A194593 -id:A194593 - cp's OEIS frontend

A077065 Semiprimes of form prime - 1.

4, 6, 10, 22, 46, 58, 82, 106, 166, 178, 226, 262, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 886, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2026, 2038, 2062, 2098, 2206, 2446, 2458, 2578, 2818, 2878, 2902

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

There are 670 semiprimes of form prime-1 below 10^5.

			A001358(16) = 46 = 2*23 is a term as 46 = A000040(15) - 1 = 47 - 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Intersection of A006093 and A001358.
Intersection of A006093 and A100484.
Cf. A077068, A232342, A064911.
Cf. A005384, A005385.
Cf. A000010, A023900, A194593.

a077065 n = a077065_list !! (n-1)
a077065_list = filter ((== 1) . a010051' . (`div` 2)) a006093_list
-- Reinhard Zumkeller, Nov 22 2013, Oct 27 2012

IsSemiprime:=func; [s: n in [2..500] | IsSemiprime(s) where s is NthPrime(n)-1]; // Vincenzo Librandi, Oct 17 2012

q:= n-> (n::even) and andmap(isprime, [n+1, n/2]):
select(q, [$1..5000])[];  # Alois P. Heinz, Jul 19 2023

Select[Range[6000],Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#+1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[Range[3000],PrimeOmega[#]==2&&PrimeQ[#+1]&] (* Harvey P. Dale, Oct 16 2012 *)
Select[ Prime@ Range@ 430 - 1, PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

[x-1|x<-primes(10^4),bigomega(x-1)==2] \\ Charles R Greathouse IV, Nov 22 2013

a(n) = A005385(n) - 1 = 2*A005384(n).
A010051(A006093(a(n))/2) = A064911(A006093(a(n))) = 1. - Reinhard Zumkeller, Nov 22 2013
a(n) = A077068(n) - A232342(n). - Reinhard Zumkeller, Dec 16 2013
a(n) = A000010(A194593(n+1)). - Torlach Rush, Aug 23 2018
A000010((a(n)*2)+2) = A023900((a(n)*2)+2). - Torlach Rush, Aug 23 2018

A176045 Numbers n such that n-1 and 2*n-1 are both prime.

Original entry on oeis.org

3, 4, 6, 12, 24, 30, 42, 54, 84, 90, 114, 132, 174, 180, 192, 234, 240, 252, 282, 294, 360, 420, 432, 444, 492, 510, 594, 642, 654, 660, 684, 720, 744, 762, 810, 912, 954, 1014, 1020, 1032, 1050, 1104, 1224, 1230, 1290, 1410, 1440, 1452, 1482, 1500, 1512, 1560

Offset: 1

Michel Lagneau, Apr 07 2010

nonn

Also numbers n such that all eigenvalues of the n X n matrix M_n defined in A176043 are prime. The eigenvalues are 2*n-1, and n-1 with multiplicity n-1.

a(n)^2 = p^2 + q, where both p and q are primes. These are the only squares of this form, and which always yields q > p with a(n) - 1 = p = A005384(n) and 2*a(n) - 1 = q = A005385(n), for the same n. Also: a(n) = q - p; p + q + a(n) = 2q = A194593(n+1); and p*q = A156592 - Richard R. Forberg, Mar 04 2015

			6-1 = 5 and 2*6-1 = 11 are both prime, so 6 is in the sequence. 7-1 = 6 and 2*7-1 = 13 are not both prime, so 7 is not in the sequence.
p = 3, q = 7; p^2 + q = 16, a(n) = sqrt(16) = 4. - _Richard R. Forberg_, Mar 04 2015

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

Cf. A176043, A005384 (Sophie Germain primes), A005385 (Safe Primes), A124485 (2*n-1 and 4*n-1 are prime).

[ n: n in [2..1600] | IsPrime(n-1) and IsPrime(2*n-1) ]; // Klaus Brockhaus, Apr 19 2010

with(numtheory):for n from 2 to 2000 do:if type((2*n-1),prime)=true and type((n-1),prime)=true then print(n):else fi:od:

Select[Prime[Range[250]],PrimeQ[2#+1]&]+1 (* Harvey P. Dale, Jul 31 2013 *)

isok(n) = isprime(n-1) && isprime(2*n-1); \\ Michel Marcus, Apr 06 2016

a(n) = A005384(n)+1.

a(n) = 2*A124485(n-1) for n > 1.

Edited and 1482 inserted by Klaus Brockhaus, Apr 19 2010

A320391 Numbers k such that phi(k - 2) = phi(k) - 2.

Original entry on oeis.org

5, 7, 8, 13, 14, 16, 19, 20, 22, 31, 43, 46, 61, 64, 73, 94, 103, 109, 118, 139, 151, 166, 181, 193, 199, 214, 229, 241, 256, 271, 283, 313, 334, 349, 358, 421, 433, 454, 463, 523, 526, 571, 601, 619, 643, 661, 694, 718, 766, 811, 823, 829, 859, 883, 934, 958

Offset: 1

Vincenzo Librandi, Oct 13 2018

nonn

			7 is in the sequence because phi(5) = 4 = phi(7) - 2.
8 is in the sequence because phi(6) = 2 = phi(8) - 2.
9 is not in the sequence because phi(7) = 6 but phi(9) - 2 = 4 instead.

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

Cf. A001838. Contains A006512 and terms > 10 in A194593.

Filtered([1..960],k->Phi(k-2)=Phi(k)-2); # Muniru A Asiru, Oct 28 2018

[n: n in [3..1000] | EulerPhi(n-2) eq EulerPhi(n)-2];

with(numtheory): select(k->phi(k-2)=phi(k)-2,[$1..960]); # Muniru A Asiru, Oct 28 2018

Select[Range@1000, EulerPhi@(# - 2) == EulerPhi[#] - 2 &]
Flatten[Position[Partition[EulerPhi[Range[1000]],3,1],?(#[[1]]==#[[3]]-2&),1,Heads->False]]+2 (* _Harvey P. Dale, Oct 24 2020 *)

isok(n) = eulerphi(n-2) == eulerphi(n)-2; \\ Michel Marcus, Oct 14 2018

a(n) = A001838(n)+2. - Robert Israel, Oct 30 2018

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A077065 Semiprimes of form prime - 1.

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A176045 Numbers n such that n-1 and 2*n-1 are both prime.

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A320391 Numbers k such that phi(k - 2) = phi(k) - 2.

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