A246547 -id:A246547 - cp's OEIS frontend

A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

2, 0, 2, 3, 0, 2, 4, 3, 4, 8, 0, 1, 8, 14, 1, 7, 7, 4, 25, 2, 15, 15, 17, 16, 10, 45, 2, 44, 20, 26, 18, 0, 2, 28, 52, 36, 42, 32, 45, 45, 47, 19, 30, 106, 36, 35, 4, 114, 28, 135, 89, 42, 87, 42, 34, 66, 192, 106, 56, 23, 39, 37, 165, 49, 37, 262, 58, 160, 22

Offset: 1

Jon Perry, Mar 07 2002

Does this sequence have any terms appearing infinitely often? In particular, are {2, 5, 11, 32, 77} the only zeros? As an example, {121, 122, 123, 124, 125} is an interval containing no primes, corresponding to a(11) = 0. - Gus Wiseman, Dec 02 2024

			The first few prime powers A246547 are 4, 8, 9, 16. The first few primes are 2, 3, 5, 7, 11, 13. We have (4), 5, 7, (8), (9), 11, 13, (16) and so the sequence begins with 2, 0, 2.
The initial terms count the following sets of primes: {5,7}, {}, {11,13}, {17,19,23}, {}, {29,31}, {37,41,43,47}, ... - _Gus Wiseman_, Dec 02 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 667 terms from Lei Zhou)

For primes between nonsquarefree numbers we have A236575.
For composite instead of prime we have A378456.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A080101 counts prime powers between primes.
A246547 lists the non prime prime powers, differences A053707.
A246655 lists the prime powers not including 1, complement A361102.
Cf. A001597, A024619, A031218, A046933, A276781, A345531, A366833, A377051, A377057, A377282, A377286-A377288.

t = {}; cnt = 0; Do[If[PrimePowerQ[n], If[FactorInteger[n][[1, 2]] == 1, cnt++, AppendTo[t, cnt]; cnt = 0]], {n, 4 + 1, 30000}]; t (* T. D. Noe, May 21 2013 *)
nn = 2^20; Differences@ Map[PrimePi, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] (* Michael De Vlieger, Oct 26 2023 *)

a(n) = A000720(A025475(n+3)) - A000720(A025475(n+2)). - David Wasserman, Dec 20 2002

More terms from David Wasserman, Dec 20 2002

Definition clarified by N. J. A. Sloane, Oct 27 2023

A025476 Prime root of n-th nontrivial prime power (A025475, A246547).

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 2, 7, 2, 3, 11, 5, 2, 13, 3, 2, 17, 7, 19, 2, 23, 5, 3, 29, 31, 2, 11, 37, 41, 43, 2, 3, 13, 47, 7, 53, 5, 59, 61, 2, 67, 17, 71, 73, 79, 3, 19, 83, 89, 2, 97, 101, 103, 107, 109, 23, 113, 11, 5, 127, 2, 7, 131, 137, 139, 3, 149, 151, 29, 157, 163, 167, 13, 31, 173, 179

Offset: 1

David W. Wilson

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A025473, A025475, A048148, A246547.

cvm := proc(n, level) local f,opf; if n < 2 then RETURN() fi;
f := ifactors(n); opf := op(1,op(2,f)); if nops(op(2,f)) > 1 or
op(2,opf) <= level then RETURN() fi; op(1,opf) end:
A025476_list := n -> seq(cvm(i,1),i=1..n); # n is search limit
A025476_list(30000);  # Peter Luschny, Sep 21 2011
# Alternative:
isA246547 := n -> n > 1 and not isprime(n) and type(n, 'primepower'):
seq(ifactors(p)[2][1][1], p in select(isA246547, [$1..30000])); # Peter Luschny, Jul 15 2023

Transpose[ Flatten[ FactorInteger[ Select[ Range[2, 30000], !PrimeQ[ # ] && Mod[ #, # - EulerPhi[ # ]] == 0 &]], 1]][[1]] (* Robert G. Wilson v *)

forcomposite(n=4,10^5,if( ispower(n, , &p) && isprime(p), print1(p,", "))) \\ Joerg Arndt, Sep 11 2021

from sympy import primepi, integer_nthroot, primefactors
def A025476(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) 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 primefactors(kmax)[0] # Chai Wah Wu, Aug 15 2024

A383295 Positions of proper prime powers (A246547) in EKG-sequence.

Original entry on oeis.org

3, 6, 8, 17, 22, 24, 31, 50, 64, 76, 112, 122, 124, 171, 232, 240, 290, 319, 359, 485, 521, 595, 696, 823, 947, 982, 1279, 1313, 1642, 1810, 1961, 2090, 2096, 2168, 2306, 2736, 3002, 3398, 3638, 3932, 4379, 4733, 4913, 5207, 6072, 6312, 6583, 6710, 7717, 7898, 9165, 9929, 10298, 11144, 11568, 11786, 12430, 14138

Offset: 1

Antti Karttunen, Apr 22 2025

nonn

Apparently also numbers k for which A265576(k) > 1 and A064413(k) is neither 2 nor 2*odd prime.

Antti Karttunen, Table of n, a(n) for n = 1..321
Index entries for sequences related to EKG sequence.

Cf. A064413, A246547, A265576.
Setwise difference A383294 \ A064955.
Conjectured to be a subsequence of A383285.

isA383295(n) = { my(x=A064413(n)); (isprimepower(x) && !isprime(x)); };

A282550 Perfect powers that are the sum of two distinct proper prime powers (A246547).

Original entry on oeis.org

25, 36, 81, 125, 144, 196, 324, 512, 576, 1089, 2304, 2744, 2916, 5041, 9216, 14884, 16641, 26244, 36864, 51984, 147456, 236196, 589824, 941192, 1196836, 2125764, 2359296, 9437184, 19131876, 37748736, 67125249, 150994944, 172186884, 322828856, 603979776

Offset: 1

Altug Alkan, Feb 18 2017

nonn

Intersection of A001597 and A225102. - Michel Marcus, Feb 18 2017

Terms t of A001597 such that A225099(t) > 0. - Felix Fröhlich, Feb 18 2017

			512 = 2^9 is a term because 2^9 = 7^3 + 13^2.

Giovanni Resta, Table of n, a(n) for n = 1..45 (terms < 2*10^11)

Cf. A001597, A225099, A225102, A225106, A246547.

Select[Union@ Map[Total, Subsets[With[{nn = 10^6}, Complement[ Select[ Range@ nn, PrimePowerQ], Prime[Range[PrimePi@ nn]]]], {2}]], # == 1 ||
GCD @@ FactorInteger[#][[All, 2]] > 1 &] (* Michael De Vlieger, Feb 18 2017, after Harvey P. Dale at A246547 *)

is(n) = if(!ispower(n), return(0), my(x=n-1, y=1); while(y < x, if(isprimepower(x) && isprimepower(y) && !ispseudoprime(x) && !ispseudoprime(y), return(1)); y++; x--)); 0 \\ Felix Fröhlich, Feb 18 2017

More terms from Felix Fröhlich, Feb 18 2017

a(28)-a(35) from Giovanni Resta, May 07 2017

A380402 Number of proper prime powers (in A246547) that do not exceed primorial A002110(n).

Original entry on oeis.org

0, 0, 1, 6, 14, 34, 75, 187, 551, 1954, 8317, 38582, 200978, 1125541, 6562122, 40444003, 266832233, 1870169623, 13424553758, 101495825622, 793832121165, 6325729776075, 52616754936494, 450157758564742, 3999323787879764, 37180986240914714, 353667558431662474

Offset: 0

Michael De Vlieger, Jan 23 2025

			Let s = A246547.
a(0) = a(1) = 0 since P(0) = 1 and P(1) = 2, and the smallest number in s is 4.
a(2) = 1 since P(2) = 6, and s(1) = 4 is the only term in s <= 6.
a(3) = 6 since P(3) = 30, and the set s(1..6) = {4, 8, 9, 16, 25, 27} contains k <= 30.
a(4) = 14 since P(4) = 210, and the set s(1..14) = {4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169} contains k <= 210, etc.

Cf. A036386, A246547.

Table[Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor@ Log2[#]}] &[Product[Prime[i], {i, n}]], {n, 0, nn}]

from sympy import primorial, primepi, integer_nthroot
def A380402(n):
    if n == 0: return 0
    m = primorial(n)
    return int(sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))) # Chai Wah Wu, Jan 24 2025

a(n) = Sum_{k=2..floor(log_2(P(n)))} pi(floor(P(n)^(1/k))), where P(n) = A002110(n).

a(24) corrected by Chai Wah Wu, Jan 25 2025

a(26) from Jinyuan Wang, Feb 25 2025

A355014 Index of A355012(n) in A246547.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 11, 7, 14, 12, 17, 10, 19, 21, 9, 24, 18, 25, 13, 28, 29, 30, 34, 15, 36, 39, 41, 43, 44, 27, 45, 48, 38, 16, 53, 54, 55, 22, 57, 49, 51, 52, 33, 71, 76, 77, 60, 63, 64, 65, 67, 23, 68, 70, 88, 20, 89, 90, 72, 75, 96, 97, 98, 42, 79, 80, 81

Offset: 1

Michael De Vlieger, Jun 15 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1009

Cf. A090252, A246547, A355012, A355013.

With[{s = Map[ToExpression@ StringSplit[#][[-1]] &, Import["https://oeis.org/A246547/b246547.txt", "Data"][[2 ;; -1]]]}, Map[FirstPosition[s, #][[1]] &, Select[Import["https://oeis.org/A090252/b090252.txt", "Data"][[1 ;; 8000, -1]], CompositeQ[#] && PrimePowerQ[#] &] ]]

A385379 The maximum possible number of distinct composite prime powers (A246547) in the factorization of n into prime powers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 27 2025

Differs from A376679 at n = 1, 48, 72, 80, ... .
The factorization includes primes if n is not a powerful number (A001694) that is larger than 1.
a(n) depends only on the prime signature of n (A118914).

			         n | a(n) | factorization
  ---------+------+----------------------------------------
         4 |  1   | 2^2
        32 |  2   | 2^2 * 2^3
       288 |  3   | 2^2 * 2^3 * 3^2
      4608 |  4   | 2^2 * 2^3 * 3^2 * 2^4
    115200 |  5   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2
   3110400 |  6   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2 * 3^3
  99532800 |  7   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2 * 3^3 * 2^5

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001694, A005117, A052146, A077761, A118914, A246547, A246655, A376679, A385378, A385380 (indices of records).

f[p_, e_] := Floor[(Sqrt[8*e + 9] - 3)/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> (sqrtint(8*x+9)-1)\2 , factor(n)[, 2]));

Additive with a(p^e) = A052146(e+1).
a(n) = 0 if and only if n is squarefree (A005117).
a(A385380(n)) = n-1.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=1} P(k*(k+3)/2) = 0.49006911093767425812..., and P is the prime zeta function.

A109314 Numbers n such that prime(n) + n is a prime power (A246547).

Original entry on oeis.org

3, 5, 8, 9, 12, 86, 230, 503, 1170, 2660, 2772, 6288, 6572, 8858, 9590, 14870, 16332, 17708, 53132, 54540, 63890, 64908, 82830, 93068, 98132, 104726, 119298, 136502, 152198, 177918, 187040, 234650, 241682, 253118, 263930, 278970, 376680, 412440, 456110, 469034

Offset: 1

Zak Seidov and Robert G. Wilson v, Jun 25 2005

nonn

			2660 is OK because prime(2660) + 2660 = 23909 + 2660 = 26569 = 163^2, 163 is prime.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A025475 = powers of a prime but not prime, also nonprime n such that sigma(n)*phi(n) > (n-1)2; A107708 = values of q, A107709 = values of k; A107710 = values of prime (A109314(n)).

Cf. A107605, A246547.

ispp:= n -> not isprime(n) and nops(numtheory:-factorset(n))=1:
p:= 1: Res:= NULL:
for n from 1 to 10^6 do
  p:= nextprime(p);
  if ispp(n+p) then Res:= Res, n fi
od:
Res; # Robert Israel, Jun 08 2016

lst = {}; fQ[n_] := Block[{pf = FactorInteger[n]}, (2-Length[pf])(pf[[1, 2]]-1) > 0]; Do[ If[ fQ[Prime[n] + n], Print[n]; AppendTo[lst, n]], {n, 456109}]; lst

isok(n) = isprimepower(n+prime(n)) >= 2; \\ Michel Marcus, Jun 18 2017

def np(n): return n+nth_prime(n)
[n for n in (1..10000) if not np(n).is_prime() and np(n).is_prime_power()] # Giuseppe Coppoletta, Jun 08 2016

prime(n) + n = q^k, q is prime and k_Integer >= 2.

A284048 a(n) is the smallest positive integer not already in the sequence such that a(n) + a(n-1) is a proper prime power (A246547), with a(1) = 1.

Original entry on oeis.org

1, 3, 5, 4, 12, 13, 14, 2, 6, 10, 15, 17, 8, 19, 30, 34, 47, 74, 7, 9, 16, 11, 21, 28, 36, 45, 76, 49, 32, 89, 39, 25, 24, 40, 41, 23, 26, 38, 43, 78, 50, 31, 18, 46, 35, 29, 20, 44, 37, 27, 22, 42, 79, 90, 153, 103, 66, 55, 70, 51, 77, 48, 33, 88, 81, 162, 94, 75, 53, 68, 57, 64, 61, 60, 65, 56, 69, 52, 73, 96, 147

Offset: 1

Ilya Gutkovskiy, Mar 19 2017

			a(5) = 12 because 1, 3, 4 and 5 have already been used in the sequence, 4 + 2 = 6, 4 + 6 = 10, 4 + 7 = 11, 4 + 8 = 12, 4 + 9 = 13, 4 + 10 = 14 and 4 + 11 = 15 are not proper prime powers while 4 + 12 = 16 is a proper prime power.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for sequences that are permutations of the natural numbers

Cf. A055265, A055266, A121878, A228730, A243625, A246547, A284049.

N:= 2000: # to get all terms before the first where a(n)+a(n-1)>N
PP:= {seq(seq(p^j, j =2..floor(log[p](N))),p = select(isprime,[2,seq(i,i=3..floor(sqrt(N)),2)]))}:
PP:= sort(convert(PP,list)):
V:= Vector(N,datatype=integer[1],1):
A[1]:= 1; V[1]:= 0;
for n from 2 do
  for pp in PP do
    t:= pp - A[n-1];
    if t > 0 and V[t] = 1 then
      A[n]:= t; V[t]:= 0; break
    fi;
  od;
  if not assigned(A[n]) then break fi
od:
seq(A[i],i=1..n-1); # Robert Israel, Apr 24 2017

f[s_List] := Block[{k = 1, a = s[[-1]]}, While[MemberQ[s, k] || ! (PrimePowerQ[a + k] && PrimeOmega[a + k] > 1), k++];Append[s, k]]; Nest[f, {1}, 80]

A320699 Numbers whose product of prime indices is a nonprime prime power (A246547).

Original entry on oeis.org

7, 9, 14, 18, 19, 21, 23, 25, 27, 28, 36, 38, 42, 46, 49, 50, 53, 54, 56, 57, 63, 72, 76, 81, 84, 92, 97, 98, 100, 103, 106, 108, 112, 114, 115, 121, 125, 126, 131, 133, 144, 147, 152, 159, 162, 168, 171, 184, 189, 194, 196, 200, 206, 212, 216, 224, 227, 228

Offset: 1

Gus Wiseman, Oct 19 2018

nonn

First differs from A320325 at a(43) = 152, A320325(43) = 151.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (4), (2,2), (4,1), (2,2,1), (8), (4,2), (9), (3,3), (2,2,2), (4,1,1), (2,2,1,1), (8,1), (4,2,1), (9,1), (4,4), (3,3,1), (16), (2,2,2,1), (4,1,1,1), (8,2), (4,2,2), (2,2,1,1,1), (8,1,1), (2,2,2,2), (4,2,1,1), (9,1,1), (25), (4,4,1), (3,3,1,1).

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A000961, A001597, A003963, A056239, A064573, A112798, A246547, A246655, A279787, A320322, A320325, A320698, A320700.

Select[Range[100],With[{x=Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]},!PrimeQ[x]&&PrimePowerQ[x]]&]

cp's OEIS Frontend

A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

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A025476 Prime root of n-th nontrivial prime power (A025475, A246547).

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A383295 Positions of proper prime powers (A246547) in EKG-sequence.

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PARI

A282550 Perfect powers that are the sum of two distinct proper prime powers (A246547).

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A380402 Number of proper prime powers (in A246547) that do not exceed primorial A002110(n).

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A355014 Index of A355012(n) in A246547.

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A385379 The maximum possible number of distinct composite prime powers (A246547) in the factorization of n into prime powers.

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A109314 Numbers n such that prime(n) + n is a prime power (A246547).

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