A106575 -id:A106575 - cp's OEIS frontend

A106571 Indices n of perfect squares n^2 which are not the difference of two primes.

5, 7, 11, 13, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 149, 151, 153, 155

Offset: 1

Alexandre Wajnberg, May 09 2005

Also, n such that 1+n^2 is a nontotient (A005277). - T. D. Noe, Sep 13 2007

			a(3)=11 because the third square which is not the difference of two primes (121=11^2) is the 11th one in the succession of the perfect squares (thus index 11).

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A106544-A106548, A106562-A106564, A106573-A106575, A106577.

Cf. A067201 (n such that n^2 + 2 is prime).

a(n) = sqrt(A106564(n)).

Extended by Ray Chandler, May 12 2005

A106564 Perfect squares which are not the difference of two primes.

Original entry on oeis.org

25, 49, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1225, 1369, 1681, 1849, 2209, 2401, 2601, 2809, 3025, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6889, 7225, 7569, 7921, 8281, 8649, 9025, 9409, 10201, 10609, 11449, 11881

Offset: 1

Alexandre Wajnberg, May 09 2005

Squares in A269345; see also the Mathematica code. - Waldemar Puszkarz, Feb 27 2016

It is conjectured (see A020483) that every even number is a difference of primes, and this is known to be true for even numbers < 10^11. If so,this sequence consists of the odd squares n such that n+2 is composite. - Robert Israel, Feb 28 2016

			a(2)=49 because it is the second perfect square which is impossible to obtain subtracting a prime from another one.
64 is not in the sequence because 64=67-3 (difference of two primes).

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

Cf. A020483, A106544-A106548, A106562-A106563, A106571, A106573-A106575, A106577.

[n^2: n in [1..150]| not IsPrime(n^2+2) and n mod 2 eq 1]; // Vincenzo Librandi, Feb 28 2016

remove(t -> isprime(t+2), [seq(i^2, i=1..1000, 2)]); # Robert Israel, Feb 28 2016

With[{lst=Union[(#[[2]]-#[[1]])&/@Subsets[Prime[Range[2000]], {2}]]}, Select[Range[140]^2, !MemberQ[lst,#]&]] (* Harvey P. Dale, Jan 04 2011 *)
Select[Range[1,174,2]^2, !PrimeQ[#+2]&]
Select[Select[Range[30000], OddQ[#]&& !PrimeQ[#]&& !PrimeQ[#+2]&], IntegerQ[Sqrt[#]]&] (* Waldemar Puszkarz, Feb 27 2016 *)

for(n=1, 174, n%2==1&&!isprime(n^2+2)&&print1(n^2, ", ")) \\ Waldemar Puszkarz, Feb 27 2016

n^2 - A106546 with 0's removed.

Extended by Ray Chandler, May 12 2005

A106544 Perfect squares n^2 which are not the sum of two primes (otherwise 0).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121, 0, 0, 0, 0, 0, 289, 0, 0, 0, 0, 0, 529, 0, 625, 0, 0, 0, 0, 0, 961, 0, 0, 0, 0, 0, 0, 0, 1521, 0, 1681, 0, 0, 0, 2025, 0, 0, 0, 0, 0, 2601, 0, 2809, 0, 0, 0, 3249, 0, 3481, 0, 0, 0, 0, 0, 4225, 0, 4489, 0, 0, 0, 0, 0, 5329, 0, 0, 0, 0, 0, 6241, 0, 6561

Offset: 1

Alexandre Wajnberg, May 08 2005

For odd n, n^2 is odd so the two primes must be opposite in parity. Lesser prime must be 2 and greater prime must be n^2-2. Thus for odd n, n^2 is the sum of two primes iff n^2-2 is prime. - Ray Chandler, May 12 2005

			a(10)=0 because 10^2=100=97+3 (sum of two primes)
a(11)=11^2=121, which is impossible to obtain summing two primes.

Cf. A106545-A106548, A106562-A106564, A106571, A106573-A106575, A106577.

a(n) = n^2 - A106545(n).

Extended by Ray Chandler, May 12 2005

A106548 Perfect squares n^2 which are both the sum and the difference of two primes (otherwise 0).

Original entry on oeis.org

0, 4, 9, 16, 0, 36, 0, 64, 81, 100, 0, 144, 0, 196, 225, 256, 0, 324, 0, 400, 441, 484, 0, 576, 0, 676, 0, 784, 0, 900, 0, 1024, 1089, 1156, 0, 1296, 0, 1444, 0, 1600, 0, 1764, 0, 1936, 0, 2116, 0, 2304, 0, 2500, 0, 2704, 0, 2916, 0, 3136, 0, 3364, 0, 3600, 0, 3844, 0

Offset: 1

Alexandre Wajnberg, May 08 2005

Cf. A106544-A106547, A106562-A106564, A106571, A106573-A106575, A106577.

a(n) = Min(A106545(n), A106546(n)).

Extended by Ray Chandler, May 12 2005

A106562 Perfect squares which are not the sum of two primes.

Original entry on oeis.org

1, 121, 289, 529, 625, 961, 1521, 1681, 2025, 2601, 2809, 3249, 3481, 4225, 4489, 5329, 6241, 6561, 6889, 7225, 7569, 8281, 9025, 9409, 9801, 10201, 11025, 11881, 12321, 12769, 13225, 15129, 15625, 16641, 17689, 18769, 19881, 20449, 21609

Offset: 1

Alexandre Wajnberg, May 09 2005

			a(2)=121 because it is the second perfect square which is impossible to obtain summing two primes.
100 is not in the sequence because 100=97+3 (sum of two primes).

Cf. A106544-A106548, A106563-A106564, A106571, A106573-A106575, A106577.

Select[Range[150]^2,Count[IntegerPartitions[#,{2}],?(AllTrue[#,PrimeQ]&)]==0&] (* _Harvey P. Dale, Aug 14 2025 *)

A106544 with 0's removed.

Extended by Ray Chandler, May 12 2005

A106573 Perfect squares which are neither the sum nor the difference of two primes.

Original entry on oeis.org

121, 289, 529, 625, 961, 1681, 2601, 2809, 3481, 4225, 4489, 5329, 6241, 6889, 7225, 7569, 8281, 9025, 9409, 10201, 11881, 12769, 13225, 15625, 16641, 17689, 18769, 19881, 20449, 22201, 22801, 23409, 24649, 25281, 26569, 27225, 27889, 30625

Offset: 1

Alexandre Wajnberg, May 09 2005

			a(2)=289 because it is the second perfect square which is impossible to obtain adding a prime to - or subtracting from - another one. 64 is not in the sequence because 64=67-3, a difference of two primes.

Cf. A106544-A106548, A106562-A106564, A106571, A106574, A106575, A106577.

A106547 with 0's removed.

Corrected and extended by Ray Chandler, May 12 2005

A106577 Indices n of perfect squares n^2 which are both the sum and the difference of two primes.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 117, 118, 120, 122

Offset: 1

Alexandre Wajnberg, May 09 2005

			a(3)=4 because the third square which is the sum and the difference of two primes (16=4^2) is the 4th one in
the succession of the perfect squares (thus: index 4).

Cf. A106544-A106548, A106562-A106564, A106571, A106573-A106575.

a(n) = SQRT(A106575(n)).

Extended by Ray Chandler, May 12 2005

A106546 a(n) = n^2 if n^2 is the difference of two primes, otherwise a(n) = 0.

Original entry on oeis.org

1, 4, 9, 16, 0, 36, 0, 64, 81, 100, 0, 144, 0, 196, 225, 256, 0, 324, 0, 400, 441, 484, 0, 576, 0, 676, 0, 784, 0, 900, 0, 1024, 1089, 1156, 0, 1296, 0, 1444, 1521, 1600, 0, 1764, 0, 1936, 2025, 2116, 0, 2304, 0, 2500, 0, 2704, 0, 2916, 0, 3136, 3249, 3364, 0, 3600, 0

Offset: 1

Alexandre Wajnberg, May 08 2005

For odd n, n^2 is odd so the two primes must be opposite in parity. Lesser prime must be 2 and greater prime must be n^2+2. Thus for odd n, n^2 is the difference of two primes iff n^2+2 is prime.

An odd difference can be obtained only by subtracting 2 from some prime > 2, hence a(n) = 0 if n is odd and n^2+2 is composite.

			a(6) = 6^2 = 36 = 41-5 (two primes).
a(5) = 0 and a(7) = 0 because 5^2+2 =27 = 3*3*3 and 7^2+2 =51 = 3*17 are composite.

Cf. A106544-A106548, A106562-A106564, A106571, A106573-A106575, A106577.

n^2 - A106546 gives perfect squares which are not the difference of two primes (otherwise 0).

Edited and extended by Klaus Brockhaus and Ray Chandler, May 12 2005

A106545 a(n) = n^2 if n^2 is the sum of two primes, otherwise a(n) = 0.

Original entry on oeis.org

0, 4, 9, 16, 25, 36, 49, 64, 81, 100, 0, 144, 169, 196, 225, 256, 0, 324, 361, 400, 441, 484, 0, 576, 0, 676, 729, 784, 841, 900, 0, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 0, 1600, 0, 1764, 1849, 1936, 0, 2116, 2209, 2304, 2401, 2500, 0, 2704, 0, 2916, 3025

Offset: 1

Alexandre Wajnberg, May 08 2005

For odd n, n^2 is odd so the two primes must be opposite in parity. Lesser prime must be 2 and greater prime must be n^2-2. Thus for odd n, n^2 is the sum of two primes iff n^2-2 is prime.

			a(2) = 2^2 = 4 = 2+2, a(5) = 5^2 = 25 = 23+2 (two primes).
a(1) = 0 because the sum of two primes is at least 4 and a(11) = 0 because 11^2 - 2 = 119 = 7*17 is composite.

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

Cf. A106544-A106548, A106562-A106564, A106571, A106573-A106575, A106577.

stpQ[n_]:=If[OddQ[n],PrimeQ[n^2-2],AnyTrue[n^2-Prime[Range[ PrimePi[ n^2]]], PrimeQ]]; Table[If[stpQ[n],n^2,0],{n,60}] (* The program uses the AnyTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 21 2018 *)

a(n) = n^2 - A106544(n).

Edited and extended by Klaus Brockhaus and Ray Chandler, May 12 2005

A106547 Perfect squares n^2 which are neither the sum nor the difference of two primes (otherwise 0).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121, 0, 0, 0, 0, 0, 289, 0, 0, 0, 0, 0, 529, 0, 625, 0, 0, 0, 0, 0, 961, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1681, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2601, 0, 2809, 0, 0, 0, 0, 0, 3481, 0, 0, 0, 0, 0, 4225, 0, 4489, 0, 0, 0, 0, 0, 5329, 0, 0, 0, 0, 0, 6241, 0, 0, 0, 6889, 0

Offset: 1

Alexandre Wajnberg, May 08 2005

Cf. A106544-A106548, A106562-A106564, A106571, A106573-A106575, A106577.

a(n) = Min(A106544(n), n^2-A106546(n)).

Corrected and extended by Ray Chandler, May 12 2005

cp's OEIS Frontend

A106571 Indices n of perfect squares n^2 which are not the difference of two primes.

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A106564 Perfect squares which are not the difference of two primes.

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A106544 Perfect squares n^2 which are not the sum of two primes (otherwise 0).

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A106548 Perfect squares n^2 which are both the sum and the difference of two primes (otherwise 0).

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A106562 Perfect squares which are not the sum of two primes.

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A106573 Perfect squares which are neither the sum nor the difference of two primes.

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A106577 Indices n of perfect squares n^2 which are both the sum and the difference of two primes.

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A106546 a(n) = n^2 if n^2 is the difference of two primes, otherwise a(n) = 0.

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A106545 a(n) = n^2 if n^2 is the sum of two primes, otherwise a(n) = 0.

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