurt, Oct 06 2020

Several incorrect terms corrected, and more terms added by Colin Barker, May 10 2014

A240838 Primes p such that prime(p) + 2*p^2 is prime.

Original entry on oeis.org

2, 3, 5, 13, 41, 43, 139, 173, 227, 239, 359, 463, 541, 691, 743, 761, 821, 823, 827, 887, 1021, 1117, 1289, 1427, 1489, 1637, 1723, 1933, 1999, 2081, 2287, 2309, 2719, 2791, 2833, 2843, 2953, 3329, 3541, 3803, 3823, 3929, 4003, 4007, 4079, 4139, 4297, 4451, 4561, 4597, 4691, 4703, 4817, 4931, 4943

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 13 2014

nonn

The associated primes are: 11, 23, 41, 379, 3541, ...

			2 is in this sequence because 2 and prime(2) + 2*2^2 = 3 + 8 = 11 are both prime.

Cf. A006450, A079704.

[n: n in {1..5000} | IsPrime(n) and IsPrime(s) where s is (2*n^2 + NthPrime(n))];

Select[Prime[Range[700]],PrimeQ[Prime[#]+2#^2]&] (* Harvey P. Dale, Mar 19 2018 *)

isok(p) = isprime(p) && isprime(prime(p) + 2*p^2); \\ Michel Marcus, Apr 13 2014

A240842 Numbers n such that n - 2*k^2 is a prime for all k > 0 with k^2 < n/2.

Original entry on oeis.org

1, 2, 4, 5, 7, 13, 15, 21, 25, 31, 49, 55, 61, 91, 181, 199

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 13 2014

nonn

No other terms found for n < 2000000. - Colin Barker, Apr 13 2014
No other terms with n < 10^17. - Charles R Greathouse IV, Apr 14 2014
All terms > 4 must be odd, since otherwise n - 2*1^2 is composite. The initial terms 1 and 2 satisfy the condition voidly (no k^2 < n/2 exists). They could be excluded explicitly, but including them can only improve search results. - M. F. Hasler, Apr 16 2014

			91 is in this sequence because 91-2*1^2 = 89, 91-2*2^2 = 83, 91-2*3^2 = 73, 91-2*4^2 = 59, 91-2*5^2 = 41, 91-2*6^2 = 19 where 89, 83, 73, 59, 41, 19 are all primes.

Cf. A039669.

isOK(n) = k=1; until(k^2>=n/2, if(!isprime(n-2*k^2), return(0)); k++); 1;
for(n=1, 20000, if(isOK(n), print1(n, ", "))) \\ Colin Barker, Apr 14 2014

A242221 Numbers n such that n - n^2/m^2 and 2n - n/m are not prime for all m.

Original entry on oeis.org

1, 25, 26, 28, 33, 35, 39, 46, 50, 58, 63, 65, 77, 78, 81, 85, 86, 88, 92, 93, 94, 95, 105, 111, 116, 118, 119, 122, 123, 124, 125, 130, 133, 134, 143, 144, 145, 146, 148, 153, 155, 160, 161, 162, 165, 170, 171, 172, 176, 178, 183, 185, 186, 188, 189, 196, 202

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, May 07 2014

nonn

Intersection of A241884 and A138666.

			26 is in this sequence because:
1) 26 - 26^2/1^2 = -650 and 2*26 - 26/1 = 26 both not prime for m = 1,
2) 26 - 26^2/2^2 = -143 and 2*26 - 26/2 = 39 both not prime for m = 2,
3) 26 - 26^2/13^2 = 22 and 2*26 - 26/13 = 50 both not prime for m = 13,
4) 26 - 26^2/26^2 = 25 and 2*26 - 26/26 = 51 both not prime for m = 26.

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

Cf. A138666, A241884.

filter:= proc(n) andmap(t -> not isprime(n - n^2/t^2) and not isprime(2*n - n/t), numtheory:-divisors(n)) end proc:
select(filter, [$1..200]); # Robert Israel, Jul 03 2017

filterQ[n_] := AllTrue[Divisors[n], !PrimeQ[n - n^2/#^2] && !PrimeQ[2n - n/#]&];
Select[Range[200], filterQ] (* Jean-François Alcover, Jul 27 2020, after Maple *)

f(n)=fordiv(n, m, if(isprime(n-n^2/m^2), return(0))); 1
g(n)=fordiv(n, m, if(isprime(2*n-n/m), return(0))); 1
for(n=1, 200, if(f(n) && g(n), print1(n, ", "))) \\ Colin Barker, May 08 2014

More terms from Colin Barker, May 08 2014

Example corrected by Colin Barker, May 09 2014

A236967 Expansion of (1+3x)^2/(1-3x)^2.

Original entry on oeis.org

1, 12, 72, 324, 1296, 4860, 17496, 61236, 209952, 708588, 2361960, 7794468, 25509168, 82904796, 267846264, 860934420, 2754990144, 8781531084, 27894275208, 88331871492, 278942752080, 878669669052, 2761533245592, 8661172452084, 27113235502176, 84728860944300

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 22 2014

Index entries for linear recurrences with constant coefficients, signature (6,-9)

Cf. Expansion of (1 + k*x)^m/(1 - k*x)^m where the values of k,m are:
......|..m = 1..|..m = 2..|..m = 3..|..m = 4..|..m = 5..|..m = 6..|
k = 1 | A040000 | A008574 | A005899 | A008412 | A008413 | A008414 |
k = 2 | A151821 | A241204 |         |         |         |         |
k = 3 | A099856 | A236967 |         |         |         |         |
k = 4 | A081654 |         |         |         |         |         |
-------------------------------------------------------------------

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)^2/(1-3*x)^2));

For n >= 1, a(n) = 4*n*3^n. - Robert Israel, May 08 2014

Edited by Wolfdieter Lang, May 07 2014

A240882 Numbers m such that m - 4*k^2 is a prime for all k > 0 with k^2 < m/4.

Original entry on oeis.org

6, 7, 9, 11, 15, 21, 23, 27, 33, 35, 47, 77, 83, 143, 167, 227, 437

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 14 2014

No other terms with m < 1000000. - Colin Barker, Apr 14 2014

If it exists, a(18) > 10^9. - Jon E. Schoenfield, Mar 17 2024

			21 is in this sequence because 21 - 4*1^2 = 17 and 21 - 4*2^2 = 5 are both prime.

Cf. A240842.

n=6;Monitor[Parallelize[While[True,If[MemberQ[PrimeQ[Table[n-4*k^2,{k,1,Floor[Sqrt[n/4]]}]],False]==False,Print[n]];n++];n],n] (* J.W.L. (Jan) Eerland, Mar 17 2024 *)

isOK(n) = k=1; until(k^2>=n/4, if(!isprime(n-4*k^2), return(0)); k++); 1;
for(n=3, 1000000, if(isOK(n), print1(n, ", "))) \\ Colin Barker, Apr 14 2014

One missing term and one additional term from Colin Barker, Apr 14 2014

A241204 Expansion of (1 + 2x)^2/(1 - 2x)^2.

Original entry on oeis.org

1, 8, 32, 96, 256, 640, 1536, 3584, 8192, 18432, 40960, 90112, 196608, 425984, 917504, 1966080, 4194304, 8912896, 18874368, 39845888, 83886080, 176160768, 369098752, 771751936, 1610612736, 3355443200, 6979321856, 14495514624, 30064771072, 62277025792

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Subsequence of A008574.

R:=PowerSeriesRing(Integers(), 41); Coefficients(R!((1+2*x)^2/(1-2*x)^2));

A241204:= n->`if`(n=0, 1, 2^(n+2)*n); seq(A241204(n), n=0..20); # Wesley Ivan Hurt, Apr 22 2014

Table[2^(n+2)*n + Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 07 2023 *)
LinearRecurrence[{4,-4},{1,8,32},30] (* Harvey P. Dale, Jun 23 2025 *)

Vec((2*x+1)^2/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Apr 22 2014

def A241204(i):
    if i==0: return 1
    else: return 2^(2+i)*i;
[A241204(n) for n in (0..30)] # Bruno Berselli, Apr 23 2014

a(n) = 2^(2+n)*n for n>0. - Colin Barker, Apr 23 2014
a(n) = 4*a(n-1)-4*a(n-2) for n>2. - Colin Barker, Apr 23 2014
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=1} 1/a(n) = log(2)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(3/2)/4. (End)
E.g.f.: 1 + 8*x*exp(x). - G. C. Greubel, Jun 07 2023

cp's OEIS Frontend

User: Ilya Lopatin

A242392 Irregular table which shows in row n the list of distinct integer arithmetic means of distinct divisors of n, or zero if n <= 2.

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A242310 Largest 1 < k <= A000005(n) such that the sum of k distinct divisors of n is equal to k*m for some m, or 1 if no such k exists.

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A241884 Numbers n such that n - n^2/k^2 is not prime for all k < n.

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Maple

PARI

A241534 Number of integer arithmetic means of 2 distinct divisors of n.

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Mathematica

PARI

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Extensions

A240838 Primes p such that prime(p) + 2*p^2 is prime.

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Magma

Mathematica

PARI

A240842 Numbers n such that n - 2*k^2 is a prime for all k > 0 with k^2 < n/2.

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PARI

A242221 Numbers n such that n - n^2/m^2 and 2n - n/m are not prime for all m.

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Maple

Mathematica

PARI

Extensions

A236967 Expansion of (1+3*x)^2/(1-3*x)^2.

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Magma

Formula

A236967 Expansion of (1+3x)^2/(1-3x)^2.

A241204 Expansion of (1 + 2x)^2/(1 - 2x)^2.