A182140 -id:A182140 - cp's OEIS frontend

A182221 Composite numbers in A182140 but not in A071700.

255, 385, 34561, 65535, 147455, 195841, 1034881, 4070401, 4746241, 5040001, 16675201, 22704001, 36067201, 47013121, 83623935, 136967041, 168720001, 271878145, 549141119, 613092481, 836567041, 1039779841, 1049759999, 1548072961, 2556902401, 2646067201

Offset: 1

José María Grau Ribas, Apr 19 2012

nonn

A182140 includes the prime numbers and A071700.

J. M. Grau, A. M. Oller-Marcen, M. Rodríguez, D. Sadornil, Fermat test with gaussian base and Gaussian pseudoprimes, arXiv preprint arXiv:1401.4708 [math.NT], 2014.

Cf. A182140, A071700, A060968, A201629.

fa=FactorInteger;isA071700[n_]:=Length@fa[n]==2&&fa[n][[1,2]]==fa[n][[2,2]]==1&&Mod[Sqrt[n+1],4]==0; A060968[p_, s_] := Which[Mod[p, 4] == 1, p^(s-1)*(p-1), Mod[p, 4]==3, p^(s-1)*(p+1), s==1, 2, True, 2^(s+1)]; A060968[1]=1; A060968[n_] := Product[A060968[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}]; A201629[n_]:=Which[Mod[n, 4]==1, (n-1), Mod[n, 4]==3, (n+1), True, n];
Select[Range[10000], !isA071700[#]&&!PrimeQ[#]&&A060968[#]==A201629[#]&]

isA071700(n)=my(k=sqrtint(n\16)); n==16*k^2+32*k+15 && isprime(4*k+3) && isprime(4*k+5)
is(n)=if(isprime(n),return(0)); my(f=factor(n)[, 1]); n*prod(i=if(n%2, 1, 2), #f, if(f[i]%4==1, 1-1/f[i], 1+1/f[i]))*if(n%4, 1, 2)==if(n%2,(n+1)\4*4,n) && !isA071700(n) \\ Charles R Greathouse IV, Jul 03 2013

a(15)-a(23) from Charles R Greathouse IV, Jul 03 2013

a(24)-a(26) from Charles R Greathouse IV, Jul 05 2013

A071700 Product of twin primes of form (4k+3,4(k+1)+1), k>=0.

Original entry on oeis.org

15, 143, 3599, 5183, 11663, 32399, 36863, 51983, 57599, 97343, 121103, 176399, 186623, 359999, 435599, 685583, 1040399, 1065023, 1192463, 1327103, 1742399, 2039183, 2108303, 2214143, 2585663, 2624399, 2782223

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

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

Subsequence of A037074, A071700 and of A182140.

Cf. A071697, a(n) = A071698(n)*A071699(n).

a071700 n = a071700_list !! (n-1)
a071700_list = [x * y | x <- [3, 7 ..], a010051' x == 1,
                        let y = x + 2, a010051' y == 1]
-- Reinhard Zumkeller, Aug 05 2014

for(k=0,1e3,if(isprime(4*k+3)&&isprime(4*k+5),print1(16*k^2+32*k +15", "))) \\ Charles R Greathouse IV, Jul 03 2013

is(n)=my(k=sqrtint(n\16)); n==16*k^2+32*k+15 && isprime(4*k+3) && isprime(4*k+5) \\ Charles R Greathouse IV, Jul 03 2013

is(n)=my(t); n%16==15 && issquare(n+1,&t) && isprime(t-1) && isprime(t+1) \\ Charles R Greathouse IV, Dec 12 2016

list(lim)=my(v=List(),p=3); forprime(q=5,sqrtint(1+lim\1)+1, if(q-p==2 && p%4==3, listput(v,p*q)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 12 2016

a(n) >> n^2 log^4 n. - Charles R Greathouse IV, Jul 03 2013

A240960 Numbers m such that sigma(m) - phi(m) = tau(m)^omega(m), where sigma=A000203, phi=A000010, tau=A000005 and omega=A001221.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Reinhard Zumkeller, Aug 05 2014

nonn

a(n) = A182140(n) for n <= 35.

All primes p are in the sequence since (p+1) - (p-1) = 2^1. The first composites are 15, 119748396, 139254850, 187768485, 1420027536, 3991789984. A182140 seems unrelated. - Jens Kruse Andersen, Aug 05 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000203, A001221, A051612, A110088, A182140, A110086.

a240960 n = a240960_list !! (n-1)
a240960_list = filter (\x -> a051612 x == a110088 x) [1..]

with(numtheory):
filter:= n -> sigma(n) - phi(n) = tau(n)^nops(factorset(n)):
select(filter, [$1..1000]); # Robert Israel, Aug 05 2014

Select[Range[300], DivisorSigma[1, #] - EulerPhi[#] == DivisorSigma[0, #]^PrimeNu[#]&] (* Jean-François Alcover, Mar 08 2019 *)

is(n)=my(f=factor(n)); sigma(f)-eulerphi(f)==numdiv(f)^omega(f) \\ Charles R Greathouse IV, Nov 26 2014

from sympy import totient,divisors,divisor_count,primefactors
filter(lambda x:sum(divisors(x))-totient(x)==divisor_count(x)**len(primefactors(x)), range(1,10**5)) # Chai Wah Wu, Aug 05 2014

A235864 G-Lehmer numbers: Composite numbers k such that A060968(k) divides A201629(k).

Original entry on oeis.org

15, 143, 255, 385, 3599, 5183, 11663, 32399, 34561, 36863, 51983, 57599, 65535, 97343, 121103, 147455, 176399, 186623, 195841, 359999, 435599, 685583, 1034881, 1040399, 1065023, 1192463, 1327103, 1742399, 2039183, 2108303, 2214143, 2585663, 2624399, 2782223, 3196943

Offset: 1

José María Grau Ribas, Jan 16 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000
José María Grau, Antonio M. Oller-Marcén, Manuel Rodríguez, and Daniel Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, Czechoslovak Mathematical Journal, Vol. 65 (2015), pp. 969-982; arXiv preprint, arXiv:1401.4708 [math.NT], 2014.

Cf. A060968, A201629, A235863, A182140.

fa=FactorInteger; phi[p_, s_] := Which[Mod[p, 4] == 1, p^(s-1)*(p-1), Mod[p, 4]==3, p^(s-1)*(p+1), s==1, 2, True, 2^(s+1)]; phi[1]=1; phi[n_] := Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}]; Select[Range[1000], IntegerQ[FU[#]/phi[#]] && PrimeQ[#] == False &]

a(29)-a(35) from Amiram Eldar, Nov 24 2023

cp's OEIS Frontend

A182221 Composite numbers in A182140 but not in A071700.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A071700 Product of twin primes of form (4*k+3,4*(k+1)+1), k>=0.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

PARI

PARI

PARI

PARI

Formula

A240960 Numbers m such that sigma(m) - phi(m) = tau(m)^omega(m), where sigma=A000203, phi=A000010, tau=A000005 and omega=A001221.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

A235864 G-Lehmer numbers: Composite numbers k such that A060968(k) divides A201629(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A071700 Product of twin primes of form (4k+3,4(k+1)+1), k>=0.