A077064 -id:A077064 - cp's OEIS frontend

A077067 Squarefree numbers of the form prime + 1.

3, 6, 14, 30, 38, 42, 62, 74, 102, 110, 114, 138, 158, 174, 182, 194, 230, 258, 278, 282, 314, 318, 354, 374, 390, 398, 402, 410, 422, 434, 458, 462, 510, 542, 570, 602, 614, 618, 642, 654, 662, 674, 678, 710, 734, 758, 762, 770, 798, 822, 830, 854, 858, 878

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

			A005117(28) = 42 = 2*3*7 is a term as 42 = A000040(13) + 1 = 41+1.

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

Cf. A000040, A005117, A008864, A049097, A077064, A077066.

Select[Prime[Range[200]]+1,SquareFreeQ] (* Harvey P. Dale, Aug 20 2017 *)

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

A077066(a(n)) = a(n).

a(n) = A049097(n)+1. - Zak Seidov, Aug 15 2006

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

A097946 a(n) = A008683(n)*A014197(n) where A008683 is the Moebius (or Mobius) function mu(n) and A014197 is the number of numbers m with Euler phi(m) = n.

Original entry on oeis.org

2, -3, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gerald McGarvey, Sep 04 2004

sign

For n < 93 and a(n) not 0, n = p - 1 where p is prime and therefore in A077064 (Squarefree numbers of form prime - 1.)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A008683, A014197, A077064, A097945.

A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))} \\ This function from M. F. Hasler, Oct 05 2009
A097946(n) = moebius(n)*A014197(n); \\ Antti Karttunen, Jul 19 2017

A145199 Nonsquarefree numbers k such that k+1 is prime.

Original entry on oeis.org

4, 12, 16, 18, 28, 36, 40, 52, 60, 72, 88, 96, 100, 108, 112, 126, 136, 148, 150, 156, 162, 172, 180, 192, 196, 198, 228, 232, 240, 250, 256, 268, 270, 276, 280, 292, 306, 312, 316, 336, 348, 352, 372, 378, 388, 396, 400, 408, 420, 432, 448, 456, 460, 486, 490

Offset: 1

Giovanni Teofilatto, Oct 04 2008

			4 is in the sequence because it is not squarefree and 5 is prime. - _Emeric Deutsch_, Oct 12 2008

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

Cf. A006093, A013929, A049092, A077064. Includes A152680.

[n: n in [1..5*10^2]| not IsSquarefree(n) and IsPrime(n+1)]; // Vincenzo Librandi, Dec 24 2015

with(numtheory): a:=proc(n) if issqrfree(n)=false and isprime(n+1)=true then n else end if end proc: seq(a(n),n=1..600); # Emeric Deutsch, Oct 12 2008
with(numtheory): a:=proc(k) if issqrfree(ithprime(k)-1)=false then ithprime(k)-1 else end if end proc: seq(a(k),k=1..110); # Emeric Deutsch, Oct 12 2008

Select[Prime[Range[120]]-1, !SquareFreeQ[ # ]&] (* T. D. Noe, Oct 06 2008 *)

is(n)=isprime(n+1) && !issquarefree(n) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A049092(n) - 1. - Amiram Eldar, Feb 10 2021

Corrected and extended by T. D. Noe, Emeric Deutsch and R. J. Mathar, Oct 05 2008

A146560 Squarefree semiprimes n such that n+1 is squarefree and n+2 is prime.

Original entry on oeis.org

21, 57, 65, 69, 77, 129, 177, 209, 221, 237, 309, 329, 365, 381, 417, 437, 497, 501, 545, 597, 681, 689, 717, 785, 905, 965, 1037, 1101, 1121, 1257, 1317, 1397, 1437, 1497, 1509, 1541, 1569, 1577, 1661, 1757, 1821, 1829, 1865, 1929, 1977, 1985, 2201, 2369, 2577, 2589, 2669, 2705, 2885, 2901

Offset: 1

Giovanni Teofilatto, Nov 01 2008

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

Cf. A077064, A146319.

Select[Range[3000],PrimeOmega[#]==2&&SquareFreeQ[#]&&SquareFreeQ[#+1]&&PrimeQ[#+2]&] (* Harvey P. Dale, Jan 20 2020 *)

(A006881 INTERSECT A040976) INTERSECT A005117

Definition corrected and more terms supplied by R. J. Mathar, Nov 02 2008

cp's OEIS Frontend

A077067 Squarefree numbers of the form prime + 1.

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A070195 Squarefree numbers sandwiched between a pair of twin primes.

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A097946 a(n) = A008683(n)*A014197(n) where A008683 is the Moebius (or Mobius) function mu(n) and A014197 is the number of numbers m with Euler phi(m) = n.

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A145199 Nonsquarefree numbers k such that k+1 is prime.

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A146560 Squarefree semiprimes n such that n+1 is squarefree and n+2 is prime.

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