cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A152180 Numbers n such that A073609(n+1)=nextprime(A073609(n)).

Original entry on oeis.org

1, 3, 145, 221, 667, 704, 882, 916, 985, 1153, 1391, 1392, 1503, 1524, 1704, 1774, 1796, 1849, 2080, 2111, 2118, 2134, 2182, 2223, 2241, 2244, 2261, 2441, 2484, 2536, 2554, 2692, 2760, 2879, 2906, 2920, 3002, 3024, 3062, 3069, 3081, 3093, 3263, 3270
Offset: 1

Views

Author

Zak Seidov, Nov 28 2008

Keywords

Crossrefs

Cf. A073609 a(n) = the smallest prime > a(n-1) such that a(n)-a(n-1) is a square, A152181 a(n)=A073609(A152180(n)).

A152181 Primes p such that q-p is a square, q=nextprime(p) and p and q are subsequent terms in A073609.

Original entry on oeis.org

2, 7, 97463, 217271, 766127, 834023, 1164287, 1204823, 1471583, 1785647, 2275067, 2275103, 2461187, 2495351, 2849087, 2979191, 3017423, 3125351, 3506483, 3554363, 3566063, 3594827, 3686411, 3823031, 3855287, 3855503, 3891287
Offset: 1

Views

Author

Zak Seidov, Nov 28 2008

Keywords

Comments

a(n)=A073609(A152180(n)).

Crossrefs

Cf. A073609 a(n) = the smallest prime > a(n-1) such that a(n)-a(n-1) is a square, A152180 Numbers n such that A073609(n+1)=nextprime(A073609(n)).

A179478 a(n) = sqrt(A073609(n+1)-A073609(n)).

Original entry on oeis.org

1, 2, 2, 6, 6, 12, 6, 18, 18, 6, 6, 6, 12, 12, 12, 6, 6, 6, 30, 30, 30, 18, 30, 6, 18, 6, 18, 24, 18, 12, 6, 18, 6, 24, 12, 48, 24, 6, 12, 30, 18, 12, 24, 18, 30, 6, 12, 30, 54, 12, 54, 6, 24, 30, 42, 18, 12, 36, 6, 6, 12, 24, 18, 12, 6, 6, 12, 12, 18, 24, 6, 12, 6, 12, 30, 48, 24, 24
Offset: 0

Views

Author

Zak Seidov, Jul 16 2010

Keywords

Comments

After 1,1,2, all terms are multiples of 6.

Crossrefs

Cf. A073609 a(0)=2, a(n) for n>0 is the smallest prime>a(n-1) that differs from a(n-1) by a square.

A076201 a(n), for n > 1, equals the least prime p such that p - a(n-1) is a cube, a(1)=2.

Original entry on oeis.org

2, 3, 11, 19, 83, 1811, 2027, 2243, 2251, 2467, 2531, 2539, 3539, 3547, 4547, 5059, 10891, 12619, 13619, 13627, 13691, 13907, 14419, 155027, 155539, 156539, 157051, 267643, 268643, 270371, 270379, 270443, 270451, 270667, 276499, 277499, 280243, 281243, 281251
Offset: 1

Views

Author

Zak Seidov, Nov 02 2002

Keywords

Examples

			a(2) = 3 because 3 - a(1) = 1^3.
a(3) = 11 because 11 - a(2) = 2^3, while neither 5 - 3 nor 7 - 3 is a cube.

Links

Crossrefs

Cf. A073609.

Programs

  • Mathematica
    p = 3; s = Join[{2, 3}, Table[x = 2; While[!PrimeQ[q = p + x^3], x = x + 2]; p = q, {29}]] (* Zak Seidov, Apr 08 2013 *)
nxt[a_]:=Module[{p=NextPrime[a]},While[!IntegerQ[CubeRoot[p-a]],p=NextPrime[p]];p]; NestList[nxt,2,40] (* Harvey P. Dale, Aug 13 2025 *)

A215761 Numbers m with property that 36m+11 is prime.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 12, 13, 16, 18, 23, 25, 26, 27, 28, 30, 32, 36, 40, 41, 42, 43, 46, 50, 51, 56, 57, 58, 61, 62, 65, 67, 68, 70, 75, 78, 81, 82, 83, 90, 92, 93, 96, 98, 103, 107, 111, 113, 118, 126, 127, 130, 133, 135, 137, 140, 141, 145, 147, 152, 153, 155
Offset: 1

Views

Author

Zak Seidov, Aug 23 2012

Keywords

Comments

36m+11 is a term in A073609.

Links

Crossrefs

Cf. A073609.

Programs

  • Mathematica
    Select[Range[0,200],PrimeQ[36#+11]&] (* Harvey P. Dale, Oct 09 2018 *)
  • PARI
    isok(n) = isprime(36*n+11); \\ Michel Marcus, Oct 19 2013

Formula

a(n) = (A073609(n+2)-11)/36 (note offset 0 in A073609).

A073770 Primes p not of the form q + s where q is prime and s > 0 is the smallest square such that q + s is prime.

Original entry on oeis.org

2, 5, 13, 19, 29, 31, 37, 43, 61, 67, 73, 79, 103, 109, 127, 139, 151, 157, 163, 179, 181, 191, 193, 199, 211, 223, 229, 241, 271, 277, 283, 313, 331, 337, 349, 359, 367, 373, 379, 397, 409, 421, 431, 433, 439, 463, 487, 499, 521, 523, 541, 547, 569, 571, 577
Offset: 1

Views

Author

Klaus Brockhaus, Aug 08 2002

Keywords

Comments

A065377 is a subsequence of this sequence. Except for the initial term 2 this sequence is disjoint to A073609.

Examples

			13 cannot be partitioned into a prime and a square > 0, so 13 is a term. The only partition of 19 into a prime and a square > 0 is (3,16), but 16 is not the smallest square s such that 3 + s is prime since 3 + 4 = 7 is also prime; therefore 19 is a term.

Crossrefs

Cf. A065377, A073609.

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

Original entry on oeis.org

2, 3, 19, 160019, 1049920019, 1050730019, 1051540019, 12910750019, 13960510019, 14167870019, 67252030019, 67252840019, 67318450019, 196918450019, 197968210019, 568118770019, 568119580019, 938270140019, 938477500019
Offset: 1

Views

Author

John L. Drost, May 31 2005

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A073609, A076201.

Programs

  • Mathematica
    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

Views

Author

John L. Drost, May 31 2005

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A073609, A076201.

Extensions

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

Views

Author

Zak Seidov, Sep 02 2014

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

A249807 a(0) = 1; afterwards a(n) is the smallest positive square that added to all previous terms produces a prime.

Original entry on oeis.org

1, 1, 1, 4, 4, 36, 36, 144, 36, 324, 324, 36, 36, 36, 144, 144, 144, 36, 36, 36, 900, 900, 900, 324, 900, 36, 324, 36, 324, 576, 324, 144, 36, 324, 36, 576, 144, 2304, 576, 36, 144, 900, 324, 144, 576, 324, 900, 36, 144, 900, 2916, 144, 2916, 36, 576, 900, 1764, 324, 144, 1296, 36, 36
Offset: 0

Views

Author

Zak Seidov, Nov 06 2014

Keywords

Comments

All terms starting with a(5) are multiples of 36.
a(n) exists for all n under the Hardy-Littlewood Conjecture F. - Charles R Greathouse IV, Nov 06 2014

Examples

			1+1+1+4=7(prime), 7+4=11(prime), 11+36=47(prime), 47+36=83(prime).

Links

Crossrefs

Cf. A073609.

Programs

  • Mathematica
    nxt[{t_,a_}]:=Module[{k=1},While[!PrimeQ[t+k^2],k++];{t+k^2,k^2}]; NestList[nxt,{1,1},70][[;;,2]] (* Harvey P. Dale, Jul 28 2023 *)
  • PARI
    first(n)=n=max(n,5); my(v=vector(n+1,i,1),k,s=11); v[4]=v[5]=4; for(i=6,#v, k=6; while(!isprime(s+k^2), k+=6); s+=v[i]=k^2); v \\ Charles R Greathouse IV, Nov 06 2014
Showing 1-10 of 10 results.

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