A024619 -id:A024619 - cp's OEIS frontend

A345531 Smallest prime power greater than the n-th prime.

3, 4, 7, 8, 13, 16, 19, 23, 25, 31, 32, 41, 43, 47, 49, 59, 61, 64, 71, 73, 79, 81, 89, 97, 101, 103, 107, 109, 113, 121, 128, 137, 139, 149, 151, 157, 163, 167, 169, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 256, 263, 269, 271, 277

Offset: 1

Dario T. de Castro, Jun 20 2021

nonn

Take the family of correlated prime-indexed conjectures appearing in A343249 - A343253, in which an alternative formula for the p-adic order of positive integers is proposed. There, the general p-indexed conjecture says that v_p(n), the p-adic order of n, is given by the formula: v_p(n) = log_p(n / L_p(k0, n)), where L_p(k0, n) is the lowest common denominator of the elements of the set S_p(k0, n) = {(1/n)*binomial(n, k), with 0 < k <= k0 such that k is not divisible by p}. Evidence suggests that the primality of p is a necessary condition in this general conjecture. So, if a composite number q is used instead of a prime p in the proposed formula for the p-adic (now, q-adic) order of n, the first counterexample (failure) is expected to occur for n = q * a(i), where i is the index of the smallest prime that divides q.

The prime-power a(n) is at most the next prime, so this sequence is strictly increasing. See also A366833. - Gus Wiseman, Nov 06 2024

			a(4) = 8 because the fourth prime number is 7, and the least power of a prime which is greater than 7 is 2^3 = 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.

Cf. A000040, A007814, A007949, A008864, A343249, A343250, A112765, A343251, A214411, A343252, A343253.
Starting with n instead of prime(n): A000015, A031218, A377468, A377780, A377782.
Opposite (greatest prime-power less than): A065514, A377289, A377781.
For squarefree instead of prime-power: A112926, opposite A112925.
The difference from prime(n) is A377281.
The prime terms have indices A377286(n) - 1.
First differences are A377703.
A version for perfect-powers is A378249.
A000961 and A246655 list the prime-powers, differences A057820.
A024619 and A361102 list the non-prime-powers, differences A375735.
Cf. A366833, A053607, A080101, A304521, A366833, A377057, A377283, A377287, A377288, A377432.

f:= proc(n) local p,x;
  p:= ithprime(n);
  for x from p+1 do
    if nops(numtheory:-factorset(x)) = 1 then return x fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Aug 25 2024

a[i_]:= Module[{j, k, N = 0, tab={}}, tab = Sort[Drop[DeleteDuplicates[Flatten[Table[ If[Prime[j]^k > Prime[i], Prime[j]^k], {j, 1, i+1}, {k, 1, Floor[Log[Prime[j], Prime[i+1]]]}]]], 1]]; N = Take[tab, 1][[1]]; N];
tabseq = Table[a[i],{i, 1, 100}];
(* second program *)
Table[NestWhile[#+1&,Prime[n]+1, Not@*PrimePowerQ],{n,100}] (* Gus Wiseman, Nov 06 2024 *)

A000015(n) = for(k=n,oo,if((1==k)||isprimepower(k),return(k)));
A345531(n) = A000015(1+prime(n)); \\ Antti Karttunen, Jul 19 2021

from itertools import count
from sympy import prime, factorint
def A345531(n): return next(filter(lambda m:len(factorint(m))<=1, count(prime(n)+1))) # Chai Wah Wu, Oct 25 2024

a(n) = A000015(1+A000040(n)). - Antti Karttunen, Jul 19 2021

a(n) = A000015(A008864(n)). - Omar E. Pol, Oct 27 2021

A065514 Largest power of a prime < prime(n).

Original entry on oeis.org

1, 2, 4, 5, 9, 11, 16, 17, 19, 27, 29, 32, 37, 41, 43, 49, 53, 59, 64, 67, 71, 73, 81, 83, 89, 97, 101, 103, 107, 109, 125, 128, 131, 137, 139, 149, 151, 157, 163, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 243, 256, 257, 263, 269, 271

Offset: 1

Reinhard Zumkeller, Nov 27 2001

nonn

Starting with n instead of prime(n) gives A031218 (A377282, A377782).
The squarefree version is A112925 (A070321, A378038).
The opposite squarefree version is A112926 (A378037, restriction of A067535).
Difference from prime(n) is A377289 (restriction of A276781, opposite A377281).
First differences are A377781.
The nonsquarefree version is A378032 (A377783 (restriction of A378033), A378034, A378040).
The perfect power version is A378035.
A000015 gives the least prime power >= n, differences A377780.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A345531 gives the least prime power > prime(n), differences A377703.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057, A377286.
Cf. A014234, A053707, A065890, A376597, A377051, A378249, A378252.

lpp[n_]:=Module[{k=n-1},While[!PrimePowerQ[k],k--];k]; Join[{1},Table[ lpp[ n],{n,Prime[Range[2,60]]}]] (* Harvey P. Dale, Nov 24 2018 *)

from sympy import factorint, prime
def A065514(n): return next(filter(lambda m:len(factorint(m))<=1, range(prime(n)-1,0,-1))) # Chai Wah Wu, Oct 25 2024

Name edited (1 is technically not a prime power even though it is a power of a prime) by Gus Wiseman, Dec 03 2024.

A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Jan 28 2003

nonn

The maximum value of terms in the sequence, through the (10^5)th term, is 2. - Harvey P. Dale, Aug 24 2014

This is conjectured to be the maximum, see also A366833. - Gus Wiseman, Nov 06 2024

			There are two prime powers between 2179 = A000040(327) and 2203 = A000040(328): 2187 = 3^7 and 2197 = 13^3, therefore a(327) = 2, A080102(327) = 2187 and A080103(327) = 2197.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A080102, A080103, A025475, A000961.
For powers of 2 instead of primes we have A244508, see also A013597, A014210, A014234, A304521.
Adding one gives A366833.
For non-prime-powers instead of prime-powers we have A368748.
Positions of positive terms are A377057, primes A053607.
Positions of 0 are A377286.
Positions of 1 are A377287.
Positions of 2 are A377288, primes A053706.
For perfect-powers (instead of prime-powers) we have A377432.
A000015 gives the least prime-power >= n, difference A377282.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, seconds A376596.
A031218 gives the greatest prime-power <= n, difference A276781.
A046933(n) counts the interval from A008864(n) to A006093(n+1).
A065514 gives the greatest prime-power < prime(n), difference A377289.
A246655 lists the prime-powers not including 1, complement A361102.
A345531 gives the least prime-power > prime(n), difference A377281.
Cf. A001597, A002808, A024619, A065890, A182908, A224363, A376597, A377051, A377054, A377436.

a := proc(n) local c, k, p: c, p := 0, ithprime(n): for k from p+1 to nextprime(p)-1 do if nops(numtheory:-factorset(k)) = 1 then c := c+1: fi: od: c: end:
seq(a(n), n = 1 .. 105); # Lorenzo Sauras Altuzarra, Jul 08 2022

prpwQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]==1&&fi[[1,2]]>1]; nn=600;With[{pwrs=Table[If[prpwQ[n],1,0],{n,nn}]},Table[Total[ Take[ pwrs,{Prime[n],Prime[n+1]}]],{n,PrimePi[nn]-1}]] (* Harvey P. Dale, Aug 24 2014 *)
Table[Length[Select[Range[Prime[n]+1,Prime[n+1]-1],PrimePowerQ]],{n,30}] (* Gus Wiseman, Nov 06 2024 *)

a(n) = A366833(n) - 1. - Gus Wiseman, Nov 06 2024

A375708 First differences of non-prime-powers (exclusive, so 1 is not a prime-power).

Original entry on oeis.org

5, 4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1

Offset: 1

Gus Wiseman, Aug 31 2024

nonn

Non-prime-powers (exclusive) are listed by A361102.

Warning: For this sequence, 1 is not a prime-power but is a non-prime-power.

			The 6th non-prime-power (exclusive) is 15, and the 7th is 18, so a(6) = 3.

For prime-powers (A000961, A246655) we have A057820, gaps A093555.
For perfect powers (A001597) we have A053289.
For nonprime numbers (A002808) we have A073783.
For squarefree numbers (A005117) we have A076259.
First differences of A361102, inclusive A024619.
Positions of 1's are A375713.
If 1 is considered a prime power we have A375735.
Runs of non-prime-powers:
- length: A110969
- first: A373676
- last: A373677
- sum: A373678
A000040 lists all of the primes, differences A001223.
A007916 lists non-perfect-powers, differences A375706.
A013929 lists the nonsquarefree numbers, differences A078147.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A046933, A061399, A176246, A251092, A375709.

Differences[Select[Range[100],!PrimePowerQ[#]&]]

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375708(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 09 2024

A366833 Number of times n appears in A362965 (number of primes <= the n-th prime power).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Paolo Xausa, Oct 25 2023

nonn

Conjecture: a(n) can be only 1, 2, or 3 (with the first occurrences of 3 appearing at n = 4, 9, 30, 327 and 3512).

One less than the number of prime powers between prime(n) and prime(n+1), inclusive. - Gus Wiseman, Jan 09 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000
Paolo Xausa, 1200 X 1200 raster image of a(n), n = 1..1440000, read left to right, top to bottom, showing a(n) = 1 in blue, a(n) = 2 in white and a(n) = 3 in red.

Run lengths of A362965.
Subtracting one gives A080101.
For non prime powers we have A368748.
Positions of terms > 1 are A377057.
Positions of 1 are A377286.
Positions of 2 are A377287.
For perfect powers we have A377432.
For squarefree we have A373198.
A000015 gives the least prime power >= n, difference A377282.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A031218 gives the greatest prime power <= n, difference A276781.
A046933(n) counts the interval from A008864(n) to A006093(n+1).
A246655 lists the prime powers not including 1.
A366835 counts primes between prime powers.
Cf. A053607, A053706, A065514, A068435, A080102, A304521, A345531, A377289.
Cf. A024620, A027883, A067871, A075526, A080769, A151800, A377436, A379157.

With[{upto=1000},Map[Length,Most[Split[PrimePi[Select[Range[upto],PrimePowerQ]]]]]] (* Considers prime powers up to 1000 *)

a(n) = A080101(n) + 1. - Gus Wiseman, Jan 09 2025

A005279 Numbers having divisors d, e with d < e < 2*d.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 77, 78, 80, 84, 88, 90, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 126, 130, 132, 135, 138, 140, 143, 144, 150, 153, 154, 156, 160, 162, 165, 168, 170, 174, 175, 176

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

The arithmetic and harmonic means of A046793(n) and a(n) are both integers.
n is in this sequence iff n is a multiple of some term in A020886.
a(n) is also a positive integer v for which there exists a smaller positive integer u such that the contraharmonic mean (uu+vv)/(u+v) is an integer c (in fact, there are two distinct values u giving with v the same c). - Pahikkala Jussi, Dec 14 2008
A174903(a(n)) > 0; complement of A174905. - Reinhard Zumkeller, Apr 01 2010
Also numbers n such that A239657(n) > 0. - Omar E. Pol, Mar 23 2014
Erdős (1948) shows that this sequence has a natural density, so a(n) ~ k*n for some constant k. It can be shown that k < 3.03, and by numerical experiments it seems that k is around 1.8. - Charles R Greathouse IV, Apr 22 2015
Numbers k such that at least one of the parts in the symmetric representation of sigma(k) has width > 1. - Omar E. Pol, Dec 08 2016
Erdős conjectured that the asymptotic density of this sequence is 1. The numbers of terms not exceeding 10^k for k = 1, 2, ... are 1, 32, 392, 4312, 45738, 476153, 4911730, 50359766, 513682915, 5224035310, ... - Amiram Eldar, Jul 21 2020
Numbers with at least one partition into two distinct parts (s,t), sWesley Ivan Hurt, Jan 16 2022
Appears to be the set of numbers x such that there exist numbers y and z satisfying the condition (x^2+y^2)/(x^2+z^2) = (x+y)/(x+z). For example, (15^2+10^2)/(15^2+3^2) = (15+10)/(15+3), so 15 is in the sequence. - Gary Detlefs, Apr 01 2023
From Bob Andriesse, Nov 26 2023: (Start)
Rewriting (x^2+y^2) / (x^2+z^2) = (x+y) / (x+z) as (x^2+y^2) / (x+y) = (x^2+z^2) / (x+z) has the advantage that the values on both sides of the = sign in the given example become integers. A possible sequence with the name: "k's for which r = (k^2+m^2) / (k+m) can be an integer while mA053629(n) and the r's being A009003(n). If (k^2+m^2) / (k+m) = r and m satisfies the divisibility condition, then r-m also does, because (k^2 + (r-m)^2) / (k + (r-m)) = r as well, confirming Pahikkala Jussi's comment about the existence of two distinct values for his u.
The fact that 15 is in the sequence is not so much because (15^2 + 10^2) / (15^2 + 3^2) = 1.3888... = (15+10) / (15+3), as indicated by Gary Detlefs, but rather because (15+10) | (15^2 + 10^2). And since r = (15^2 + 10^2) / (15+10) = 13, the second value that satisfies the divisibility condition is 13-10 = 3, so (15^2 + 3^2) / (15+3) = 13 as well.
Since (k+m)| (k^2 + m^2) is equivalent to (k+m) | 2*k^2 as well as to (k+m) | 2*m^2, both of these alternative divisibility conditions can be used to generate the same sequence too. (End)

R. K. Guy, Unsolved Problems in Number Theory, E3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Thomas Bloom, The density of integers which have two divisors d_1, d_2 such that d_1 < d_2 < 2*d_1 exists and is equal to 1, Erdős Problems.
Paul Erdős, On the density of some sequences of numbers, Bull. Amer. Math. Soc. 54 (1948), 685--692 MR10,105b; Zbl 32,13 (see Theorem 3).
Paul Erdős, Some unconventional problems in number theory, Journées Arithmétiques de Luminy, Astérisque 61 (1979), p. 73-82.
Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.
Paul Erdős, On some applications of probability to analysis and number theory, J. London Math. Soc., Vol. 39, No. 1 (1964), pp. 692-696, alternative link.
Planet Math, Integer Contraharmonic Means, Proposition 4.
Planet Math, Contraharmonic proportion
Terence Tao, Erdős problem database, see no. 144.
Robert G. Wilson v, Letter, N.D.

Subsequence of A024619 and hence of A002808.

Cf. A010814, A089341, A020886, A046793, A174903, A174905, A237271, A237593, A239657.

a005279 n = a005279_list !! (n-1)
a005279_list = filter ((> 0) . a174903) [1..]
-- Reinhard Zumkeller, Sep 29 2014

isA005279 := proc(n) local divs,d,e ; divs := numtheory[divisors](n) ; for d from 1 to nops(divs)-1 do for e from d+1 to nops(divs) do if divs[e] < 2*divs[d] then RETURN(true) ; fi ; od: od: RETURN(false) : end; for n from 3 to 300 do if isA005279(n) then printf("%d,",n) ; fi ; od : # R. J. Mathar, Jun 08 2006

aQ[n_] := Select[Partition[Divisors[n], 2, 1], #[[2]] < 2 #[[1]] &] != {}; Select[Range[178], aQ] (* Jayanta Basu, Jun 28 2013 *)

is(n)=my(d=divisors(n));for(i=3,#d,if(d[i]<2*d[i-1],return(1)));0 \\ Charles R Greathouse IV, Apr 22 2015

from sympy import divisors
def is_A005279(n): D=divisors(n)[1:]; return any(e<2*d  for d,e in zip(D, D[1:]))
# M. F. Hasler, Mar 20 2025

a(n) = A010814(n)/2. - Omar E. Pol, Dec 04 2016

A110969 Length of the runs of ones in A014963.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 4, 3, 1, 3, 1, 3, 5, 1, 2, 2, 3, 1, 5, 1, 1, 5, 7, 3, 1, 3, 1, 3, 7, 3, 1, 2, 5, 1, 9, 1, 5, 5, 3, 1, 3, 5, 1, 9, 1, 3, 1, 11, 11, 3, 1, 3, 5, 1, 1, 7, 4, 5, 5, 1, 5, 3, 1, 5, 3, 13, 3, 1, 3, 13, 5, 5, 3, 1, 3, 5, 1, 5, 5, 5, 3, 5, 7, 3, 7

Offset: 1

Franz Vrabec, Sep 27 2005

nonn

Unbounded sequence.
From A373669 we see that 10 first appears at a(28195574) = 10.
Also run-lengths of non-prime-powers (assuming 1 is not a prime-power), where a run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one. Also nonzero differences of consecutive prime-powers minus one. - Gus Wiseman, Jun 18 2024

			a(5)=2 because the fifth run of ones in A014963 is of length 2.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A014963.
Positions of first appearances are A373670, sorted A373669.
For runs of prime-powers:
- length A174965, antiruns A373671
- min A373673, antiruns A120430
- max A373674, antiruns A006549
- sum A373675, antiruns A373576
For runs of non-prime-powers:
- length A110969 (this sequence), antiruns A373672
- min A373676, antiruns A373575
- max A373677, antiruns A255346
- sum A373678, antiruns A373679
A000961 lists all powers of primes. A246655 lists just prime-powers.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A356068 counts non-prime-powers up to n.
A361102 lists all non-prime-powers (A024619 if not including 1).
Various run-lengths: A053797, A120992, A175632, A176246.
Various antirun-lengths: A027833, A373127, A373403, A373409.
Cf. A001359, A008864, A067774, A251092.

Length /@ SplitBy[Table[Exp[MangoldtLambda[n]], {n, 400}], # != 1 &][[ ;; -1 ;; 2]] (* Michael De Vlieger, Mar 21 2024 *)
DeleteCases[Differences[Select[Range[100],PrimePowerQ]]-1,0] (* Gus Wiseman, Jun 18 2024 *)

\\ b(n) returns boolean of A014963(n) == 1.
b(n)={my(t); !isprime(if(ispower(n, ,&t), t, n))}
seq(n)={my(k=1, i=0, L=List()); while(#Lk, listput(L, i-k)); k = i+1)); Vec(L)} \\ Andrew Howroyd, Jan 02 2020

Terms a(41) and beyond from Andrew Howroyd, Jan 02 2020

A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

Original entry on oeis.org

1, 36, 100, 144, 196, 216, 225, 324, 400, 441, 484, 576, 676, 784, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1728, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3600, 3844, 3969, 4225, 4356, 4624

Offset: 1

Daniel Forgues, May 27 2008

nonn

Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])
From Michael De Vlieger, Aug 11 2025: (Start)
This sequence is A001597 \ A246547, i.e., perfect powers without proper prime powers.
Union of {1} with the intersection of A001597 and A126706, where A126706 is the sequence of numbers that are neither prime powers nor squarefree.
Union of {1} and A286708 \ A052486, i.e., powerful numbers that are not prime powers, without Achilles numbers, but including the empty product. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (Terms a(n), n = 1..1323 from Klaus Brockhaus, and n = 1324..8649 from Daniel Forgues.)

Cf. A000961, A001597, A024619, A025475, A052486, A072102, A126706, A136141, A246547, A286708.

With[{nn = 2^20}, {1}~Join~Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[Length[#2] > 1, GCD @@ #2 > 1] & @@ {#, FactorInteger[#][[;; , -1]]} &] ] (* Michael De Vlieger, Aug 11 2025 *)

isok(n) = if (n == 1, return (1), return (ispower(n, ,&np) && (! isprime(np)))); \\ Michel Marcus, Jun 12 2013

from sympy import mobius, integer_nthroot, primepi
def A131605(n):
    def f(x): return int(n-2+x+sum(mobius(k)*((a:=integer_nthroot(x,k)[0])-1)+primepi(a) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Sum_{n>=1} 1/a(n) = 1 + A072102 - A136141 = 1.10130769935514973882... . - Amiram Eldar, Aug 15 2025

A119288 a(n) is the second smallest prime factor of n, or 1 if n is a prime power.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 3, 1, 5, 7, 11, 1, 3, 1, 13, 1, 7, 1, 3, 1, 1, 11, 17, 7, 3, 1, 19, 13, 5, 1, 3, 1, 11, 5, 23, 1, 3, 1, 5, 17, 13, 1, 3, 11, 7, 19, 29, 1, 3, 1, 31, 7, 1, 13, 3, 1, 17, 23, 5, 1, 3, 1, 37, 5, 19, 11, 3, 1, 5, 1, 41, 1, 3, 17, 43, 29, 11, 1, 3, 13, 23

Offset: 1

Reinhard Zumkeller, May 13 2006

nonn

Least prime factor of {n divided by the maximal power of the least prime factor of n}. - after the original name of the sequence.
a(n) = A020639(A028234(n)).
a(n) = 1 iff n is a prime power: a(A000961(n))=1 and a(A024619(n))>1.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A014673, A020639, A028234, A292269.

Join[{1},Table[Which[PrimePowerQ[n],1,True,FactorInteger[n][[2,1]]],{n,2,100}]] (* Harvey P. Dale, Feb 08 2020 *)

a(n) = if (isprimepower(n) || (n==1), 1, my(f=factor(n)[,1]); f[2]); \\ Michel Marcus, Mar 01 2023

from sympy import primefactors
def A119288(n): return 1 if len(s:=primefactors(n)) <= 1 else sorted(s)[1] # Chai Wah Wu, Mar 31 2023

A010055(n) = 0^(a(n)-1). - Reinhard Zumkeller, May 13 2006

Name changed by Antti Karttunen, Oct 04 2017

A377432 Number of perfect-powers x in the range prime(n) < x < prime(n+1).

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Gus Wiseman, Oct 31 2024

nonn

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

			Between prime(4) = 7 and prime(5) = 11 we have perfect-powers 8 and 9, so a(4) = 2.

For prime-powers instead of perfect-powers we have A080101.
Non-perfect-powers in the same range are counted by A377433.
Positions of 1 are A377434.
Positions of 0 are A377436.
Positions of terms > 1 are A377466.
For powers of 2 instead of primes we have A377467, for prime-powers A244508.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.
A377468 gives the least perfect-power > n.
Cf. A000015, A002808, A024619, A031218, A053707, A064113, A065514, A065890, A080769, A377051, A377282.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[Prime[n]+1, Prime[n+1]-1],perpowQ]],{n,100}]

a(n) + A377433(n) = A046933(n) = prime(n+1) - prime(n) - 1.

cp's OEIS Frontend

A345531 Smallest prime power greater than the n-th prime.

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A065514 Largest power of a prime < prime(n).

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A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

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A375708 First differences of non-prime-powers (exclusive, so 1 is not a prime-power).

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A366833 Number of times n appears in A362965 (number of primes <= the n-th prime power).

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A005279 Numbers having divisors d, e with d < e < 2*d.

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A110969 Length of the runs of ones in A014963.

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