A164336 -id:A164336 - cp's OEIS frontend

A352518 Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves prime numbers.

225, 675, 1089, 1125, 2601, 3025, 3267, 3375, 6075, 7225, 7803, 8649, 11979, 15125, 15129, 24025, 25947, 27225, 28125, 29403, 30375, 31329, 33275, 34969, 35937, 36125, 40401, 42025, 44217, 45387, 54675, 62001, 65025, 70227, 81675, 84375, 87025, 93987

Offset: 1

Gus Wiseman, Mar 24 2022

nonn

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 terms together with their prime indices (not factors) begin:
     225: {2,2,3,3}
     675: {2,2,2,3,3}
    1089: {2,2,5,5}
    1125: {2,2,3,3,3}
    2601: {2,2,7,7}
    3025: {3,3,5,5}
    3267: {2,2,2,5,5}
    3375: {2,2,2,3,3,3}
    6075: {2,2,2,2,2,3,3}
    7225: {3,3,7,7}
    7803: {2,2,2,7,7}
    8649: {2,2,11,11}
   11979: {2,2,5,5,5}
   15125: {3,3,3,5,5}
   15129: {2,2,13,13}
   24025: {3,3,11,11}
   25947: {2,2,2,11,11}
   27225: {2,2,3,3,5,5}
   28125: {2,2,3,3,3,3,3}
For example, 7803 = prime(1)^3 prime(4)^2.

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

These partitions are counted by A352493.
This is the restriction of A346068 to numbers that are not a prime power.
The prime-power version is A352519, counted by A230595.
A000040 lists the primes.
A000961 lists prime powers.
A001694 lists powerful numbers, counted by A007690.
A038499 counts partitions of prime length.
A053810 lists all numbers p^q for p and q prime, counted by A001221.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A001597, A002035, A007821, A007916, A101436, A117958, A164336, A268335, A330945.

Select[Range[10000],!PrimePowerQ[#]&& And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&& And@@PrimeQ/@Last/@FactorInteger[#]&]

Sum_{n>=1} 1/a(n) = (Product_{p prime-indexed prime} (1 + Sum_{q prime} 1/p^q)) - (Sum_{p prime-indexed prime} Sum_{q prime} 1/p^q) - 1 = 0.0106862606... . - Amiram Eldar, Aug 04 2024

A352519 Numbers of the form prime(p)^q where p and q are primes. Prime powers whose prime index and exponent are both prime.

Original entry on oeis.org

9, 25, 27, 121, 125, 243, 289, 961, 1331, 1681, 2187, 3125, 3481, 4489, 4913, 6889, 11881, 16129, 24649, 29791, 32041, 36481, 44521, 58081, 68921, 76729, 78125, 80089, 109561, 124609, 134689, 160801, 161051, 177147, 185761, 205379, 212521, 259081, 299209

Offset: 1

Gus Wiseman, Mar 26 2022

nonn

Alternatively, numbers of the form prime(prime(i))^prime(j) for some positive integers i, j.

			The terms together with their prime indices begin:
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    125: {3,3,3}
    243: {2,2,2,2,2}
    289: {7,7}
    961: {11,11}
   1331: {5,5,5}
   1681: {13,13}
   2187: {2,2,2,2,2,2,2}
   3125: {3,3,3,3,3}
   3481: {17,17}
   4489: {19,19}
   4913: {7,7,7}
   6889: {23,23}
  11881: {29,29}
  16129: {31,31}
  24649: {37,37}
  29791: {11,11,11}

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

Numbers of the form p^q for p and q prime are A053810, counted by A001221.
These partitions are counted by A230595.
This is the prime power case of A346068.
For numbers that are not a prime power we have A352518, counted by A352493.
A000040 lists the primes.
A000961 lists prime powers.
A001597 lists perfect powers.
A001694 lists powerful numbers, counted by A007690.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A164336 lists all possible power-towers of prime numbers.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A002035, A005117, A007821, A038499, A101436, A181819, A181821.

N:= 10^7: # for terms <= N
M:=numtheory:-pi(numtheory:-pi(isqrt(N))):
PP:= {seq(ithprime(ithprime(i)),i=1..M)}:
R:= NULL:
for p in PP do
  q:= 1:
  do
    q:= nextprime(q);
    t:= p^q;
    if t > N then break fi;
    R:= R, t;
  od;
od:
sort([R]); # Robert Israel, Dec 08 2022

Select[Range[10000],PrimePowerQ[#]&&MatchQ[FactorInteger[#],{{?(PrimeQ[PrimePi[#]]&),k?PrimeQ}}]&]

from sympy import primepi, integer_nthroot, primerange
def A352519(n):
    def f(x): return int(n+x-sum(primepi(primepi(integer_nthroot(x,p)[0])) for p in primerange(x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

A096165 Prime powers with exponents that are themselves prime powers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Reinhard Zumkeller, Jul 25 2004

nonn

A000040, A053810, A050376 and A082522 are subsequences;
a(n) = A000961(n+1) for 1<=n<=26.
Complement of A164345 with respect to A000961.

			512=2^9=2^(3^2), A000961(118)=A000040(1)^A000961(118), therefore 512 is a term;
64=2^6, but 6 is not a prime power, therefore 64 is not a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A000961, A010055, A001222, A050376, A053810, A082522, A164336, A164345.

a096165 n = a096165_list !! (n-1)
a096165_list = filter ((== 1) . a010055 . a001222) $ tail a000961_list
-- Reinhard Zumkeller, Nov 17 2011

F:= proc(t) local P;
P:= ifactors(t)[2];
nops(P) = 1 and (P[1][2]=1 or nops(numtheory:-factorset(P[1][2]))=1)
end proc:
select(F, [$2..1000]); # Robert Israel, Jul 20 2015

Select[Range@ 240, Or[PrimeQ@ #, PrimePowerQ@ # && PrimePowerQ@ FactorInteger[#][[1, 2]]] &] (* Michael De Vlieger, Jul 20 2015 *)

is(n)=while(1,if(!(n=isprimepower(n)),return(0),if(n==1,return(1)))) \\ Anders Hellström, Jul 19 2015

ispp(n)=n==1 || isprimepower(n)
is(n)=ispp(isprimepower(n)) \\ Charles R Greathouse IV, Oct 19 2015

from sympy import primepi, integer_nthroot, factorint
def A096165(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length()) if len(factorint(k))<=1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

a(n) ~ n log n. - Charles R Greathouse IV, Oct 19 2015

A287620 a(n) = product, with multiplicity, of the prime numbers appearing at leaf positions in the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 3, 2, 10, 11, 6, 13, 14, 15, 2, 17, 4, 19, 10, 21, 22, 23, 9, 2, 26, 3, 14, 29, 30, 31, 5, 33, 34, 35, 4, 37, 38, 39, 15, 41, 42, 43, 22, 10, 46, 47, 6, 2, 4, 51, 26, 53, 6, 55, 21, 57, 58, 59, 30, 61, 62, 14, 6, 65, 66, 67, 34, 69, 70, 71

Offset: 1

Rémy Sigrist, May 28 2017

The prime tower factorization of a number is defined in A182318.
a(n) <= n.
a(n) = n iff n is squarefree (A005117).
a(n) is noncomposite iff n belongs to A164336.
This sequence is surjective; see A287621 for the least k such that a(k) = n.
For n>1, A001222(a(n)) = A064372(n).

			See illustration of the first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms

Cf. A001222, A005117, A064372, A164336, A182318, A287621.

a(n) = my (f=factor(n)); return (prod(i=1, #f~, if (f[i,2]==1, f[i,1], a(f[i,2]))))

Multiplicative with:
- a(p) = p for any prime p,
- a(p^k) = a(k) for any prime p and k > 1.

A323129 a(1) = 1, and for any n > 1, let p be the greatest prime factor of n, and e be its exponent, then a(n) = p^a(e).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 49, 25, 17, 13, 53, 27, 11, 7, 19, 29, 59, 5, 61, 31, 7, 8, 13, 11, 67, 17, 23, 7, 71, 9, 73

Offset: 1

Rémy Sigrist, Jan 05 2019

This sequence is a recursive variant of A053585.

All terms belong to A164336.

			a(1458) = a(2 * 3^6) = 3^a(6) = 3^a(2*3) = 3^3 = 27.

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

See A323130 for the variant involving the least prime factor.

Cf. A006530, A053585, A071178, A164336.

f:= proc(n) option remember;
  local F,t;
  F:= ifactors(n)[2];
  t:= F[max[index](map(t -> t[1],F))];
  t[1]^procname(t[2]);
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Jan 07 2019

Nest[Append[#, Last@ FactorInteger[Length[#] + 1] /. {p_, e_} :> p^#[[e]] ] &, {1}, 72] (* Michael De Vlieger, Jan 07 2019 *)

a(n) = if (n==1, 1, my (f=factor(n)); f[#f~,1]^a(f[#f~,2]))

a(n) <= n with equality iff n belongs to A164336.

a(n) = A006530(n)^a(A071178(n)) for any n > 1.

A383309 Numbers whose prime indices are prime powers > 1 with a common sum of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 17, 19, 23, 25, 27, 31, 35, 41, 49, 53, 59, 67, 81, 83, 97, 103, 109, 121, 125, 127, 131, 157, 175, 179, 191, 209, 211, 227, 241, 243, 245, 277, 283, 289, 311, 331, 343, 353, 361, 367, 391, 401, 419, 431, 461, 509, 529, 547, 563, 587, 599

Offset: 1

Gus Wiseman, Apr 25 2025

nonn

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. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The systems with these MM-numbers begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  35: {{2},{1,1}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  59: {{7}}
  67: {{8}}
  81: {{1},{1},{1},{1}}
  83: {{9}}
  97: {{3,3}}

Twice-partitions of this type are counted by A279789.
For just a common sum we have A326534.
For just constant blocks we have A355743.
Numbers without a factorization of this type are listed by A381871, counted by A381993.
The multiplicative version is A381995.
This is the odd case of A382215.
For strict instead of constant blocks we have A382304.
A001055 counts factorizations, strict A045778.
A023894 counts partitions into prime-powers.
A034699 gives maximal prime-power divisor.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A317141 counts coarsenings of prime indices, refinements A300383.
A353864 counts rucksack partitions, ranked by A353866.
A355742 chooses a prime-power divisor of each prime index.
Cf. A000688, A000720, A001222, A006171, A038041, A279784, A302242, A302493, A321455, A326518, A381719.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Total/@prix/@prix[#]&&And@@PrimePowerQ/@prix[#]&]

Equals A326534 /\ A355743.

A323130 a(1) = 1, and for any n > 1, let p be the least prime factor of n, and e be its exponent, then a(n) = p^a(e).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 4, 13, 2, 3, 16, 17, 2, 19, 4, 3, 2, 23, 8, 25, 2, 27, 4, 29, 2, 31, 32, 3, 2, 5, 4, 37, 2, 3, 8, 41, 2, 43, 4, 9, 2, 47, 16, 49, 2, 3, 4, 53, 2, 5, 8, 3, 2, 59, 4, 61, 2, 9, 4, 5, 2, 67, 4, 3, 2, 71, 8, 73, 2, 3, 4, 7, 2, 79

Offset: 1

Rémy Sigrist, Jan 05 2019

nonn

This sequence is a recursive variant of A028233.

All terms belong to A164336.

			a(320) = a(2^6 * 5) = 2^a(6) = 2^a(2*3) = 2^2 = 4.

See A323129 for the variant involving the greatest prime factor.

Cf. A020639, A028233, A067029, A164336.

Nest[Append[#, First@ FactorInteger[Length[#] + 1] /. {p_, e_} :> p^#[[e]] ] &, {1}, 78] (* Michael De Vlieger, Jan 07 2019 *)

a(n) = if (n==1, 1, my (f=factor(n)); f[1,1]^a(f[1,2]))

a(n) <= n with equality iff n belong to A164336.

a(n) = A020639(n)^a(A067029(n)) for any n > 1.

A356067 Number of integer partitions of n into relatively prime prime-powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 11, 7, 18, 16, 26, 27, 43, 41, 65, 65, 92, 100, 137, 142, 194, 210, 270, 295, 379, 410, 519, 571, 699, 782, 947, 1046, 1267, 1414, 1673, 1870, 2213, 2465, 2897, 3230, 3757, 4210, 4871, 5427, 6265, 6997

Offset: 0

Gus Wiseman, Jul 28 2022

nonn

			The a(5) = 1 through a(12) = 7 partitions:
  (32)  .  (43)   (53)   (54)    (73)    (74)     (75)
           (52)   (332)  (72)    (433)   (83)     (543)
           (322)         (432)   (532)   (92)     (552)
                         (522)   (3322)  (443)    (732)
                         (3222)          (533)    (4332)
                                         (542)    (5322)
                                         (722)    (33222)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

This is the relatively prime case of A023894, facs A000688, w/ 1's A023893.
For strict instead of coprime: A054685, facs A050361, with 1's A106244.
The version for factorizations instead of partitions is A354911.
A000041 counts partitions, strict A000009.
A072233 counts partitions by sum and length.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A279784 counts twice-partitions where the latter partitions are constant.
A289509 lists numbers whose prime indices are relatively prime.
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A001970, A055887, A063834, A076610, A085970, A355737, A355742.

Table[Length[Select[IntegerPartitions[n],And@@PrimePowerQ/@#&&GCD@@#==1&]],{n,0,30}]

cp's OEIS Frontend

A352518 Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves prime numbers.

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A352519 Numbers of the form prime(p)^q where p and q are primes. Prime powers whose prime index and exponent are both prime.

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A096165 Prime powers with exponents that are themselves prime powers.

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A287620 a(n) = product, with multiplicity, of the prime numbers appearing at leaf positions in the prime tower factorization of n.

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A323129 a(1) = 1, and for any n > 1, let p be the greatest prime factor of n, and e be its exponent, then a(n) = p^a(e).

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A383309 Numbers whose prime indices are prime powers > 1 with a common sum of prime indices.

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A323130 a(1) = 1, and for any n > 1, let p be the least prime factor of n, and e be its exponent, then a(n) = p^a(e).

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