A001771 -id:A001771 - cp's OEIS frontend

A011774 Nonprimes k that divide sigma(k) + phi(k).

1, 312, 560, 588, 1400, 23760, 59400, 85632, 147492, 153720, 556160, 569328, 1590816, 2013216, 3343776, 4563000, 4695456, 9745728, 12558912, 22013952, 23336172, 30002960, 45326160, 52021242, 75007400, 113315400, 137617728, 153587720, 402831360, 699117024

Offset: 1

R. K. Guy

2*k = sigma(k) + phi(k) if and only if k is 1 or a prime.
If 7*2^j - 1 is prime then m = 2^(j+2)*3*(7*2^j - 1) is in the sequence. Because phi(m) = 2^(j+2)*(7*2^j - 2); sigma(m) = 7*2^(j+2)*(2^(j+3) - 1) so phi(m) + sigma(m) = 2^(j+2)*((7*2^j - 2) + (7*2^(j+3) - 7)) = 2^(j+2)* (63*2^(j+2) - 9) = 3*(2^(j+2)*3*(7*2^j - 1)) = 3*m, hence m is a term of A011251 and consequently m is a term of this sequence. A112729 gives such m's. - Farideh Firoozbakht, Dec 01 2005
Conjecture: For n > 1, a(n) is a Zumkeller number (A083207). Verified for all n in [2,63]. - Ivan N. Ianakiev, Jan 25 2023

			a(26) = 113315400: sigma = 426535200, phi = 26726400, quotient = 4.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B42, p. 151.
Zhang Ming-Zhi, typescript submitted to Unsolved Problems section of Monthly, 96-01-10.

Giovanni Resta, Table of n, a(n) for n = 1..63 (terms < 10^13; first 53 terms from Donovan Johnson)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Q.-X. Jin and M. Tang, The 4-Nicol Numbers Having Five Different Prime Divisors, J. Int. Seq. 14 (2011) # 11.7.2.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A065387, A011251, A011254, A055681, A001771, A112729.

Do[If[Mod[DivisorSigma[1, n]+EulerPhi[n], n]==0, Print[n]], {n, 1, 2*10^7}]
Do[ If[ ! PrimeQ[n] && Mod[ DivisorSigma[1, n] + EulerPhi[n], n] == 0, Print[n] ], {n, 1, 10^8} ]

sp(n)=my(f=factor(n));n*prod(i=1, #f[,1], 1-1/f[i,1]) + prod(i=1, #f[,1], (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))
p=2;forprime(q=3, 1e6, for(n=p+1, q-1, if(sp(n)%n==0, print1(n", ")));p=q) \\ Charles R Greathouse IV, Mar 19 2012

More terms from David W. Wilson

Corrected by Labos Elemer, Feb 12 2004

A268061 Numbers k such that 7*8^k - 1 is prime.

Original entry on oeis.org

3, 7, 15, 59, 6127, 8703, 11619, 23403, 124299

Offset: 1

Robert Price, Jan 25 2016

a(10) > 2*10^5.

Terms are A001771(n)/3 that are integers.

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. similar sequences of the form k*(k+1)^n-1: A003307 (k=2), ... (k=3), A046865 (k=4), A079906 (k=5), A046866 (k=6), this sequence (k=7), ... (k=8), A056725 (k=9), A046867 (k=10), A079907 (k=11).

Cf. A001771, A002235, A005541, A247260.

Select[Range[0, 200000], PrimeQ[7*8^# - 1] &]

lista(nn) = for(n=1, nn, if(ispseudoprime(7*8^n-1), print1(n, ", "))) \\ Altug Alkan, Jan 25 2016

A238797 Smallest k such that 2^k - (2n+1) and (2n+1)2^k - 1 are both prime, k <= 2n+1, or 0 if no such k exists.

Original entry on oeis.org

0, 3, 4, 0, 0, 0, 0, 5, 6, 5, 7, 6, 9, 5, 0, 7, 6, 6, 0, 0, 10, 0, 6, 0, 7, 9, 6, 7, 8, 0, 17, 8, 0, 0, 7, 0, 0, 18, 0, 0, 0, 8, 0, 10, 8, 9, 18, 0, 0, 7, 0, 0, 8, 12, 0, 7, 0, 11, 16, 0, 21, 0, 0, 0, 8, 14, 0, 0, 18, 9, 10, 8, 77, 0, 0, 0, 12, 8, 0, 11, 18, 0

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 05 2014

nonn

Numbers n such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime:
For k = 0: 2, 3, 5, 7, 13, 17, ...   Intersection of A000043 and A000043
for k = 1: 3, 4, 6, 94, ...          Intersection of A050414 and A002235
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 3:                           Intersection of A059609 and A001771
for k = 4: 21, ...                   Intersection of A059610 and A002236
for k = 5:                           Intersection of A096817 and A001772
for k = 6:                           Intersection of A096818 and A001773
for k = 7: 5, 10, 14, ...            Intersection of A059612 and A002237
for k = 8: 6, 16, 20, 36, ...        Intersection of A059611 and A001774
for k = 9: 5, 21, ...                Intersection of A096819 and A001775
for k = 10: 7, 13, ...               Intersection of A096820 and A002238
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...

			a(1) = 3 because 2^3 - (2*1+1) = 5 and (2*1+1)*2^3 - 1 = 23 are both prime, 3 = 2*1+1,
a(2) = 4 because 2^4 - (2*2+1) = 11 and (2*2+1)*2^4 - 1 = 79 are both prime, 4 < 2*2+1 = 5.

Giovanni Resta, Table of n, a(n) for n = 0..1000

Cf. A238748, A238904 (smallest k such that 2^k + (2n+1) and (2n+1)*2^k + 1 are both prime, k <= n, or -1 if no such k exists).

a[n_] := Catch@ Block[{k = 1}, While[k <= 2*n+1, If[2^k - (2*n + 1) > 0 && PrimeQ[2^k - (2*n+1)] && PrimeQ[(2*n + 1)*2^k-1], Throw@k]; k++]; 0]; a/@ Range[0, 80] (* Giovanni Resta, Mar 15 2014 *)

a(0), a(19), a(20) corrected by Giovanni Resta, Mar 13 2014

A245241 Integers n such that 6 * 7^n + 1 is prime.

Original entry on oeis.org

0, 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572

Offset: 1

Robert Price, Nov 14 2014

All terms correspond to verified primes, that is, not merely probable primes.

a(14) > 2*10^5.

			4 is in this sequence because 6 * 7^4 + 1 = 14407, which is prime.

Cf. A003307, A002235, A046865, A079906, A001771, A005541, A056725, A046867, A079907.

Cf. A005537, A005538, A005539, A216889, A216890, A247260.

Select[Range[0,200000], PrimeQ[6 * 7^# + 1] &]

A112729 Numbers of the form 2^(k+2)3(72^k-1) where 72^k-1 is prime.

Original entry on oeis.org

312, 85632, 22013952, 1443107438592, 369435881766912, 24211351590301335552, 103986963299971520879061368832

Offset: 1

Farideh Firoozbakht, Dec 01 2005

This sequence is a subsequence of A011251 and A011774, namely if m is in the sequence then phi(m)+sigma(m)=3*m (see Comments line of A011251).

Number of digits of all the 20 known terms of this sequence are respectively 3, 5, 8, 13, 15, 20, 30, 109, 11069, 13566, 14787, 15722, 20988, 25263, 40594, 42272, 101802, 104453, 107155 and 219110.

			103986963299971520879061368832 is in the sequence because 103986963299971520879061368832=2^(45+2)*3*(7*2^45-1) and 7*2^45-1 is prime.

Cf. A001771. A011251, A011774.

Do[If[PrimeQ[7*2^n-1], Print[3*2^(n+2)*(7*2^n-1)]], {n, 177}]

A050523 Primes of the form 7*2^k - 1.

Original entry on oeis.org

13, 223, 3583, 917503, 14680063, 3758096383, 246290604621823, 1340933598257652751063553648756520535666396731910651903

Offset: 1

N. J. A. Sloane, Dec 29 1999

Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime.

See A001771 for more terms.

Cf. A086224, A158795 [Vincenzo Librandi, Mar 28 2009]

Select[Table[7*2^n-1, {n,0,200}], PrimeQ] (* Jean-François Alcover, Mar 20 2011 *)

a(n) = A086224(A001771(n)). - Elmo R. Oliveira, Apr 22 2025

13 inserted by R. J. Mathar, Apr 01 2009

A238749 Exponents of third Mersenne prime pair: numbers n such that 2^n - 5 and 5*2^n - 1 are both prime.

Original entry on oeis.org

4, 8, 10, 12, 18, 32

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 04 2014

a(7) > 350028.
Intersection of A059608 and A001770.
Exponents of Mersenne prime pairs {2^n - (2k + 1), (2k + 1)*2^n - 1}:
for k = 0: 2, 3, 5, 7, 13, 17, ...   Intersection of A000043 and A000043
for k = 1: 3, 4, 6, 94, ...          Intersection of A050414 and A002235
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 3:                           Intersection of A059609 and A001771
for k = 4: 21, ...                   Intersection of A059610 and A002236
for k = 5:                           Intersection of A096817 and A001772
for k = 6:                           Intersection of A096818 and A001773
for k = 7: 5, 10, 14, ...            Intersection of A059612 and A002237
for k = 8: 6, 16, 20, 36, ...        Intersection of A059611 and A001774
for k = 9: 5, 21, ...                Intersection of A096819 and A001775
for k = 10: 7, 13, ...               Intersection of A096820 and A002238
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...
for k = 14:

			a(1) = 4 because 2^4 - 5 = 11 and 5*2^4 - 1 = 79 are both primes.

Cf. A000043, A001770, A059608, A237422, A238694, A238251, A238751, A238797.

[n: n in [0..100] | IsPrime(2^n-5) and IsPrime(5*2^n-1)]; // Vincenzo Librandi, May 17 2015

fQ[n_] := PrimeQ[2^n - 5] && PrimeQ[5*2^n - 1]; k = 1; While[ k < 15001, If[fQ@ k, Print@ k]; k++] (* Robert G. Wilson v, Mar 05 2014 *)
Select[Range[1000], PrimeQ[2^# - 5] && PrimeQ[5 2^# - 1] &] (* Vincenzo Librandi, May 17 2015 *)

isok(n) = isprime(2^n - 5) && isprime(5*2^n - 1); \\ Michel Marcus, Mar 04 2014

A377248 Numbers k such that 8191 * 2^k + 1 is prime.

Original entry on oeis.org

12, 20, 412, 712, 2092, 4704, 10176, 33396, 41124, 105604, 139780, 142924

Offset: 1

Arsen Vardanyan, Oct 21 2024

8191 is the 5th Mersenne prime: 8191 = 2^13 - 1 (a term of A000668).

			12 is a term, because 8191 * 2^12 + 1 = 8191 * 4096 + 1 = 33550337 is prime. (also a term of A061644).

Cf. A000668, A002253, A002254, A001771, A061644.

PARI
```
is(k) = isprime(8191 * 2^k + 1);
```

a(8)-a(9) from Hugo Pfoertner, Oct 21 2024

a(10)-a(12) from Michael S. Branicky, Nov 05 2024

cp's OEIS Frontend

A011774 Nonprimes k that divide sigma(k) + phi(k).

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A268061 Numbers k such that 7*8^k - 1 is prime.

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A238797 Smallest k such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime, k <= 2*n+1, or 0 if no such k exists.

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A245241 Integers n such that 6 * 7^n + 1 is prime.

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A112729 Numbers of the form 2^(k+2)*3*(7*2^k-1) where 7*2^k-1 is prime.

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A050523 Primes of the form 7*2^k - 1.

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A238749 Exponents of third Mersenne prime pair: numbers n such that 2^n - 5 and 5*2^n - 1 are both prime.

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Magma

Mathematica

PARI

A377248 Numbers k such that 8191 * 2^k + 1 is prime.

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Author

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Comments

A238797 Smallest k such that 2^k - (2n+1) and (2n+1)2^k - 1 are both prime, k <= 2n+1, or 0 if no such k exists.

A112729 Numbers of the form 2^(k+2)3(72^k-1) where 72^k-1 is prime.