A049098 -id:A049098 - cp's OEIS frontend

A122870 Primes congruent to 3 or 7 mod 20.

3, 7, 23, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 263, 283, 307, 347, 367, 383, 443, 463, 467, 487, 503, 523, 547, 563, 587, 607, 643, 647, 683, 727, 743, 787, 823, 827, 863, 883, 887, 907, 947, 967, 983, 1063, 1087, 1103, 1123, 1163, 1187, 1223

Offset: 1

Alexander Adamchuk, Sep 16 2006

nonn

The old name was "Primes p that divide Lucas((p+1)/2) = A000032((p+1)/2)".

Note that F(p+1) = F((p+1)/2)*Lucas((p+1)/2), where F = A000045. Since gcd(F(n),Lucas(n)) = 1 or 2 (because Lucas(n)^2 - 5*F(n)^2 = 4*(-1)^n), this sequence (under the old definition above) lists primes p such that p divides F(p+1) but does not divides F((p+1)/2). By Propositions 1.1 and 1.2 (the k = 3 case) of my link below, this is primes p == 3, 7 (mod 20). - Jianing Song, Jun 20 2025

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jianing Song, Lucas sequences and entry point modulo p
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Gaussian Prime.

Cf. A000032, A000045, A122869, A216815.

Subseqeunce of A002145, A003631, A049098, A053027. Essentially the same as A106865.

[p: p in PrimesUpTo(1500) | p mod 20 in [3, 7]]; // Vincenzo Librandi, Jan 06 2013

Select[Prime[Range[1000]],IntegerQ[(Fibonacci[(#1+1)/2-1]+Fibonacci[(#1+1)/2+1])/#1]&]
Select[Prime[Range[300]], MemberQ[{3, 7}, Mod[#, 20]]&] (* Vincenzo Librandi, Jan 06 2013 *)

I merged A216816 into this entry at the suggestion of Jianing Song, Jun 20 2025. - N. J. A. Sloane, Jun 22 2025

A089199 Primes p such that p+1 is divisible by a cube.

Original entry on oeis.org

7, 23, 31, 47, 53, 71, 79, 103, 107, 127, 151, 167, 191, 199, 223, 239, 263, 269, 271, 311, 359, 367, 383, 431, 439, 463, 479, 487, 499, 503, 593, 599, 607, 631, 647, 701, 719, 727, 743, 751, 809, 823, 839, 863, 887, 911, 919, 967, 971, 983, 991

Offset: 1

Cino Hilliard, Dec 08 2003

This sequence is infinite and its relative density in the sequence of primes is equal to 1 - Product_{p prime} (1-1/(p^2*(p-1))) = 1 - A065414 = 0.302498... (Mirsky, 1949). - Amiram Eldar, Apr 07 2021

Robert Israel, Table of n, a(n) for n = 1..10000
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

Includes A007522 and A141965.

Cf. A049098, A065414.

filter:= proc(p)
  isprime(p) and ormap(t -> t[2]>=3, ifactors(p+1)[2])
end proc:
select(filter, [seq(i,i=3..2000,2)]); # Robert Israel, Jan 11 2019

f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p+1]>=3,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)

ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) }
powerfreep3(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(ispowerfree(x+k,p)==0, c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }

A160696 Largest k such that k^2 divides prime(n)+1.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 3, 2, 2, 1, 4, 1, 1, 2, 4, 3, 2, 1, 2, 6, 1, 4, 2, 3, 7, 1, 2, 6, 1, 1, 8, 2, 1, 2, 5, 2, 1, 2, 2, 1, 6, 1, 8, 1, 3, 10, 2, 4, 2, 1, 3, 4, 11, 6, 1, 2, 3, 4, 1, 1, 2, 7, 2, 2, 1, 1, 2, 13, 2, 5, 1, 6, 4, 1, 2, 8, 1, 1, 1, 1, 2, 1, 12, 1, 2, 2, 15, 1, 1, 4, 6, 4, 2, 2, 10, 6, 1, 3, 2, 1, 2, 3, 2

Offset: 1

Reinhard Zumkeller, May 24 2009

nonn

A160697 and A160698 give record values and where they occur.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000188, A008864, A049097, A049098.

a(n) = core(prime(n)+1, 1)[2]; \\ Michel Marcus, Nov 06 2022

a(A049097(n)) = 1; a(A049098(n)) > 1;

a(n) = A000188(A008864(n)).

A089495 a(n) = mu(prime(n)+1), where mu is the Moebius function.

Original entry on oeis.org

-1, 0, 1, 0, 0, 1, 0, 0, 0, -1, 0, 1, -1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 1, -1, -1, 0, 1, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, -1, 0, 0, -1, 0, 0

Offset: 1

T. D. Noe, Nov 04 2003

sign

Eric Weisstein's World of Mathematics, Moebius Function

Cf. A049098 (primes p such that mu(p+1) = 0), A078329 (primes p such that mu(p+1) = -1), A089523 (primes p such that mu(p+1) = 1), A089451 (mu(p-1) for prime p), A089496 (mu(p+1)+mu(p-1) for prime p), A089497 (mu(p+1)-mu(p-1) for prime p).

A067461(n) - 1.

Mathematica
```
Table[MoebiusMu[Prime[n]+1], {n, 150}]
```

A089523 Primes p such that mu(p+1) = 1; that is, p+1 is squarefree and has an even number of distinct prime factors, where mu is the Moebius function.

Original entry on oeis.org

5, 13, 37, 61, 73, 157, 193, 277, 313, 389, 397, 421, 457, 461, 509, 541, 569, 613, 661, 673, 733, 757, 769, 797, 857, 877, 929, 997, 1093, 1109, 1153, 1201, 1213, 1217, 1229, 1237, 1289, 1301, 1321, 1381, 1409, 1429, 1453, 1481, 1553, 1609, 1621, 1657

Offset: 1

T. D. Noe, Nov 06 2003

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Function

Cf. A089495 (mu(p+1) for prime p), A049098 (primes p with mu(p+1)=0), A078329 (primes p with mu(p+1)=-1).

[p: p in PrimesUpTo(2000) | MoebiusMu(p+1) eq 1]; // Vincenzo Librandi, Aug 17 2018

select(n -> isprime(n) and numtheory:-mobius(n+1)=1, [seq(i,i=1..2000,4)]); # Robert Israel, Aug 16 2018

Select[Prime[Range[300]], MoebiusMu[ #+1]==1&]

isok(p) = isprime(p) && (moebius(p+1) == 1); \\ Michel Marcus, Aug 16 2018

A319050 Primes p such that neither p + 1 nor p + 2 is squarefree.

Original entry on oeis.org

7, 23, 43, 47, 79, 97, 151, 167, 223, 241, 331, 349, 359, 367, 439, 523, 547, 619, 691, 727, 773, 823, 839, 907, 1051, 1087, 1123, 1223, 1231, 1249, 1303, 1367, 1423, 1447, 1483, 1523, 1571, 1627, 1663, 1699, 1723, 1811, 1823, 1847, 1861, 1879, 1951, 1987, 2131, 2203, 2207

Offset: 1

Seiichi Manyama, Sep 08 2018

nonn

			8 = 2^3 and 9 = 3^2. So 7 is a term.
24 = 2^3*3 and 25 = 5^2. So 23 is a term.

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

Cf. A000040, A049098, A257108, A318959, A319051.

Select[Prime[Range[400]],NoneTrue[#+{1,2},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 05 2019 *)

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

A319051 Primes p such that none of p + 1, p + 2 and p + 3 are squarefree.

Original entry on oeis.org

47, 97, 241, 349, 547, 773, 1249, 1447, 1663, 1847, 1861, 2347, 2887, 3049, 3547, 3607, 3623, 3697, 4111, 4373, 4597, 5237, 5273, 5749, 6173, 6857, 7549, 8467, 8647, 8719, 9161, 9349, 9547, 9749, 11149, 11321, 11447, 12049, 12473, 12689, 12823, 12941, 13147, 13291

Offset: 1

Seiichi Manyama, Sep 08 2018

nonn

			48 = 2^4*3, 49 = 7^2 and 50 = 2*5^2. So 47 is a term.
98 = 2*7^2, 99 = 3^2*11 and 100 = 2^2*5^2. So 97 is a term.

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

Cf. A000040, A049098, A257108, A319049, A319050.

forprime(p=2, 1e5, if(!issquarefree(p+1) && !issquarefree(p+2) && !issquarefree(p+3), print1(p", ")))

cp's OEIS Frontend

A122870 Primes congruent to 3 or 7 mod 20.

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A089199 Primes p such that p+1 is divisible by a cube.

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A160696 Largest k such that k^2 divides prime(n)+1.

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Formula

A089495 a(n) = mu(prime(n)+1), where mu is the Moebius function.

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A089523 Primes p such that mu(p+1) = 1; that is, p+1 is squarefree and has an even number of distinct prime factors, where mu is the Moebius function.

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A319050 Primes p such that neither p + 1 nor p + 2 is squarefree.

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A319051 Primes p such that none of p + 1, p + 2 and p + 3 are squarefree.

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PARI