A077067 -id:A077067 - cp's OEIS frontend

A049097 Primes p such that p+1 is squarefree.

2, 5, 13, 29, 37, 41, 61, 73, 101, 109, 113, 137, 157, 173, 181, 193, 229, 257, 277, 281, 313, 317, 353, 373, 389, 397, 401, 409, 421, 433, 457, 461, 509, 541, 569, 601, 613, 617, 641, 653, 661, 673, 677, 709, 733, 757, 761, 769, 797, 821, 829, 853, 857

Offset: 1

Labos Elemer

nonn

Numbers k such that core(sigma(k)) = k + 1 where core(k) is the squarefree part of k (A007913). - Benoit Cloitre, May 01 2002

This sequence is infinite and its relative density in the sequence of primes is equal to Artin's constant (A005596): Product_{p prime} (1-1/(p*(p-1))) = 0.373955... (Mirsky, 1949). - Amiram Eldar, Dec 29 2020

			29 is included since 29 + 1 = 30 = 2*3*5 is squarefree.
17 is not here because 18 is divisible by a square, 9.

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.

Cf. A000040, A005117, A039787.

Cf. A000203, A005596, A007913, A077067, A160696.

[ p: p in PrimesUpTo(900) | IsSquarefree(p+1) ]; // Vincenzo Librandi, Dec 25 2010

N:= 10000; # to get all entries up to N
A049097:= select(t -> isprime(t) and numtheory:-issqrfree(t+1), [2, seq(1+2*k,k=1..floor((N-1)/2))]); # Robert Israel, May 11 2014

Select[Prime[Range[100]], SquareFreeQ[# + 1] &] (* Zak Seidov, Feb 08 2016 *)

lista(nn) = forprime(p=1, nn, if (issquarefree(p+1), print1(p, ", "))); \\ Michel Marcus, Jan 08 2015

A160696(a(n)) = 1. - Reinhard Zumkeller, May 24 2009

a(n) = A077067(n)-1. - Zak Seidov, Mar 19 2016

A077068 Semiprimes of the form prime + 1.

Original entry on oeis.org

4, 6, 14, 38, 62, 74, 158, 194, 278, 314, 398, 422, 458, 542, 614, 662, 674, 734, 758, 878, 998, 1094, 1154, 1202, 1214, 1238, 1322, 1382, 1454, 1622, 1658, 1754, 1874, 1934, 1994, 2018, 2138, 2342, 2474, 2558, 2594, 2798, 2858, 2918, 3062, 3218, 3254

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

a(n) = A005383(n)+1 = 2*A005382(n).
There are 672 semiprimes of form prime+1 below 100000.
a(n) = A232342(n) + A077065(n). - Reinhard Zumkeller, Dec 16 2013

			A001358(25)=74=2*37 is a term as 74=A000040(21)+1=73+1.

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

Cf. A008864, A001358, A000040, A077065, A077067.

a077068 n = a077068_list !! (n-1)
a077068_list = filter ((== 1) . a010051 . (`div` 2)) a008864_list
-- Reinhard Zumkeller, Nov 22 2013

Select[Range[6000],Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#-1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[ Prime@ Range@ 460 +1,PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

[x+1|x<-primes(10^5),bigomega(x+1)==2] \\ Charles R Greathouse IV, Nov 22 2013

is(n)=isprime(n-1) && isprime(n\2) \\ Charles R Greathouse IV, Mar 20 2016

A010051(A008864(n)/2) = A064911(A008864(n)) = 1. - Reinhard Zumkeller, Nov 22 2013

A077064 Squarefree numbers of form prime - 1.

Original entry on oeis.org

1, 2, 6, 10, 22, 30, 42, 46, 58, 66, 70, 78, 82, 102, 106, 130, 138, 166, 178, 190, 210, 222, 226, 238, 262, 282, 310, 330, 346, 358, 366, 382, 418, 430, 438, 442, 462, 466, 478, 498, 502, 546, 562, 570, 586, 598, 606, 618, 642, 646, 658, 682, 690, 718, 742

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

This sequence is infinite and its relative density in the sequence of primes is equal to Artin's constant (A005596): Product_{p prime} (1-1/(p*(p-1))) = 0.373955... (Victorovich, 2013). - Amiram Eldar, Dec 29 2020

			A005117(44) = 70 = 2*5*7 is a term as 70 = A000040(20)-1 = 71-1.

Harvey P. Dale, Table of n, a(n) for n = 1..10000
Radoslav Tsvetkov, On the distribution of k-free numbers and r-tuples of k-free numbers. A survey, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 3 (2019), pp. 207-222. See section 3.4, p. 210.
G. D. Victorovich, On additive property of arithmetic functions (in Russian), Thesis, Moscow State University, 2013.

Cf. A005596, A006093, A005117, A000040, A077063, A077067.

Select[Prime[Range[200]]-1,SquareFreeQ] (* Harvey P. Dale, Feb 09 2015 *)

isok(n) = issquarefree(n) && isprime(n+1); \\ Michel Marcus, Mar 22 2016

lista(nn) = forprime(p=2, nn, if (issquarefree(p-1), print1(p-1, ", "))); \\ Michel Marcus, Mar 22 2016

Wrong formula removed by Amiram Eldar, Dec 29 2020

A070195 Squarefree numbers sandwiched between a pair of twin primes.

Original entry on oeis.org

6, 30, 42, 102, 138, 282, 462, 570, 618, 642, 822, 858, 1230, 1290, 1302, 1482, 1698, 1722, 1878, 2082, 2130, 2238, 2310, 2382, 2658, 2730, 2802, 3390, 3462, 3558, 3918, 3930, 4002, 4218, 4242, 4422, 4638, 4722, 5010, 5442, 5478, 5502, 5658, 6090, 6198

Offset: 1

Benoit Cloitre and Labos Elemer, May 06 2002

Numbers k such that k is squarefree, k-1 and k+1 are primes.
Intersection of A005117 and A014574. - Michel Marcus, Mar 06 2014
Also, intersection of A077064 and A077067. - Zak Seidov, Mar 20 2016
All terms are multiples of 6. - Zak Seidov, Mar 20 2016
All terms == 6 (mod 12). - Robert Israel, Mar 21 2016

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

Cf. A005117, A014574, A077064, A077067. - Zak Seidov, Mar 20 2016

select(t -> numtheory:-issqrfree(t) and isprime(t+1) and isprime(t-1),
[seq(i, i=6..10000, 12)]); # Robert Israel, Mar 21 2016

Select[12 * Range[0, 500] + 6, PrimeQ[#-1] && PrimeQ[#+1] && SquareFreeQ[#] &] (* Amiram Eldar, May 23 2022 *)

{forstep(n=6,8000,12,if(issquarefree(n)&&isprime(n-1)&&isprime(n+1),print1(n",")));} \\ Zak Seidov, Mar 20 2016

A077066 Squarefree kernel of prime(n) + 1.

Original entry on oeis.org

3, 2, 6, 2, 6, 14, 6, 10, 6, 30, 2, 38, 42, 22, 6, 6, 30, 62, 34, 6, 74, 10, 42, 30, 14, 102, 26, 6, 110, 114, 2, 66, 138, 70, 30, 38, 158, 82, 42, 174, 30, 182, 6, 194, 66, 10, 106, 14, 114, 230, 78, 30, 22, 42, 258, 66, 30, 34, 278, 282, 142, 42, 154, 78, 314, 318, 166

Offset: 1

Reinhard Zumkeller, Oct 23 2002

			a(25) = rad(prime(25)+1) = rad(97+1) = rad(2*7^2) = 14.

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

Cf. A007947, A008864, A077063, A077067.

a077066 = a007947 . a008864  -- Reinhard Zumkeller, Sep 04 2013

a[n_] := Times @@ FactorInteger[Prime[n] + 1][[;;, 1]]; Array[a, 100] (* Amiram Eldar, Apr 10 2025 *)

a(n)=my(f=factor(prime(n)+1)[,1]); prod(i=1,#f,f[i]) \\ Charles R Greathouse IV, Aug 21 2013

a(n) = A007947(A008864(n)).

a(A077067(n)) = A077067(n).

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A049097 Primes p such that p+1 is squarefree.

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A077068 Semiprimes of the form prime + 1.

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A077064 Squarefree numbers of form prime - 1.

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A070195 Squarefree numbers sandwiched between a pair of twin primes.

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A077066 Squarefree kernel of prime(n) + 1.

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