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.

Previous Showing 11-20 of 20 results.

A214633 a(1)=2; a(n) is smallest prime of the form k*a(n-1) + 3, k>0.

Original entry on oeis.org

2, 5, 13, 29, 61, 491, 3931, 15727, 157273, 314549, 4403689, 17614759, 387524701, 5425345817, 119357607977, 9787323854117, 78298590832939, 1722568998324661, 68902759932986443, 4685387675443078127, 318606361930129312639, 637212723860258625281
Offset: 1

Views

Author

Robin Garcia, Jul 23 2012

Keywords

Links

Crossrefs

Cf. A061092, A059411, A214523, A214632.

Programs

  • Maple
    A214633 := proc(n)
    option remember;
    local k;
    if n =  1 then
        2;
    else
        for k from 1 do
            if isprime(k*procname(n-1)+3) then
                return k*procname(n-1)+3 ;
            end if;
        end do:
    end if;
end proc:
seq(A214633(n),n=1..20) ; # R. J. Mathar, Jul 23 2012

A071258 a(1) = 4; a(n) = smallest composite number of form k*a(n-1) + 1 with k > 1.

Original entry on oeis.org

4, 9, 28, 57, 115, 231, 694, 1389, 2779, 5559, 16678, 33357, 66715, 133431, 400294, 800589, 1601179, 3202359, 9607078, 19214157, 38428315, 76856631, 153713263, 307426527, 614853055, 1229706111, 3689118334, 7378236669, 14756473339
Offset: 0

Views

Author

Amarnath Murthy, May 30 2002

Keywords

Links

Crossrefs

Cf. A061766, A061092, A059411.

Programs

  • Mathematica
    nxt[n_]:=Module[{k=2},While[PrimeQ[k*n+1],k++];k*n+1]; NestList[nxt,4,30] (* Harvey P. Dale, Mar 26 2014 *)

Extensions

More terms from Vladeta Jovovic, Jun 03 2002

A075341 a(1) = 1, a(2n) is the smallest composite number == 1 mod (a(2n-1)) and a(2n+1) is the smallest prime == 1 (mod a(2n)).

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 17, 18, 19, 20, 41, 42, 43, 44, 89, 90, 181, 182, 547, 548, 1097, 1098, 7687, 7688, 15377, 15378, 30757, 30758, 276823, 276824, 553649, 553650, 2768251, 2768252, 8304757, 8304758, 99657097, 99657098, 199314197, 199314198
Offset: 1

Views

Author

Amarnath Murthy, Sep 18 2002

Keywords

Comments

a(k) = b(k-5) for k > 8 where b(r) is the r-th term of A075340.

Crossrefs

Cf. A075340.
See also: Always look for prime: A061092. Always look for composite: A061766.

Programs

  • Mathematica
    a[1] = 1; a[2] = 4; a[n_] := a[n] = Block[{k = a[n - 1] + 1, m = a[n - 1]}, If[OddQ@n, While[ !PrimeQ@k || Mod[k, m] != 1, k += m]; k, While[PrimeQ@k || Mod[k, m] != 1, k += m]; k]]; Array[a, 40] (* Robert G. Wilson v Sep 21 2006 *)

Extensions

More terms from David Wasserman, Jan 16 2005

A085065 a(1) = 4, a(n) = smallest number of the form k*a(n-1) +1 with the same prime signature p^2, where p is a prime.

Original entry on oeis.org

4, 9, 289, 332929, 8867310888979, 78629202401805543263662441, 24730205881376410453776804740727484868533062121664161
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 29 2003

Keywords

Comments

a(8) = (28*A(7)+1)^2, which has 108 digits and is too large to include. - David Wasserman, Jan 11 2005

Crossrefs

A061092 gives the sequence with a(1) = 2 i.e. terms of the form k*a(n-1)+1 which are primes.( prime signature p^1)
Cf. A061092.

Extensions

More terms from David Wasserman, Jan 11 2005

A085066 a(1) = 6, a(n) = smallest number of the form k*a(n-1) +1 with the same prime signature p*q (6 = 2*3), where p and q are primes.

Original entry on oeis.org

6, 55, 111, 334, 335, 671, 1343, 16117, 16118, 64473, 64474, 257897, 2063177, 8252709, 41263546, 123790639, 371371918, 1485487673, 2970975347, 59419506941, 356517041647, 5704272666353, 11408545332707, 262396542652262
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 29 2003

Keywords

Comments

Is this the same as A085067? - R. J. Mathar, Aug 28 2025

Crossrefs

A061092 gives the sequence with a(1) = 2 i.e. terms of the form k*a(n-1)+1 which are primes.( prime signature p^1)
Cf. A061092, A085065.

Extensions

Corrected and extended by David Wasserman, Jan 11 2005

A214634 a(1) = 7; a(n) is smallest prime of the form k*a(n-1) + 3, k>0.

Original entry on oeis.org

7, 17, 37, 151, 607, 1217, 2437, 4877, 39019, 78041, 624331, 6243313, 174812767, 1398502139, 19579029949, 39158059901, 1957902995053, 15663223960427, 156632239604273, 3132644792085463, 181693397940956857, 726773591763827431, 7267735917638274313, 1148302274986847341457, 4593209099947389365831
Offset: 1

Views

Author

Robin Garcia, Jul 23 2012

Keywords

Examples

			a(2) = 17 = 2 * 7 + 3.
a(3) = 37 = 2 * 17 + 3.
a(4) = 151 = 4 * 37 + 3.

Links

Crossrefs

Cf. A061092, A059411, A214523, A214632, A214633

Programs

  • Maple
    A214634 := proc(n)
    option remember;
    local k;
    if n =  1 then
        7;
    else
        for k from 1 do
            if isprime(k*procname(n-1)+3) then
                return k*procname(n-1)+3 ;
            end if;
        end do:
    end if;
end proc:
seq(A214634(n),n=1..20) ; # R. J. Mathar, Jul 23 2012
  • Mathematica
    spf[n_]:=Module[{k=1},While[!PrimeQ[k*n+3],k++];k*n+3]; NestList[spf,7,25] (* Harvey P. Dale, Aug 02 2017 *)
  • PARI
    a=7;for(n=1,200,b=a*n+3;if(isprime(b),a=b;print1(a,", ");next(n=1)))

Extensions

More terms from Robert Israel, Nov 23 2016

A214680 a(1)=3; a(n) is the smallest prime of the form k*a(n-1) + 2.

Original entry on oeis.org

3, 5, 7, 23, 71, 73, 367, 1103, 7723, 131293, 3807499, 19037497, 57112493, 1427812327, 15705935599, 141353420393, 989473942753, 44526327423887, 311684291967211, 4675264379508167, 4675264379508169, 275840598390981973, 4137608975864729597
Offset: 1

Views

Author

Robin Garcia, Jul 25 2012

Keywords

Comments

Up to the end of the b-file, there are only 4 pairs of twin primes in the sequence, with lesser twin primes 3, 5, 71 and 4675264379508167. - Editors, Feb 20 2018

Links

Crossrefs

Cf. A061092, A059411, A214523.

Programs

  • Mathematica
    t = {3}; Do[k = 1; While[p = k*t[[-1]] + 2; ! PrimeQ[p], k++]; AppendTo[t, p], {25}]; t (* T. D. Noe, Jul 26 2012 *)

A359340 The primes associated with A339174.

Original entry on oeis.org

2, 3, 7, 43, 3613, 65250781, 38318979202732621, 8810065002836730577256726488782121, 6131762382982476362788562753503495060507087787406616806191258317645081
Offset: 1

Views

Author

Jeppe Stig Nielsen, Dec 27 2022

Keywords

Comments

a(1)=2; for n > 0, a(n+1) is the first prime of the form k*(a(n) - 1)*a(n) + 1. It exists by Dirichlet's theorem on arithmetic progressions.
It is simple to reconstruct a(n) from A339174, which has the more compact representation.

Crossrefs

Cf. A061092, A071580, A339174.

Programs

  • PARI
    p=2; k=1; print1(p, ", "); while(1, runningP=k*(p-1)*p+1; if(ispseudoprime(runningP), k=1; p=runningP; print1(p, ", ") , k++))

A105050 a(0)=2, a(n) is the smallest prime of the form a(n-1)*k^2 + 1.

Original entry on oeis.org

2, 3, 13, 53, 30529, 122117, 17584849, 70339397, 22789964629, 11030342880437, 9927308592393301, 91490075987496662017, 23421459452799145476353, 918214896387537699254943013, 528891780319221714770847175489, 8665362928750128574805560123211777, 311953065435004628693000164435623973
Offset: 0

Views

Author

John L. Drost, Apr 04 2005

Keywords

Examples

			a(5) = 30529 = 53*24^2 + 1 is the smallest prime of the form 53*k^2 + 1.

Crossrefs

Cf. A061092, A093490.

Programs

  • PARI
    lista(nn) = my(k, p=2); print1(p); for(n=1, nn, k=1; while(!isprime(p*k^2+1), k++); print1(", ", p=p*k^2+1)); \\ Jinyuan Wang, Aug 02 2021

Extensions

a(15)-a(16) from Jinyuan Wang, Aug 02 2021

A162278 a(0)=1. For n >= 1, a(n) = the smallest prime either of the form a(n-1)*k - 1 or of the form a(n-1)*k + 1, for some k >= 2.

Original entry on oeis.org

1, 2, 3, 5, 11, 23, 47, 281, 563, 2251, 22511, 225109, 450217, 2701301, 10805203, 43220813, 2333923901, 18671391209, 560141736269, 5601417362689, 22405669450757, 448113389015141, 3584907112121129, 186415169830298707
Offset: 0

Views

Author

Leroy Quet, Jun 29 2009

Keywords

Crossrefs

A061092, A072532

Extensions

Extended by R. J. Mathar, Jul 04 2009
Previous Showing 11-20 of 20 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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