A007918 -id:A007918 - cp's OEIS frontend

A378249 Least perfect power > prime(n).

4, 4, 8, 8, 16, 16, 25, 25, 25, 32, 32, 49, 49, 49, 49, 64, 64, 64, 81, 81, 81, 81, 100, 100, 100, 121, 121, 121, 121, 121, 128, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 196, 196, 216, 216, 216, 225, 243, 243, 243, 243, 243, 256, 289, 289, 289

Offset: 1

Gus Wiseman, Nov 21 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

Which terms appear only once? Just 128, 225, 256, 64009, 1295044?

			The first number line below shows the perfect powers. The second shows each prime.
-1-----4-------8-9------------16----------------25--27--------32------36------------------------49--
===2=3===5===7======11==13======17==19======23==========29==31==========37======41==43======47======

A version for prime powers (but starting with prime(k) + 1) is A345531.
Positions of last appearances are A377283, complement A377436.
Restriction of A377468 to the primes, for prime powers A000015.
The opposite is A378035, restriction of A081676.
The union is A378250.
Run lengths are A378251.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists numbers that are not perfect powers, differences A375706, seconds A376562.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, zeros A377436, postpositives A377466.
Cf. A007918, A023055, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A377434.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]

f(p) = p++; while(!ispower(p), p++); p;
lista(nn) = apply(f, primes(nn)); \\ Michel Marcus, Dec 19 2024

A378251 Number of primes between consecutive perfect powers, zeros omitted.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 1, 3, 5, 5, 3, 1, 5, 1, 7, 5, 2, 4, 6, 7, 7, 5, 2, 6, 9, 8, 7, 8, 9, 8, 8, 6, 4, 9, 10, 9, 10, 7, 2, 9, 12, 11, 12, 6, 5, 9, 12, 11, 3, 10, 8, 2, 13, 15, 10, 11, 15, 7, 9, 12, 13, 11, 12, 17, 2, 11, 16, 16, 13, 17, 15, 14, 16, 15

Offset: 1

Gus Wiseman, Nov 23 2024

First differences of A377283 and A378365. Run-lengths of A378035 and A378249.

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

			The first number line below shows the perfect powers. The second shows each prime. To get a(n) we count the primes between consecutive perfect powers, skipping the cases where there are none.
-1-----4-------8-9------------16----------------25--27--------32------36----
===2=3===5===7======11==13======17==19======23==========29==31==========37==

Robert Israel, Table of n, a(n) for n = 1..10000

Same as A080769 with 0's removed (which were at positions A274605).
First differences of A377283 and A378365 (union of A378356).
Run-lengths of A378035 (union A378253) and A378249 (union A378250).
The version for nonprime prime powers is A378373, with zeros A067871.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, run-lengths of A377468.
A007916 lists the non-perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
Cf. A007918, A023055, A045542, A052410, A076412, A081676, A188951.
Cf. A216765, A377057, A377431, A378355.

N:= 10^6: # to use perfect powers up to N
PP:= {1,seq(seq(i^j,j=2..ilog[i](N)),i=2..isqrt(N))}:
PP:= sort(convert(PP,list)):
M:= map(numtheory:-pi, PP):
subs(0=NULL, M[2..-1]-M[1..-2]): # Robert Israel, Jan 23 2025

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]

A378250 Perfect-powers x > 1 such that it is not possible to choose a prime y and a perfect-power z satisfying x > y > z.

Original entry on oeis.org

4, 8, 16, 25, 32, 49, 64, 81, 100, 121, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936

Offset: 1

Gus Wiseman, Nov 21 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

			The first number line below shows the perfect-powers. The second shows the primes. The third is a(n).
-1-----4-------8-9------------16----------------25--27--------32------36----
===2=3===5===7======11==13======17==19======23==========29==31==========37==
       4       8              16                25            32
The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    25: {3,3}
    32: {1,1,1,1,1}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   128: {1,1,1,1,1,1,1}
   144: {1,1,1,1,2,2}
   169: {6,6}
   196: {1,1,4,4}
   216: {1,1,1,2,2,2}
   225: {2,2,3,3}
   243: {2,2,2,2,2}
   256: {1,1,1,1,1,1,1,1}

A version for prime-powers (but starting with prime(k) + 1) is A345531.
The opposite is union of A378035, restriction of A081676.
Union of A378249, run-lengths are A378251.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A069623 counts perfect-powers <= n.
A076411 counts perfect-powers < n.
A131605 lists perfect-powers that are not prime-powers.
A377432 counts perfect-powers between primes, zeros A377436, positive A377283, postpositive A377466.
Cf. A000015, A007918, A023055, A045542, A052410, A076412, A188951, A216765, A377431, A377434, A377468.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Union[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]

A013633 nextprime(n) - prevprime(n).

Original entry on oeis.org

3, 2, 4, 2, 6, 4, 4, 4, 6, 2, 6, 4, 4, 4, 6, 2, 6, 4, 4, 4, 10, 6, 6, 6, 6, 6, 8, 2, 8, 6, 6, 6, 6, 6, 10, 4, 4, 4, 6, 2, 6, 4, 4, 4, 10, 6, 6, 6, 6, 6, 12, 6, 6, 6, 6, 6, 8, 2, 8, 6, 6, 6, 6, 6, 10, 4, 4, 4, 6, 2, 8, 6, 6, 6, 6, 6, 10, 4, 4

Offset: 3

N. J. A. Sloane

nonn

			a(3) = nextprime(3) - prevprime(3) = 5 - 2 = 3: This shows that here the variants 2 (A151800 and A151799) of the nextprime and precprime functions are used, rather than the variants A007918 and A007917. - _M. F. Hasler_, Sep 09 2015

T. D. Noe, Table of n, a(n) for n=3..2000

Cf. A151800, A007918; A151799, A007917.

[ seq(nextprime(i)-prevprime(i),i=3..100) ];

A013633(n)=nextprime(n+1)-precprime(n-1) \\ M. F. Hasler, Sep 09 2015

a(n) = A151800(n) - A151799(n) = A007918(n+1) - A007917(n-1). - M. F. Hasler, Sep 09 2015

A199333 Triangle read by rows: T(n,0) = T(n,n) = 1, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 7, 5, 1, 1, 7, 13, 13, 7, 1, 1, 11, 23, 29, 23, 11, 1, 1, 13, 37, 53, 53, 37, 13, 1, 1, 17, 53, 97, 107, 97, 53, 17, 1, 1, 19, 71, 151, 211, 211, 151, 71, 19, 1, 1, 23, 97, 223, 367, 431, 367, 223, 97, 23, 1, 1, 29, 127

Offset: 0

Reinhard Zumkeller, Nov 09 2011

T(n,k) = T(n,n-k);
T(n,0) = 1, cf. A000012;
T(n,1) = A008578(n), n > 0;
A199424(n) = first row in triangle A199302 containing n-th prime;
A199425(n) = number of distinct primes in rows 0 through n;
large terms in the b-file are probable primes only, row number > 50.

			0:                 1
1:               1   1
2:             1   2   1
3:           1   3   3   1
4:         1   5   7   5   1
5:       1   7  13  13   7   1
6:     1  11  23  29  23  11   1
7:   1  13  37  53  53  37  13   1
8: 1  17  53  97 107  97  53  17   1
primes in 8th row:
T(7,0) + T(7,1) = 1+13 = 14 --> T(8,1) = T(8,7) = 19;
T(7,1) + T(7,2) = 13+37 = 50 --> T(8,2) = T(8,6) = 53, already in row 7;
T(7,2) + T(7,3) = 37+53 = 90 --> T(8,3) = T(8,5) = 97;
T(7,3) + T(7,4) = 53+53 = 106 --> T(8,4) = 107.

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A159477; A199581 & A199582 (central terms), A199694 (row sums), A199695 & A199696 (row products); A007318.

Cf. A132403, A362034.

a199333 n k = a199333_tabl !! n !! k
a199333_row n = a199333_tabl !! n
a199333_list = concat a199333_tabl
a199333_tabl = iterate
   (\row -> map a159477 $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]

T[n_, k_] := T[n, k] = Switch[k, 0|n, 1, _, With[{m = T[n-1, k] + T[n-1, k-1]}, If[PrimeQ[m], m, NextPrime[m]]]];
Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

T(n,k) = A007918(T(n-1,k) + T(n-1,k-1)), 0 < k < n, T(n,0) = T(n,n) = 1.

A055498 a(0)=0, a(1)=1, a(n) = smallest prime >= a(n-1) + a(n-2).

Original entry on oeis.org

0, 1, 2, 3, 5, 11, 17, 29, 47, 79, 127, 211, 347, 563, 911, 1481, 2393, 3877, 6271, 10151, 16427, 26591, 43019, 69623, 112643, 182279, 294923, 477209, 772139, 1249361, 2021501, 3270863, 5292367, 8563237, 13855607, 22418849, 36274471, 58693331, 94967809, 153661163

Offset: 0

N. J. A. Sloane, Jul 08 2000

nonn

			After 3, 5, the next prime >=8 is 11.

Zak Seidov and Robert G. Wilson v, Table of n, a(n) for n = 0..1001 (first 101 terms from Zak Seidov)

Cf. A073021, A073022, A055499, A055500, A055501, A055502.

Cf. A007918.

a055498 n = a055498_list !! n
a055498_list = 0 : 1 : map a007918
    (zipWith (+) a055498_list $ tail a055498_list)
-- Reinhard Zumkeller, Nov 13 2014

a[0] = 0; a[1] = 1; a[n_] := a[n] = NextPrime[a[n - 1] + a[n - 2] -1]; Array[a, 37, 0] (* Robert G. Wilson v, Mar 13 2013 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==NextPrime[a[n-1]+a[n-2]-1]},a,{n,50}] (* Harvey P. Dale, May 08 2013 *)

a(n)=local(v);if(n<2,n>=0,n++;v=vector(n,i,1);for(i=3,n,v[i]=nextprime(v[i-1]+v[i-2]));v[n]) /* Michael Somos, Feb 01 2004 */

a(n+1) = nextprime(a(n) + a(n-1)) where nextprime(n) is smallest prime >= n.
a(n) is asymptotic to c*phi^n where phi = (1 + sqrt(5))/2 and c = 1.086541275044988562375... - Benoit Cloitre, May 02 2004
a(n) = A055499(n-1) for n>3. - Robert G. Wilson v, Mar 13 2013
a(n) = A007918(a(n-1) + a(n-2)) for n > 1. - Reinhard Zumkeller, Nov 13 2014

A074984 m^p-n, for smallest m^p>=n.

Original entry on oeis.org

0, 2, 1, 0, 3, 2, 1, 0, 0, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 0, 4, 3, 2, 1, 0, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8

Offset: 1

Zak Seidov, Oct 07 2002

nonn

a(n) = 0 if n = m^p that is if n is a full power (square, cube etc.).

This is the distance between n and the next perfect power. The previous perfect power is A081676, which differs from n by A069584. After a(8) = a(9) this sequence is an anti-run (no adjacent equal terms). - Gus Wiseman, Dec 02 2024

Wikipedia, Catalan's conjecture

Sequences obtained by subtracting n from each term are placed in parentheses below.
Positions of 0 are A001597.
Positions of 1 are A375704.
The version for primes is A007920 (A007918).
The opposite (greatest perfect power <= n) is A069584 (A081676).
The version for perfect powers is A074984 (this) (A377468).
The version for squarefree numbers is A081221 (A067535).
The version for non perfect powers is A378357 (A378358).
The version for nonsquarefree numbers is A378369 (A120327).
The version for prime powers is A378370 (A000015).
The version for non prime powers is A378371 (A378372).
A001597 lists the perfect powers, differences A053289.
A007916 lists the non perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, zeros A377436.
Cf. A014210, A023055, A045542, A052410, A076412, A151800, A188951, A216765.

powerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; powerQ[1] = True; a[n_] := For[k = n, True, k++, If[powerQ[k], Return[k-n]]]; Table[a[n], {n, 1, 92}] (* Jean-François Alcover, Apr 19 2013 *)

a(n) = { if (n==1, return (0)); my(nn = n); while(! ispower(nn), nn++); return (nn - n);} \\ Michel Marcus, Apr 19 2013

a(n) = A377468(n) - n. - Gus Wiseman, Dec 02 2024

A124123 Primes not of the form nextprime(f(p)) with p prime, where f(p)=p/2 if p=2 (mod 3), f(p)=2p otherwise (cf. A138750).

Original entry on oeis.org

5, 19, 61, 73, 83, 103, 107, 109, 113, 139, 151, 167, 173, 191, 199, 229, 269, 271, 277, 313, 337, 349, 359, 379, 397, 439, 463, 503, 523, 563, 571, 601, 607, 619, 733, 773, 823, 827, 829, 859, 883, 887, 911, 971, 983, 997, 1013, 1031, 1063, 1091, 1093, 1103

Offset: 1

Jacques Tramu, Dec 13 2006

These are the primes which cannot be part of a gb-sequence (except as seed).
Is this sequence finite or infinite?
From M. F. Hasler, Mar 27 2008: (Start)
The last comment above probably refers not to this sequence but to the "gb-sequences" themselves, e.g., the one starting with 4499221 which reaches a peak of approximately 10^110, cf. Formula and Links.
The function f(p)=p/2 if p == 2 (mod 3), f(p)=2p otherwise, yields a half-integer for primes p=6k-1 and an even number for primes p=6k+1; in all cases nextprime(f(p)) is defined without ambiguity: f(p) will never be equal to a prime.
This sequence lists primes p' not in the range of the map p -> nextprime(f(p)), defined on the primes.
Equivalently, p' is listed iff: (i) no even number between p' and the next lower prime is of the form 2p with p=0 or p == 1 (mod 3), AND (ii) no half-integer between p' and the next lower prime is of the form p/2 with p == 2 (mod 3) and p prime (in both conditions).
This characterization allows easy computation of the sequence, cf. PARI code.
Experimentally, it does not appear that this sequence is finite. Instead, its (local) density within the primes seems to increase, from roughly 25% for the first terms to about 50% at 10^30. (End)
The function f is discussed in A138750. Composed with the nextprime function and restricted to the primes (cf. A138751), it yields a ("natural") variant of the Collatz function on the set of the primes, with (mod 2) replaced by (mod 3). The gb-sequences are the orbits under that function. - M. F. Hasler, Nov 18 2018

			Example: a(1) = 5 because there is no prime gb(n) such that gb(n+1) = 5.

Communication paper by Georges Brougnard.

Vincenzo Librandi, Table of n, a(n) for n = 1..6470
Georges Brougnard, Definition of GB-sequences.
Georges Brougnard, GB-sequence of length 96, obtained for gb[0]=1381.
Georges Brougnard, GB-sequence of length 63337, obtained for gb[0]=4499221.

Cf. A007918 (nextprime), A138750 (function f), A138751, A138752, A138753, A138754.

lim = PrimePi[1000]; f[p_ /; Mod[p, 3] == 2] := p/2; f[p_] := 2*p; Complement[Prime[Range[lim]], Table[ NextPrime[ f[Prime[k]]], {k, 1, 2*lim}]] (* Jean-François Alcover, Sep 20 2011 *)

{forprime( p=3,10^3, for( i=precprime(p-1)+1,p, (2*i)%3==0 & isprime(2*i-1) & next(2); i%2==0 & ( i/2 )%3!=2 & isprime( i/2 ) & next(2)); print1( p", " ))}
nextA124123(p)={ while( p=nextprime(p+1), for( i=precprime(p-1)+1,p, (2*i)%3==0 & isprime(2*i-1) & next(2); i%2==0 & ( i/2 )%3!=2 & isprime( i/2 ) & next(2)); return( p )) }
t=2;vector(200,i,t=nextA124123(t)) \\ 60% of the first 200 terms are in 1+3Z:
t=[0,0];vector(#%,i,t[%[i]%3]++);t \\ yields [120, 80]
t=10^11;vector(200,i,t=nextA124123(t)) \\ exactly 50% of these terms are in 1+3Z:
t=[0,0];vector(#%,i,t[%[i]%3]++);t \\ yields [100, 100]
t=10^30;vector(200,i,t=nextA124123(t+1));t-10^30 \\ yields 31773 = distance of 200th term beyond 10^30
t=10^30;vector(200,i,t=nextprime(t+1));(t-1e30)/% \\ yields 0.52..., approx. local density in the primes. (End)

Complement of A007918(A138750(A000040)) = nextprime(f({primes})).
Recurrence for a gb-sequence starting with gb(0) = a prime > 2 (the seed):
| If gb(n) = 2 (mod 3) then gb(n+1) := least prime > gb(n)/2;
| otherwise gb(n+1) := least prime > gb(n)*2.
A gb-sequence of length L ends in the loop 7, 17, 11, 7, ... ; gb(L-1) = 7.

Edited by M. F. Hasler, Mar 27 2008, Nov 18 2018

A138750 a(n) = ceiling(n/2) if n == 2 (mod 3), a(n) = 2n otherwise.

Original entry on oeis.org

0, 2, 1, 6, 8, 3, 12, 14, 4, 18, 20, 6, 24, 26, 7, 30, 32, 9, 36, 38, 10, 42, 44, 12, 48, 50, 13, 54, 56, 15, 60, 62, 16, 66, 68, 18, 72, 74, 19, 78, 80, 21, 84, 86, 22, 90, 92, 24, 96, 98, 25, 102, 104, 27, 108, 110, 28, 114, 116, 30, 120, 122, 31, 126, 128, 33, 132, 134, 34

Offset: 0

M. F. Hasler, Mar 28 2008

This map is inspired by A124123, which hides in fact a variation of the Collatz problem, defined on the set of primes and working mod 3 instead of mod 2. See A138751 for more information.
The use of ceiling() is here equivalent to round().
The main reason for defining this function is to write A124123 as complement of A007918(A138750(A000040)), and to express the recursion function occurring there in terms of this map.
It might have been more natural to define this map as a(n) = 2n if n == 1 (mod 3), a(n) = ceiling(n/2) otherwise, which is equivalent for all primes > 3 (which are either == 1 or == 2 (mod 3)) and would have "better" properties regarding the analysis of orbits of all integers under this map.
However, for the prime n=3 it does make a difference, and in order to reproduce the map occurring in A124123, we had to adopt the present convention.

			a(0) = 2*0 = 0, a(1) = 2*1 = 2, a(3) = 2*3 = 6, a(4) = 2*4 = 8, ... since these indices are not congruent to 2 (mod 3).
a(2) = ceiling(2/2) = 1, a(5) = ceiling(5/2) = 3, a(8) = ceiling(8/2) = 4, a(11) = ceiling(11/2) = 6, ... since these indices are congruent to 2 (mod 3).

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A001281, A124123, A138751, A138752, A138753, A008588 (trisection), A016933 (trisection), A032766 (trisection)

Table[If[Mod[n,3]==2,Ceiling[n/2],2n],{n,0,70}] (* or *) LinearRecurrence[{0,0,1,0,0,1,0,0,-1},{0,2,1,6,8,3,12,14,4},70] (* Harvey P. Dale, Nov 20 2013 *)

A138750(n) = if( n%3==2, ceil(n/2), 2*n )

G.f.: x*(2 + x + 6*x^2 + 6*x^3 + 2*x^4 + 6*x^5 + 4*x^6) / ( (1+x)*(x^2-x+1)*(x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Oct 16 2013
a(n) = a(n-3) + a(n-6) - a(n-9); a(0)=0, a(1)=2, a(2)=1, a(3)=6, a(4)=8, a(5)=3, a(6)=12, a(7)=14, a(8)=4. - Harvey P. Dale, Nov 20 2013
Sum_{n>=1} (-1)^n/a(n) = log(3)/2 - log(2)/3 = log(27/4)/6. - Amiram Eldar, Jul 26 2024

A179278 Largest nonprime integer <= n.

Original entry on oeis.org

1, 1, 1, 4, 4, 6, 6, 8, 9, 10, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 22, 22, 24, 25, 26, 27, 28, 28, 30, 30, 32, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72

Offset: 1

Reinhard Zumkeller, Jul 08 2010

			From _Gus Wiseman_, Dec 04 2024: (Start)
The nonprime integers <= n:
  1  1  1  4  4  6  6  8  9  10  10  12  12  14  15  16
           1  1  4  4  6  8  9   9   10  10  12  14  15
                 1  1  4  6  8   8   9   9   10  12  14
                       1  4  6   6   8   8   9   10  12
                          1  4   4   6   6   8   9   10
                             1   1   4   4   6   8   9
                                     1   1   4   6   8
                                             1   4   6
                                                 1   4
                                                     1
(End)

Cf. A014684, A113523, A113638, A062298.
For prime we have A007917.
For nonprime we have A179278 (this).
For squarefree we have A070321.
For nonsquarefree we have A378033.
For prime power we have A031218.
For non prime power we have A378367.
For perfect power we have A081676.
For non perfect power we have A378363.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A018252 lists the nonprimes, differences A065310.
A095195 has row n equal to the k-th differences of the prime numbers.
A113646 gives least nonprime >= n.
A151800 gives the least prime > n, weak version A007918.
A377033 has row n equal to the k-th differences of the composite numbers.

Array[# - Boole[PrimeQ@ #] - Boole[# == 3] &, 72] (* Michael De Vlieger, Oct 13 2018 *)
Table[Max@@Select[Range[n],!PrimeQ[#]&],{n,30}] (* Gus Wiseman, Dec 04 2024 *)

a(n) = if (isprime(n), if (n==3, 1, n-1), n); \\ Michel Marcus, Oct 13 2018

For n > 3: a(n) = A113523(n) = A014684(n);
For n > 0: a(n) = A113638(n). - Georg Fischer, Oct 12 2018
A005171(a(n)) = 1; A010051(a(n)) = 0.
a(n) = A018252(A062298(n)). - Ridouane Oudra, Aug 22 2025

Inequality in the name reversed by Gus Wiseman, Dec 05 2024

cp's OEIS Frontend

A378249 Least perfect power > prime(n).

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A378251 Number of primes between consecutive perfect powers, zeros omitted.

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Maple

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A378250 Perfect-powers x > 1 such that it is not possible to choose a prime y and a perfect-power z satisfying x > y > z.

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Mathematica

A013633 nextprime(n) - prevprime(n).

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A199333 Triangle read by rows: T(n,0) = T(n,n) = 1, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).

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Haskell

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A055498 a(0)=0, a(1)=1, a(n) = smallest prime >= a(n-1) + a(n-2).

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A074984 m^p-n, for smallest m^p>=n.

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A124123 Primes not of the form nextprime(f(p)) with p prime, where f(p)=p/2 if p=2 (mod 3), f(p)=2p otherwise (cf. A138750).

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