A076201 -id:A076201 - cp's OEIS frontend

A108022 a(1)=2; a(n) is the smallest prime such that a(n)-a(n-1) is a 4th power (>0).

2, 3, 19, 160019, 1049920019, 1050730019, 1051540019, 12910750019, 13960510019, 14167870019, 67252030019, 67252840019, 67318450019, 196918450019, 197968210019, 568118770019, 568119580019, 938270140019, 938477500019

Offset: 1

John L. Drost, May 31 2005

nonn

All members after 19 will end in '0019'.

Also, for n > 3, a(n) - a(n - 1) = k^4, k is a multiple of 30. - Zak Seidov, Apr 09 2013

			a(3)=19, for 19 +k^4 to be prime, k must be even and divisible by 5. 19+10^4=10019=43*233,but 19+20^4 is prime.

Cf. A073609, A076201.

Join[{2,3,19,p=160019},Table[x=30;While[!PrimeQ[q=p+x^4],x=x+30];p=q,{19}]] (* Zak Seidov, Apr 09 2013 *)

A108023 a(1)=2; a(n) is the smallest prime such that a(n)-a(n-1) is a 6th power (>0).

Original entry on oeis.org

2, 3, 67, 131, 2176782467, 22485250805891, 132514367714796227, 132514373203827971, 1472610013828827971, 3552822265021773233027, 3552822910800868882883, 3552824349717606382019, 3552824349723095413763

Offset: 1

John L. Drost, May 31 2005

nonn

Since a(5) is 6 mod 7, all entries after a(5) are congruent to a(5) mod 14^6

			a(4)=131 which is 2 mod 3 so if 131 +k^6 is prime, k must be divisible by 6. 131+6^6 and 131+24^6 are divisible by 13, 131 +12^6 and 131+18^6 are divisible by 5, 131+30^6 is divisible by 41, 131+36^6 is prime.

Cf. A073609, A076201.

Corrected by T. D. Noe, Nov 15 2006

A246760 a(1) = 5; a(n) for n > 1 is the smallest prime > a(n-1) that differs from a(n-1) by a square.

Original entry on oeis.org

5, 41, 617, 653, 797, 941, 977, 1013, 1049, 1193, 1229, 1373, 1409, 1553, 1697, 1733, 1877, 1913, 1949, 2273, 2309, 2633, 2777, 3677, 3821, 4397, 4721, 5297, 5333, 5477, 5801, 6701, 6737, 8501, 8537, 8573, 8609, 8753, 11057, 11093, 13397, 13721, 13757, 13901, 18257, 18401, 19301, 20201, 21101, 22397, 22433, 22469, 22613, 22937, 22973, 23117, 24413, 24989

Offset: 1

Zak Seidov, Sep 02 2014

nonn

All terms are congruent to 5 mod 36.

For sequences of this type, once you get a(n) == 5, 11, 17, 23, 29, or 35 mod 36, all later terms stay in the same congruence class mod 36. Sequences in the same congruence class are likely to merge after a few terms. Thus with a(1) = 77 you get 77, 113, 149, 293, 617 and from then on it's the same as the present sequence. - Robert Israel, Sep 05 2014

			41 - 5 = 6^2, 617 - 41 = 24^2, 653 - 617 = 6^2.

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

Cf. A073609, A073770, A076201, A108022, A108023.

sps[n_]:=Module[{p=NextPrime[n]},While[!IntegerQ[Sqrt[p-n]],p= NextPrime[ p]];p]; NestList[sps,5,60] (* Harvey P. Dale, Jul 28 2016 *)

print1(p=5",");for(k=1,100,x=1;while(!isprime(q=p+36*x^2),x=x+1);print1(q",");p=q)

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A108022 a(1)=2; a(n) is the smallest prime such that a(n)-a(n-1) is a 4th power (>0).

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A108023 a(1)=2; a(n) is the smallest prime such that a(n)-a(n-1) is a 6th power (>0).

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A246760 a(1) = 5; a(n) for n > 1 is the smallest prime > a(n-1) that differs from a(n-1) by a square.

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