A106571 -id:A106571 - cp's OEIS frontend

A005277 Nontotients: even numbers k such that phi(m) = k has no solution.

14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, 302, 304, 308, 314, 318

Offset: 1

N. J. A. Sloane

nonn

If p is prime then the following two statements are true. I. 2p is in the sequence iff 2p+1 is composite (p is not a Sophie Germain prime). II. 4p is in the sequence iff 2p+1 and 4p+1 are composite. - Farideh Firoozbakht, Dec 30 2005
Another subset of nontotients consists of the numbers j^2 + 1 such that j^2 + 2 is composite. These numbers j are given in A106571. Similarly, let b be 3 or a number such that b == 1 (mod 4). For any j > 0 such that b^j + 2 is composite, b^j + 1 is a nontotient. - T. D. Noe, Sep 13 2007
The Firoozbakht comment can be generalized: Observe that if k is a nontotient and 2k+1 is composite, then 2k is also a nontotient. See A057192 and A076336 for a connection to Sierpiński numbers. This shows that 271129*2^j is a nontotient for all j > 0. - T. D. Noe, Sep 13 2007

			There are no values of m such that phi(m)=14, so 14 is a term of the sequence.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 44 at p. 91.
R. K. Guy, Unsolved Problems in Number Theory, B36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 91.

Robert G. Wilson v, Table of n, a(n) for n = 1..29750 (terms 1..10000 from T. D. Noe).
Lambert A'Campo, Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible L-adic Galois Representation, arXiv:2006.06737 [math.NT], 2020.
Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, arXiv:1801.04542 [hep-th], 2018.
K. Ford, S. Konyagin, and C. Pomerance, Residue classes free of values of Euler's function, arXiv:2005.01078 [math.NT] (1999).
L. Havelock, A Few Observations on Totient and Cototient Valence.
Eric Weisstein's World of Mathematics, Nontotient.
Wikipedia, Nontotient.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

See A007617 for all numbers k (odd or even) such that phi(m) = k has no solution.
All even numbers not in A002202. Cf. A000010.
Cf. A005384, A006093, A014197, A057192, A076336, A079695, A106571.

a005277 n = a005277_list !! (n-1)
a005277_list = filter even a007617_list
-- Reinhard Zumkeller, Nov 22 2015

[n: n in [2..400 by 2] | #EulerPhiInverse(n) eq 0]; // Marius A. Burtea, Sep 08 2019

A005277 := n -> if type(n,even) and invphi(n)=[] then n fi: seq(A005277(i),i=1..318); # Peter Luschny, Jun 26 2011

searchMax = 320; phiAnsYldList = Table[0, {searchMax}]; Do[phiAns = EulerPhi[m]; If[phiAns <= searchMax, phiAnsYldList[[phiAns]]++ ], {m, 1, searchMax^2}]; Select[Range[searchMax], EvenQ[ # ] && (phiAnsYldList[[ # ]] == 0) &] (* Alonso del Arte, Sep 07 2004 *)
totientQ[m_] := Select[ Range[m +1, 2m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; (* after Jean-François Alcover, May 23 2011 in A002202 *) Select[2 Range@160, ! totientQ@# &] (* Robert G. Wilson v, Mar 20 2023 *)

is(n)=n%2==0 && !istotient(n) \\ Charles R Greathouse IV, Mar 04 2017

a(n) = 2*A079695(n). - R. J. Mathar, Sep 29 2021

{k: k even and A014197(k) = 0}. - R. J. Mathar, Sep 29 2021

More terms from Jud McCranie, Oct 13 2000

A067201 Numbers k such that k^2 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, 105, 111, 117, 123, 147, 171, 219, 225, 237, 243, 249, 255, 273, 297, 303, 309, 321, 345, 351, 363, 369, 375, 387, 417, 423, 429, 441, 447, 453, 477, 501, 513, 549, 555, 561, 573, 603, 609, 651, 675, 681, 699, 711, 753

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

All terms > 1 are divisible by 3. - Robert Israel, Sep 05 2014

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Equals 6*A056900(n-2) + 3, n>1.
Cf. A106571, A040976.
Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), this sequence (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

select(t -> isprime(t^2+2), [0,1,seq(3*i,i=1..1000)]); # Robert Israel, Sep 05 2014

lst={};Do[If[PrimeQ[n^2+2], AppendTo[lst, n]], {n, 3*10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Join[{0, 1}, Select[Range[3, 1000, 6], PrimeQ[#^2 + 2] &]] (* Zak Seidov, Jan 30 2014 *)

select(n -> isprime(n^2+2),[1..500]) \\ Edward Jiang, Sep 05 2014

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

A106575 Perfect squares which are both the sum and the difference of two primes.

Original entry on oeis.org

4, 9, 16, 36, 64, 81, 100, 144, 196, 225, 256, 324, 400, 441, 484, 576, 676, 784, 900, 1024, 1089, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 6084, 6400, 6724, 7056

Offset: 1

Alexandre Wajnberg, May 09 2005

Equals A106548 with 0's removed.
Appears to contain all even squares.
By well-known conjectures, every even integer > 2 is both the sum and the difference of two primes; this would be a special case. - Franklin T. Adams-Watters, Sep 13 2015

			2^2 = 4 is in the sequence because it is the sum of two primes (2+2) and the difference of two primes (7-3). 10^2 = 100 is in the sequence because it is the sum and the difference of two primes: 97+3 (or 89+11) and 103-3. 11^2 = 121 is not in the sequence because it is neither the sum nor the difference of two primes. 13^2 = 169 is the sum of two primes (167+2), but it doesn't figure here since it is not the difference of two primes.

Cf. A106544-A106548, A106562-A106564, A106571, A106573, A106574, A106577.

[ s: n in [1..85] | exists(t){ k: k in [1..s] | s-k gt 0 and IsPrime(k) and IsPrime(s-k) } and exists(u){ k: k in [1..s] | IsPrime(k) and IsPrime(s+k) } where s is n^2 ]; /* Klaus Brockhaus, Nov 17 2010 */

Extended by Ray Chandler, May 12 2005

Edited by Klaus Brockhaus, Nov 17 2010

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

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A005277 Nontotients: even numbers k such that phi(m) = k has no solution.

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A067201 Numbers k such that k^2 + 2 is prime.

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

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A106575 Perfect squares which are both the sum and 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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