A055496 -id:A055496 - cp's OEIS frontend

A175627 a(1) = 1; a(n) is the smallest square > 2*a(n-1).

1, 4, 9, 25, 64, 144, 289, 625, 1296, 2601, 5329, 10816, 21904, 44100, 88209, 177241, 355216, 710649, 1423249, 2849344, 5702544, 11410884, 22829284, 45670564, 91355364, 182736324, 365497924, 730999369, 1462068169, 2924213776, 5848578576, 11697287716, 23394620209, 46789583481

Offset: 1

Ctibor O. Zizka, Dec 04 2010

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

Cf. A055496.

NestList[(Floor[Sqrt[2#]]+1)^2&,1,40] (* Harvey P. Dale, Jul 26 2017 *)

{a=1; print1(a, ", "); for(i=1, 40, a=(ceil(sqrt(2*a)))^2; print1(a, ", "))} /* Zak Seidov, Dec 04 2010 */

a(n) = (ceiling(sqrt(2*a(n-1))))^2.

lim_{n->oo} (a(n+1)/a(n)) = 2.

A207890 a(0)=1; for n>=1,- the minimal increasing sequence, such that, for n>=1, the row sums of Pascal-like triangle with left side {1,1,1,...} and right side {a(0), a(1), a(2),...} form an increasing sequence of primes.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 11, 14, 17, 20, 29, 44, 55, 66, 69, 72, 77, 86, 149, 152, 167, 172, 183, 198, 229, 230, 233, 254, 267, 276, 285, 316, 355, 370, 377, 402, 423, 458, 469, 478, 517, 570, 623, 704, 725, 730, 753, 762, 801, 818, 839, 858, 861, 938, 943, 982

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Feb 21 2012

			Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....2.....2
.3..|..1.....3.....4.....3
.4..|..1.....4.....7.....7.....4
.5..|..1.....5....11....14....11.....5
.6..|..1.....6....16....25....25....16.....8
.7..|..1.....7....22....41....50....41....24.....11
.8..|
The row sums for n >= 1 form sequence A055496.

Cf. A055496.

rows={{1},{1,1}}; Table[(x=Flatten[{1,2 MovingAverage[rows[[n]],2]}]; sx=Apply[Plus,x]; z=NextPrime[sx,NestWhile[#+1&,1,NextPrime[sx,#]-sxA207890=Map[Last[#]&,rows]

A083005 a(1)=1, a(n)=2a(n-1)+1 if a(n-1) is prime, a(n)=a(n-1)+1 otherwise.

Original entry on oeis.org

1, 2, 5, 11, 23, 47, 95, 96, 97, 195, 196, 197, 395, 396, 397, 795, 796, 797, 1595, 1596, 1597, 3195, 3196, 3197, 3198, 3199, 3200, 3201, 3202, 3203, 6407, 6408, 6409, 6410, 6411, 6412, 6413, 6414, 6415, 6416, 6417, 6418, 6419, 6420, 6421, 12843, 12844

Offset: 1

Benoit Cloitre, May 31 2003

nonn

Primes in this sequence are A055496.- Robert Israel, May 23 2017

Robert Israel, Table of n, a(n) for n = 1..10197

Cf. A055496.

P[0]:= 0:
for n from 2 to 20 do P[n]:= nextprime(2*P[n-1]) od:
seq($2*P[i]+1..P[i+1],i=0..19); # Robert Israel, May 23 2017

NestList[If[PrimeQ[#],2#+1,#+1]&,1,50] (* Harvey P. Dale, Jul 02 2021 *)

Conjecture : limit n ->oo log(a(n))/sqrt(n)=c= 1.3....

Offset corrected by Robert Israel, May 23 2017

A118476 a(0) = 1; a(n) is least k with n prime factors and k > n*a(n-1).

Original entry on oeis.org

1, 2, 6, 20, 81, 408, 2480, 17376, 139040, 1251450, 12514816, 137663064, 1651956992, 21475443200, 300656206080, 4509843098112, 72157489576704, 1226677322842112, 22080191811166208, 419523644412176256, 8390472888243683328, 176199930653117513728

Offset: 0

Jonathan Vos Post, May 04 2006

This is a super-polynomial function, as for positive n, a(n) > n!.

Prime factors counted with multiplicity. - Harvey P. Dale, Aug 25 2019

			a(1) = 2 because 2 is the smallest prime (integer with 1 prime factor) greater than 1 * 1 = 1.
a(2) = 6 because 6 = 2 * 3 is the smallest semiprime (integer with 2 prime factors) greater than 2 * 2 = 4.
a(3) = 20 because 20 = 2^2*5 is the smallest 3-almost prime (integer with 3 prime factors) greater than 3 * 6 = 18.

Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000040, A001358, A055496, A076656.

A118476 := proc(n) option remember; local k; if n = 0 then 1; else for k from n*procname(n-1)+1 do if numtheory[bigomega](k) = n then return k; end if; end do: end if; end proc:
seq(A118476(n),n=0..14) ; # R. J. Mathar, Dec 22 2010

lkpf[{n_,a_}]:=Module[{k=a(n+1)+1},While[PrimeOmega[k]!=n+1,k++];{n+1,k}]; NestList[lkpf,{0,1},21][[All,2]] (* Harvey P. Dale, Aug 25 2019 *)

a(0) = 1; a(n) least n-almost prime > n*a(n-1).

Terms corrected from a(4) on by R. J. Mathar, Dec 22 2010

a(15)-a(21) from Donovan Johnson, Jan 06 2011

A118912 a(1) = 2; a(n) is greatest prime < a(n-1)^4.

Original entry on oeis.org

2, 13, 28559, 665230244078823349, 195833931687186822327230545227550596864953022841534058316595001440791433

Offset: 1

Jonathan Vos Post, May 05 2006

Exponent-4 analog of A059785 a(n+1)=prevprime(a(n)^2), with exponent 3 being A118910 a(1) = 2; a(n) is greatest prime < a(n-1)^3.

			a(1) = 2, by definition.
a(2) = 13 = 2^4 - 3.
a(3) = 28559 = 13^4 - 2.
a(4) = 665230244078823349 = 28559^4 - 12.
a(5) = 195833931687186822327230545227550596864953022841534058316595001440791433 = 665230244078823349^4 - 168.
a(6) is too large to include.

Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908-A118913.

NestList[NextPrime[#^4,-1]&,2,5] (* Harvey P. Dale, Feb 18 2025 *)

a(1) = 2; a(n) is greatest prime < a(n-1)^4.

A156321 a(1)=0, a(n+1) is smallest nonprime > 2*a(n).

Original entry on oeis.org

0, 1, 4, 9, 20, 42, 85, 171, 343, 687, 1375, 2751, 5504, 11009, 22019, 44039, 88080, 176162, 352325, 704651, 1409303, 2818607, 5637215, 11274431, 22548863, 45097727, 90195455, 180390911, 360781823, 721563647, 1443127295, 2886254591

Offset: 1

Juri-Stepan Gerasimov, Feb 08 2009

nonn

			a(5) is the smallest nonprime > 2*a(4)=2*9=18, hence a(5)=20.

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

Cf. A000027, A055496, A141468.

np[n_]:=Module[{k=2n+1},If[PrimeQ[k],k+1,k]]; NestList[np,0,40] (* Harvey P. Dale, May 23 2012 *)

More terms from R. J. Mathar, Feb 10 2009

A156324 a(1)=0, a(n+1) is smallest nonprime >= a(n)+n.

Original entry on oeis.org

0, 1, 4, 8, 12, 18, 24, 32, 40, 49, 60, 72, 84, 98, 112, 128, 144, 161, 180, 200, 220, 242, 264, 287, 312, 338, 364, 391, 420, 450, 480, 511, 543, 576, 610, 645, 681, 718, 756, 795, 835, 876, 918, 961, 1005, 1050, 1096, 1143, 1191, 1240, 1290, 1341, 1393, 1446

Offset: 1

Juri-Stepan Gerasimov, Feb 08 2009

nonn

			a(4) is the smallest nonprime >= a(3) + 3 = 4 + 3 = 7, hence a(4)=8.

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

Cf. A000040, A055496, A141468.

A141468 := proc(n) option remember ; local a ; if n = 1 then 0 ; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) ; fi; od: fi; end: A156324 := proc(n) option remember ; local a; if n = 1 then 0; else for a from procname(n-1)+n-1 do if not isprime(a) then RETURN(a) ; fi; od: fi; end: seq(A156324(n),n=1..80) ; # R. J. Mathar, Feb 10 2009

nxt[{n_,a_}]:=Module[{k=0,c=a+n+1},While[PrimeQ[c+k],k++];{n+1,c+k}]; Transpose[NestList[nxt,{0,0},60]][[2]] (* Harvey P. Dale, Dec 29 2015 *)

Terms beginning at a(31) corrected by R. J. Mathar, Feb 10 2009

A164962 a(n) is the least prime from the union {2,3} and A164333, beginning with which the n-th prime p_n is obtained by some number of iterations of the S operator g(see A164960).

Original entry on oeis.org

2, 3, 2, 3, 2, 13, 3, 19, 2, 13, 31, 3, 19, 43, 2, 53, 13, 61, 31, 71, 73, 3, 19, 43, 2, 101, 103, 53, 109, 113, 13, 131, 31

Offset: 1

Vladimir Shevelev, Sep 02 2009

The sequence is connected with our sieve selecting the primes of the union {2,3} and A164333 from all primes. Note that a(n)=n iff p_n is in the considered union, which corresponds to 0's iterations of g.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A164333, A055496, A164960, A104272, A080359, A164918, A006992, A164917, A164368, A164288.

A175953 Let a(1)=1; for n>1 a(n)=nextprime(a(n-1)+(a(n-1)+1)/4).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 17, 23, 29, 37, 47, 59, 79, 101, 127, 163, 211, 269, 337, 431, 541, 677, 853, 1069, 1361, 1709, 2137, 2677, 3347, 4201, 5261, 6577, 8231, 10289, 12889, 16127, 20161, 25219, 31531, 39419, 49277, 61603, 77017, 96281, 120371, 150473

Offset: 1

Juri-Stepan Gerasimov, Oct 29 2010

The following definition of nextprime(q) is used: if q is an integer and prime, nextprime(q)=q. If q is an integer and composite or rational, nextprime(q) is the smallest prime >q. [R. J. Mathar, Oct 30 2010]

Cf. A008578, A055496, A089571, A162336.

nprime := proc(n) if type(n,'integer') then if isprime(n) then return n; else return nextprime(n) ; end if; else return nextprime(floor(n)) ; end if; end proc:
A175953 := proc(n) option remember; if n= 1 then 1; else p := procname(n-1)+(procname(n-1)+1)/4 ; return nprime(p) ; end if; end proc:
seq(A175953(n),n=1..120) ; # R. J. Mathar, Oct 30 2010

More terms from R. J. Mathar, Oct 30 2010

A117880 a(1) = 4; a(n) is smallest semiprime > 2*a(n-1).

Original entry on oeis.org

4, 9, 21, 46, 93, 187, 377, 755, 1513, 3027, 6059, 12127, 24257, 48529, 97059, 194127, 388257, 776515, 1553033, 3106083, 6212177, 12424355, 24848723, 49697447, 99394909, 198789819, 397579639, 795159283, 1590318573, 3180637153

Offset: 1

Jonathan Vos Post, May 04 2006

a(1)=4, a(n)=2*a(n-1)+k, where k is least positive integer chosen so that a(n) is the product of two primes. Corresponding k's are 1, 3, 4, 1, 1, 3, 1, 3, 1, 5, 9, 3, 15, 1, 9, 3, 1, 3, 17, 11, 1, 13, 1, 15, 1, 1, 5, 7, 7, 11, 5, 5, 15, 1, 3, 9, 9, 5, 7, 8, ... - Zak Seidov, Dec 24 2007

			a(1)=4, then
k=1, a(2)=2*4+1=9,
k=3, a(3)=2*9+3=21,
k=4, a(4)=2*21+4=46,
k=1, a(5)=2*46+1=93,
k=1, a(6)=2*93+1=187.

Semiprime analog of A055496.

Cf. A001358, A076656.

a=0;Do[Do[b=2a+n;If[2==Plus@@FactorInteger[b][[All,2]],Print[b];Break[]],{n,1000}];a=b,{40}] (* Zak Seidov, Dec 24 2007 *)
ssp[n_]:=Module[{k=2n+1},While[PrimeOmega[k]!=2,k++];k]; NestList[ssp,4,30] (* Harvey P. Dale, Apr 14 2022 *)

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A175627 a(1) = 1; a(n) is the smallest square > 2*a(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A207890 a(0)=1; for n>=1,- the minimal increasing sequence, such that, for n>=1, the row sums of Pascal-like triangle with left side {1,1,1,...} and right side {a(0), a(1), a(2),...} form an increasing sequence of primes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A083005 a(1)=1, a(n)=2a(n-1)+1 if a(n-1) is prime, a(n)=a(n-1)+1 otherwise.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A118476 a(0) = 1; a(n) is least k with n prime factors and k > n*a(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A118912 a(1) = 2; a(n) is greatest prime < a(n-1)^4.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A156321 a(1)=0, a(n+1) is smallest nonprime > 2*a(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A156324 a(1)=0, a(n+1) is smallest nonprime >= a(n)+n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A164962 a(n) is the least prime from the union {2,3} and A164333, beginning with which the n-th prime p_n is obtained by some number of iterations of the S operator g(see A164960).

Views

Author

Keywords

Comments