A039787 -id:A039787 - cp's OEIS frontend

A075430 Primes with a squarefree neighbor.

2, 3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 101, 103, 107, 109, 113, 131, 137, 139, 157, 167, 173, 179, 181, 191, 193, 211, 223, 227, 229, 239, 257, 263, 277, 281, 283, 311, 313, 317, 331, 347, 353, 359, 367, 373, 383, 389, 397

Offset: 1

Reinhard Zumkeller, Sep 15 2002

nonn

			a(3) = 5 because 6 (the number above) is squarefree.
a(13) = 47 because 46 (the number below) is squarefree.
53 is not in the sequence because both 52 and 54 have squares among their divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)

Union of A039787 and A049097.
Complement of A075432 in A000040.
Cf. A005117.

Select[Prime[Range[100]], Or @@ SquareFreeQ /@ (# + {-1, 1}) &] (* Amiram Eldar, May 07 2025 *)

isok(p) = isprime(p) && (issquarefree(p-1) || issquarefree(p+1)); \\ Michel Marcus, Feb 20 2023

def is_A075430(n): return is_prime(n) and (is_squarefree(n-1) or is_squarefree(n+1)) # D. S. McNeil, Jan 16 2011

A049198 Numbers that are not squarefree and whose Euler totient function is squarefree.

Original entry on oeis.org

4, 9, 18, 49, 98, 121, 242, 529, 961, 1058, 1849, 1922, 2209, 3481, 3698, 4418, 4489, 5041, 6241, 6889, 6962, 8978, 10082, 10609, 11449, 12482, 13778, 17161, 19321, 21218, 22898, 27889, 32041, 34322, 36481, 38642, 44521, 49729, 51529, 55778, 57121, 64082, 69169

Offset: 1

Labos Elemer

nonn

Numbers k such that abs(mu(phi(k))) = 1 and abs(mu(k)) = 0.

Contains all the squares p^2 of primes p such that p-1 is squarefree (A039787). - Amiram Eldar, Mar 18 2025

			a(27) = 13778 = 2*83*83 is divisible by a square, but phi(13778) = 6806 = 2*41*83 is squarefree.

Donovan Johnson, Table of n, a(n) for n = 1..5000

Intersection of A049149 and A013929.

Cf. A000010, A005117, A008683, A039787, A049199.

Select[Range[70000], Abs[ MoebiusMu[ EulerPhi[ # ] ] ] == 1 && Abs[ MoebiusMu[ # ] ] == 0 &]

isok(k)=!issquarefree(k) && issquarefree(eulerphi(k)) \\ Donovan Johnson, Jun 20 2012

A175625 Numbers k such that gcd(k, 6) = 1, 2^(k-1) == 1 (mod k), and 2^(k-3) == 1 (mod (k-1)/2).

Original entry on oeis.org

7, 11, 23, 31, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1123, 1187, 1283, 1291, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543

Offset: 1

Alzhekeyev Ascar M, Jul 28 2010, Jul 30 2010

nonn

All composites in this sequence are 2-pseudoprimes, A001567. That subsequence begins with 536870911, 46912496118443, 192153584101141163, with no other composites below 2^64 (the first two were found by 'venco' from the dxdy.ru forum), and contains the terms of A303448 that are not multiples of 3. Correspondingly, composite terms include those of the form A007583(m) = (2^(2m+1) + 1)/3 for m in A303009. The only known composite member not of this form is a(1018243) = 536870911.
Intended as a pseudoprimality test; note that many primes do not pass the third condition either.
Conjecture: The prime values belong to A039787. - Bill McEachen, Dec 27 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001567, A007583, A303009, A303448.

Cf. A039787.

Select[Array[(6 # + (-1)^# - 3)/2 &, 3000], And[PowerMod[2, (# - 1), #] == 1, PowerMod[2, (# - 3), (# - 1)/2] == 1] &] (* Michael De Vlieger, Dec 27 2023 *)

isA175625(n) = gcd(n,6)==1 && Mod(2,n)^(n-1)==1 && Mod(2,n\2)^(n-3)==1

Partially edited by N. J. A. Sloane, Jul 29 2010
Entry rewritten by Charles R Greathouse IV, Aug 04 2010
Comment and b-file from Charles R Greathouse IV, Sep 06 2010
Edited by Max Alekseyev, May 28 2014, Apr 24 2018

A224718 Primes p such that p^2 + 1 is not squarefree.

Original entry on oeis.org

7, 41, 43, 107, 157, 193, 239, 251, 257, 293, 307, 443, 457, 557, 577, 593, 607, 643, 743, 757, 829, 857, 907, 1093, 1193, 1303, 1307, 1451, 1483, 1493, 1543, 1607, 1657, 1693, 1723, 1789, 1907, 1993, 2143, 2207, 2243, 2267, 2293, 2309, 2357, 2393, 2543

Offset: 1

Zak Seidov, Apr 16 2013

nonn

			7^2 + 1 = 50 = 2*5^2, 41^2 + 1 = 1681 = 2*29^2.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A039787.

Select[Prime[Range[300]], ! SquareFreeQ[#^2 + 1] &]

A089188 Lesser member p of a pair of twin primes such that p-1 is squarefree.

Original entry on oeis.org

3, 11, 59, 71, 107, 179, 191, 227, 239, 311, 347, 419, 431, 599, 659, 827, 1019, 1031, 1091, 1319, 1427, 1487, 1607, 1619, 1787, 1871, 1931, 2027, 2087, 2111, 2267, 2339, 2591, 2687, 2711, 2999, 3119, 3167, 3299, 3359, 3371, 3467, 3527, 3539, 3671, 3767

Offset: 1

Cino Hilliard, Dec 07 2003

			71 is a term because it is a prime, 71 + 2 = 73 is a prime, and 71 - 1 = 70 = 2 * 5 * 7 is squarefree.
17 is not a term because 17 - 1 = 2^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1001 from Harvey P. Dale)

Intersection of A001359 and A039787.

Cf. A005117.

Select[Transpose[Select[Partition[Prime[Range[600]],2,1],#[[2]]-#[[1]]==2&]][[1]],SquareFreeQ[#-1]&] (* Harvey P. Dale, Aug 10 2013 *)

pm1th(n) = { c=0; pc=0; forprime(x=2,n, pc++; y=x-1; if(isprime(x+2), if(issquarefree(y), c++; print1(x","); ) ) ); print(); print(c","pc","c/pc+.0) }

Offset corrected by Amiram Eldar, Jun 29 2024

A153215 Primes p such that none of p-2, p-1, p+1, and p+2 is squarefree.

Original entry on oeis.org

727, 1423, 1861, 3719, 6173, 9749, 11321, 13183, 19073, 20873, 23227, 23473, 23827, 26981, 27883, 34351, 35323, 41263, 42677, 44449, 45127, 45523, 47527, 48751, 49727, 52391, 53623, 53849, 68749, 71993, 72559, 78823, 83609, 89227, 92779

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 20 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049282, A049097, A039787, A075432, A153207, A153208, A153209, A153210, A153213, A153214.

<< NumberTheory`NumberTheoryFunctions` lst={}; Do[p=Prime[n];If[ !SquareFreeQ[p-1]&&!SquareFreeQ[p+1]&&!SquareFreeQ[p-2]&&!SquareFreeQ[p+2],AppendTo[lst,p]],{n,4*7!}]; lst

A318959 Primes p (> 2) such that p - 2 and p - 1 are nonsquarefree.

Original entry on oeis.org

29, 101, 127, 137, 149, 173, 277, 281, 317, 353, 389, 461, 509, 541, 569, 577, 641, 677, 727, 821, 857, 877, 929, 977, 1109, 1129, 1181, 1217, 1277, 1289, 1361, 1423, 1433, 1451, 1613, 1667, 1721, 1777, 1861, 1877, 1901, 1913, 1973, 2081, 2153, 2297, 2333, 2351

Offset: 1

Seiichi Manyama, Sep 06 2018

nonn

			21 (= 23 - 2) is squarefree. So 23 is not a term.
27 = 3^3 and 28 = 2^2*7. So 29 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A039787, A049231, A240473, A257545, A319049.

[p: p in PrimesInInterval(3, 2500)| not IsSquarefree(p-2) and not IsSquarefree(p-1)]; // Vincenzo Librandi, Sep 06 2018

Select[Prime[Range[500]], !SquareFreeQ[# - 2] && !SquareFreeQ[# - 1] &] (* Vincenzo Librandi, Sep 06 2018 *)

forprime(p=2, 1e4, if(!issquarefree(p-1)&&!issquarefree(p-2), print1(p, ", "))); \\ Altug Alkan, Sep 06 2018

A319049 Primes p such that none of p - 1, p - 2 and p - 3 are squarefree.

Original entry on oeis.org

101, 127, 353, 727, 1277, 1423, 1451, 1667, 2153, 2351, 2647, 3187, 3251, 3511, 3701, 3719, 3727, 4421, 4951, 5051, 5393, 5527, 6427, 6653, 6959, 7517, 7867, 8527, 9127, 9551, 9803, 9851, 10243, 10253, 10487, 10831, 11273, 11351, 11777, 11827, 12007, 12251, 12277

Offset: 1

Seiichi Manyama, Sep 08 2018

nonn

If p is a term, so that there are primes q,r,s such that q^2|p-3, r^2|p-2 and s^2|p-1, then the sequence includes all primes == p (mod q^2*r^2*s^2). In particular, the sequence is infinite, and a(n)/(n*log(n)) is bounded above and below by constants. - Robert Israel, Sep 09 2018

			98 = 2*7^2, 99 = 3^2*11 and 100 = 2^2*5^2. So 101 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A039787, A049231, A240473, A257545, A318959.

[p: p in PrimesUpTo(13000) | not IsSquarefree(p-1) and not IsSquarefree(p-2) and not IsSquarefree(p-3)]; // Vincenzo Librandi, Sep 17 2018

Res:= NULL: count:= 0:
p:= 1;
while count < 100 do
  p:= nextprime(p);
  if not ormap(numtheory:-issqrfree, [p-1,p-2,p-3]) then
    count:= count+1; Res:= Res, p
  fi
od:
Res; # Robert Israel, Sep 09 2018

Select[Prime[Range[2000]], !SquareFreeQ[# - 1] && !SquareFreeQ[# - 2] && !SquareFreeQ[# - 3]&] (* Jean-François Alcover, Sep 17 2018 *)
Select[Prime[Range[1500]],NoneTrue[#-{1,2,3},SquareFreeQ]&] (* Harvey P. Dale, Apr 11 2022 *)

isok(p) = isprime(p) && !issquarefree(p-1) && !issquarefree(p-2) && !issquarefree(p-3); \\ Michel Marcus, Sep 09 2018

A123269 Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ].

Original entry on oeis.org

1, 28, 7625731729896, 13407807929942597099574024998205985135931742965325158317510351105024878248924471298029103219186757034747676158536830429928105045387310278568778808509188348

Offset: 1

Alexander Adamchuk, Oct 09 2006

nonn

The next term is too large to include.

Prime p divides a(p) for p = {2, 3, 7, 11, 23, 31, 43, 47, 59, 67, 71, 79, ...} = A039787[n] Primes p such that p-1 is squarefree. p^2 divides a(p) for prime p = {2,3}.

Cf. A039787. Cf. A086787 - Sum[ i^j, {i, 1, n}, {j, 1, n} ].

Numbers n that divide a(n) are listed in A124391(n) = {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 21, 22, 23, 24, 27, 28, 31, ...}.

Table[Sum[i^j^k,{i,1,n},{j,1,n},{k,1,n}],{n,1,5}]

a(n)=sum(i=1,n,sum(j=1,n,sum(k=1,n,i^j^k))) \\ Charles R Greathouse IV, May 15 2013

a(n) = Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ].

A124391 Numbers m that divide A123269(m) = Sum_{i=1..m, j=1..m, k=1..m} i^j^k.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 40, 42, 43, 44, 46, 47, 48, 49, 54, 56, 59, 60, 62, 63, 64, 66, 67, 69, 71, 72, 77, 79, 80, 81, 83, 84, 86, 88, 92, 93, 94, 96, 98, 99, 100, 103

Offset: 1

Alexander Adamchuk, Oct 30 2006

A123269(m) = Sum_{i=1..m, j=1..m, k=1..m} i^j^k = {1, 28, 7625731729896, ...}.

Primes terms are listed in A039787.

Cf. A123269, A039787, A086787.

Do[f=Sum[Mod[Sum[Mod[Sum[PowerMod[i,j^k,n], {i, 1, n}],n], {j, 1, n}],n], {k, 1, n}];If[IntegerQ[f/n],Print[n]],{n,1,103}]

cp's OEIS Frontend

A075430 Primes with a squarefree neighbor.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

A049198 Numbers that are not squarefree and whose Euler totient function is squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A175625 Numbers k such that gcd(k, 6) = 1, 2^(k-1) == 1 (mod k), and 2^(k-3) == 1 (mod (k-1)/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A224718 Primes p such that p^2 + 1 is not squarefree.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A089188 Lesser member p of a pair of twin primes such that p-1 is squarefree.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A153215 Primes p such that none of p-2, p-1, p+1, and p+2 is squarefree.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A318959 Primes p (> 2) such that p - 2 and p - 1 are nonsquarefree.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A319049 Primes p such that none of p - 1, p - 2 and p - 3 are squarefree.

Views

Author

Keywords

Comments

Examples

Links