A109080 -id:A109080 - cp's OEIS frontend

A074844 Largest difference between consecutive divisors of n is equal to the sum of divisors of n except 1 and n.

4, 345, 6489, 88473

Offset: 1

Jason Earls, Sep 10 2002

No other term < 600000. - Emeric Deutsch, Aug 04 2005
No more terms < 10^9. - Lars Blomberg, Jun 04 2013
If p = 5^k - 2 is a prime > 3, then 3*p*(p+2)/5 is in this sequence (see A109080). - Charlie Neder, Oct 13 2018
a(5) > 10^13. - Giovanni Resta, Feb 15 2020

			The divisors of 345 are [1, 3, 5, 15, 23, 69, 115, 345] and the largest difference between consecutive divisors is 345-115 = 230; the sum of divisors except 1 and 345 are 3+5+15+23+69+115 = 230.

Cf. A048050, A060681.

with(numtheory): a:=proc(n) local div: div:=divisors(n): if max(seq(div[j]-div[j-1],j=2..tau(n)))=sigma(n)-1-n then n else fi end: seq(a(n),n=1..100000); # Emeric Deutsch, Aug 04 2005

More terms from Emeric Deutsch, Aug 04 2005

A133856 Least number k > (2n-1) such that (2n-1)^k - 2 is prime, or 0 if no such number exists.

Original entry on oeis.org

0, 4, 14, 8, 11, 22420, 78, 17, 24, 20, 25, 24, 63, 30, 42, 69, 128, 50, 119, 204, 2816, 76, 52, 288, 64, 66, 184, 153, 67, 268, 78, 210, 438, 295, 96, 74, 136, 128, 2900, 1898, 130, 92, 381, 106, 18626, 97, 98, 1650, 747, 109, 214, 113, 312, 354, 1702, 560, 2798, 123, 171, 554, 11210, 834, 208, 990, 9271

Offset: 1

Alexander Adamchuk, Oct 01 2007

nonn

a(66) > 40000. - Robert Price, Mar 02 2015

Henri Lifchitz & Renaud Lifchitz: PRP Records. Probable Primes Top 10000.

Cf. A128472 (smallest prime of the form (2n-1)^k - 2 for k > (2n-1), or 0 if no such number exists).
Cf. A084714 (smallest prime of the form (2n-1)^k - 2, or 0 if no such number exists).
Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.
Cf. A133858, A133982.

A128472(n) = (2n-1)^a(n) - 2 for n > 1.

a(6) = 22420 was found by Rick L. Shepherd, Sep 29 2009
a(21)-a(44) from Max Alekseyev, Oct 04 2007
a(45)-a(65) from Robert Price, Mar 02 2015

A155897 Square matrix T(m,n)=1 if (2m+1)^n-2 is prime, 0 otherwise; read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

M. F. Hasler, Feb 01 2009

In some sense a "minimal" possible generalization of the pattern of Mersenne primes (cf. A000043) is to consider powers of odd numbers (> 1) minus 2. Since even powers obviously correspond to an odd power of the base squared, it is sufficient to consider only odd powers, cf. A155899.

Cf. A084714, A128472, A094786, A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.

T = matrix( 19,19,m,n, isprime((2*m+1)^n-2)) ;
A155897 = concat( vector( vecmin( matsize(T)), i, vector( i, j, T[j,i-j+1])))

A155898 Square matrix T(m,n)=1 if (2m+1)^(2n)-2 is prime, 0 otherwise; read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 1

M. F. Hasler, Feb 01 2009

In some sense the "minimal" possible generalization of the pattern of Mersenne primes (cf. A000043) is to consider powers of odd numbers minus 2. Here only even powers are considered (which obviously correspond to an odd power of the base squared).

Cf. A084714, A128472, A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.

T = matrix( 19,19,m,n, isprime((2*m+1)^(2*n)-2)) ;
A155898 = concat( vector( vecmin( matsize(T)), i, vector( i, j, T[j,i-j+1])))

A204578 Primes of the form 5^k-2.

Original entry on oeis.org

3, 23, 6103515623, 1490116119384765623, 88817841970012523233890533447265623, 11754943508222875079687365372222456778186655567720875215087517062784172594547271728515623, 44841550858394146269559346665277316200968382140048504696226185084473314645947539247572422027587890623

Offset: 1

M. F. Hasler, Jan 30 2012

See the sequence A109080 for the corresponding exponents k.

The number a(3) = 6103515623 is also in A095304, A104090 and A128472.

Cf. A109080.

for(i=0,999, ispseudoprime(t=5^i-2) & print1(t","))

a(n) = 5^A109080(n)-2.

A217134 Numbers n such that 5^n - 8 is prime.

Original entry on oeis.org

2, 4, 10, 14, 88, 112, 140, 764, 3040, 11096, 24934, 25616, 54584, 93400

Offset: 1

Vincenzo Librandi, Oct 01 2012

nonn

a(15) > 10^5. - Robert Price, Feb 03 2014

Cf. A059613, A087885, A089142, A109080, A124621, A165701.

Select[Range[2, 5000], PrimeQ[5^# - 8] &]

for(n=2, 5*10^3, if(isprime(5^n-8), print1(n", ")))

a(10)-a(14) from Robert Price, Feb 03 2014

A248546 Numbers k such that 75^k - 2 is prime.

Original entry on oeis.org

1, 2, 25, 32, 62, 128, 848, 2091, 2882, 11761, 25915

Offset: 1

Vaclav Kotesovec, Oct 08 2014

Dedicated to N. J. A. Sloane for his 75th birthday!

Cf. A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A376850.

[n: n in [0..200] | IsPrime(75^n-2)]; // Vincenzo Librandi, Oct 08 2014

Mathematica
```
Select[Range[1000],PrimeQ[75^#-2]&]
```

is(n)=ispseudoprime(75^n-2) \\ Charles R Greathouse IV, Jun 13 2017

a(10) from Michael S. Branicky, Apr 02 2023

a(11) from Michael S. Branicky, Oct 10 2024

A258043 Smallest k such that prime(k)^n - 2 is prime.

Original entry on oeis.org

3, 1, 8, 2, 2, 2, 4, 4, 2, 16, 193, 4, 8, 3, 4, 21, 11, 18, 8, 8, 11, 2, 8, 7, 70, 3, 95, 4, 172, 7, 4, 94, 143, 90, 193, 17, 2, 8, 46, 41, 2, 10, 254, 90, 74, 75, 371, 85, 70, 3, 177, 53, 85, 91, 18, 24, 84, 103, 34, 95, 34, 111, 80, 253, 84, 224, 397, 1002, 11, 33, 773, 29, 647, 20

Offset: 1

Juri-Stepan Gerasimov, May 22 2015

nonn

Primes of the form prime(n)^n - 2: 3, 7, 61, 67, 71, 73, 127, ...

			a(1) = 3 because prime(3)^1 - 2 = 3 and 3 is prime,
a(2) = 1 because prime(1)^2 - 2 = 2 and 2 is prime,
a(3) = 8 because prime(8)^3 - 2 = 6857 and 6857 is prime.

Cf. A014224, A014232, A109080, A204578.

a(n) = my(k = 1); while(! isprime(prime(k)^n-2), k++); k; \\ Michel Marcus, May 23 2015

A359695 Numbers k such that 29^k - 2 is prime.

Original entry on oeis.org

2, 4, 8, 14, 42, 420, 1344

Offset: 1

Arsen Vardanyan, Mar 07 2023

a(8) > 10^4, if it exists. - Amiram Eldar, Mar 10 2023
All terms in this sequence are even. - Yifan Xie, Mar 12 2023
a(8) > 5*10^4, if it exists. - Michael S. Branicky, Sep 14 2024

			4 is a term because 29^4 - 2 = 707279 is a prime number.

Cf. A087886 (29^k + 2 is prime).

Cf. A128460, A128459, A128457, A109076, A090669, A105772, A109080, (and similar others).

Select[Range[1400], PrimeQ[29^# - 2] &] (* Amiram Eldar, Mar 10 2023 *)

PARI
```
is(k) = isprime(29^k - 2);
```

A378868 Numbers k such that 5^k - 22 is prime.

Original entry on oeis.org

2, 3, 31, 79, 491, 3019, 3623, 4175, 9957, 21963, 71637, 80551, 80831

Offset: 1

Robert Price, Dec 09 2024

a(14) > 10^5. - Michael S. Branicky, Dec 24 2024

			3 is a term because 5^3 - 22 = 103 is prime.

Cf. A059613, A109080, A165701, A217134.

Magma
```
[k: k in [0..1000] |IsPrime (5^k-22)];
```
Mathematica
```
Select[Range[0,5000],PrimeQ[5^#-22]&]
```

a(8)-a(10) from Michael S. Branicky, Dec 17 2024

a(11)-a(13) from Michael S. Branicky, Dec 22 2024

cp's OEIS Frontend

A074844 Largest difference between consecutive divisors of n is equal to the sum of divisors of n except 1 and n.

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A133856 Least number k > (2n-1) such that (2n-1)^k - 2 is prime, or 0 if no such number exists.

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A155897 Square matrix T(m,n)=1 if (2m+1)^n-2 is prime, 0 otherwise; read by antidiagonals.

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A155898 Square matrix T(m,n)=1 if (2m+1)^(2n)-2 is prime, 0 otherwise; read by antidiagonals.

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A204578 Primes of the form 5^k-2.

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A217134 Numbers n such that 5^n - 8 is prime.

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A248546 Numbers k such that 75^k - 2 is prime.

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Magma

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Extensions

A258043 Smallest k such that prime(k)^n - 2 is prime.

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PARI

A359695 Numbers k such that 29^k - 2 is prime.

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