A164336 -id:A164336 - cp's OEIS frontend

A064372 Additive function a(n) defined by the recursive formula a(1)=1 and a(p^k)=a(k) for any prime p.

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

Offset: 1

Steven Finch, Sep 26 2001

That is, if i, j, k, ... are relatively prime, then a(i*j*k*...) = a(i) + a(j) + a(k) + ... - N. J. A. Sloane, Nov 20 2007
Starts almost the same as A001221 (the number of distinct primes dividing n): the first twelve terms which are different are a(1), a(64), a(192), a(320), a(448), a(576), a(704), a(729), a(832), a(960), a(1024) and a(1088), since the first non-unitary values of n are a(6) and(10). - Henry Bottomley, Sep 23 2002
a(A164336(n)) = 1. - Reinhard Zumkeller, Aug 27 2011

			a(30) = a(5^1 * 3^1 * 2^1) = a(1) + a(1) + a(1) = 3.

Cf. A001221, A079553, A106444, A106490, A106491, A106495, A124010, A164336.

a064372 1 = 1
a064372 n = sum $ map a064372 $ a124010_row n
-- Reinhard Zumkeller, Aug 27 2011

a:= proc(n) option remember; `if`(n=1, 1,
      add(a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Aug 23 2020

a[1] = 1; a[n_] := a[n] = Plus @@ a /@ FactorInteger[n][[All, 2]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Sep 19 2012 *)

a(n) = A106491(n) - A106490(n) = A106495(A106444(n)). - Antti Karttunen, May 09 2005

a(1) = 1, a(n) = Sum_{k=1..A001221(n)} a(A124010(n,k)) for n > 1. - Reinhard Zumkeller, Aug 27 2011

A356065 Squarefree numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

The multiplicative version (factorizations) is A050361, non-strict A000688.
Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.
Counting twice-partitions of this type gives A279786, non-strict A279784.
Counting twice-factorizations gives A295935, non-strict A296131.
These are the odd products of distinct elements of A302493.
Allowing prime index 1 gives A302496, non-strict A302492.
The case of primes (instead of prime-powers) is A302590, non-strict A076610.
These are the squarefree positions of 1's in A355741.
This is the squarefree case of A355743, complement A356066.
A001222 counts prime-power divisors.
A005117 lists the squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

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

Intersection of A005117 and A355743.

A356068 Number of integers ranging from 1 to n that are not prime-powers (1 is not a prime-power).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 22, 23, 24, 25, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42

Offset: 1

Gus Wiseman, Jul 31 2022

nonn

			The a(30) = 14 numbers: 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30.

The complement is counted by A025528, with 1's A065515.
For primes instead of prime-powers we have A062298, with 1's A065855.
The version treating 1 as a prime-power is A085970.
One more than the partial sums of A143731.
A000688 counts factorizations into prime-powers.
A001222 counts prime-power divisors.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A000720, A023894, A036234, A050361, A054685, A069513, A355743, A356064, A356065, A356066.

Table[Length[Select[Range[n],!PrimePowerQ[#]&]],{n,100}]

a(n) = A085970(n) + 1.

A294337 Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 7, 2, 4, 4, 10, 2, 7, 2, 7, 4, 4, 2, 10, 4, 4, 6, 7, 2, 8, 2, 12, 4, 4, 4, 12, 2, 4, 4, 10, 2, 8, 2, 7, 7, 4, 2, 15, 4, 7, 4, 7, 2, 10, 4, 10, 4, 4, 2, 13, 2, 4, 7, 16, 4, 8, 2, 7, 4, 8, 2, 16, 2, 4, 7, 7, 4, 8, 2, 15, 10, 4, 2, 13, 4, 4, 4, 10, 2, 13, 4, 7, 4, 4, 4, 18, 2, 7, 7, 12, 2, 8, 2, 10, 8

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

			The a(12) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294338, A294339.

a[n_]:=1+Sum[a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Table[a[2^n],{n,100}]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d)));
A294337(n) = sumdiv(n,d,A294336(d));
\\ Or alternatively, after Mathematica-code as:
A294337(n) = A294336(2^n); \\ Antti Karttunen, Jun 12 2018

a(n) = Sum_{d|n} A294336(d) = A294336(A000079(n)). - Antti Karttunen, Jun 12 2018

More terms from Antti Karttunen, Jun 12 2018

A351982 Number of integer partitions of n into prime parts with prime multiplicities.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 3, 0, 1, 1, 3, 3, 3, 0, 1, 4, 5, 5, 3, 3, 5, 8, 5, 5, 6, 8, 8, 11, 7, 8, 10, 17, 14, 14, 12, 17, 17, 21, 18, 23, 20, 28, 27, 31, 27, 36, 32, 35, 37, 46, 41, 52, 45, 60, 58, 63, 59, 78, 71, 76, 81, 87, 80, 103, 107, 113, 114, 127

Offset: 0

Gus Wiseman, Mar 18 2022

nonn

			The partitions for n = 4, 6, 10, 19, 20, 25:
  (22)  (33)   (55)     (55333)     (7733)       (55555)
        (222)  (3322)   (55522)     (77222)      (77722)
               (22222)  (3333322)   (553322)     (5533333)
                        (33322222)  (5522222)    (5553322)
                                    (332222222)  (55333222)
                                                 (55522222)
                                                 (333333322)
                                                 (3333322222)

The version for just prime parts is A000607, ranked by A076610.
The version for just prime multiplicities is A055923, ranked by A056166.
For odd instead of prime we have A117958, ranked by A352142.
The constant case is A230595, ranked by A352519.
Allowing any multiplicity > 1 gives A339218, ranked by A352492.
These partitions are ranked by A346068.
The non-constant case is A352493, ranked by A352518.
A000040 lists the primes.
A001221 counts constant partitions of prime length, ranked by A053810.
A001694 lists powerful numbers, counted A007690, weak A052485.
A038499 counts partitions of prime length.
A101436 counts parts of prime signature that are themselves prime.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A257994 counts prime indices that are prime, nonprime A330944.
Cf. A000720, A000961, A001156, A011757, A052335, A164336, A320628.

Table[Length[Select[IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]

A352492 Powerful numbers whose prime indices are all prime numbers.

Original entry on oeis.org

1, 9, 25, 27, 81, 121, 125, 225, 243, 289, 625, 675, 729, 961, 1089, 1125, 1331, 1681, 2025, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 5625, 6075, 6561, 6889, 7225, 7803, 8649, 9801, 10125, 11881, 11979, 14641, 15125, 15129, 15625, 16129, 16875

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 prime factors) begin:
    1: {}
    9: {2,2}
   25: {3,3}
   27: {2,2,2}
   81: {2,2,2,2}
  121: {5,5}
  125: {3,3,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  289: {7,7}
  625: {3,3,3,3}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  961: {11,11}
For example, 675 = prime(2)^3 prime(3)^2 = 3^3 * 5^2.

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

Powerful numbers are A001694, counted by A007690.
The version for prime exponents instead of indices is A056166, counted by A055923.
This is the powerful case of A076610 (products of A006450), counted by A000607.
The partitions with these Heinz numbers are counted by A339218.
A000040 lists primes.
A031368 lists primes of odd index, products A066208.
A101436 counts exponents in prime factorization that are themselves prime.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A053810 lists all numbers p^q with p and q prime, counted by A230595.
A257994 counts prime indices that are themselves prime, complement A330944.
Cf. A000720, A000961, A001597, A007821, A007916, A052485, A109297, A140319, A164336, A181819, A325131, A330945, A346068.

Select[Range[1000],#==1||And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&&Min@@Last/@FactorInteger[#]>1&]

Intersection of A001694 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/(p*(p-1))) = 1.24410463... - Amiram Eldar, May 04 2022

A294338 Number of ways to write n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

			The a(16) = 5 ways are: 16, 4^2, (2^2)^2, 2^4, 2^(2^2).

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294337, A294339.

A294338 := proc(n)
    local expo,g,a,d ;
    if n =1 then
        return 1;
    end if;
    # compute gcd of the set of prime power exponents (A052409)
    ifactors(n)[2] ;
    [ seq(op(2,ep),ep=%)] ;
    igcd(op(%)) ;
    # set of divisors of A052409 (without the 1)
    g := numtheory[divisors](%) minus {1} ;
    a := 0 ;
    for d in g do
        # recursive (sort of convolution) call
        a := a+ procname(d)*procname(root[d](n)) ;
    end do:
    1+a ;
end proc:
seq(A294338(n),n=1..120) ; # R. J. Mathar, Nov 27 2017

a[n_]:=1+Sum[a[n^(1/g)]*a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Array[a,100]

A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

Original entry on oeis.org

13, 26, 29, 37, 39, 43, 47, 52, 58, 61, 65, 71, 73, 74, 78, 79, 86, 87, 89, 91, 94, 101, 104, 107, 111, 113, 116, 117, 122, 129, 130, 137, 139, 141, 142, 143, 145, 146, 148, 149, 151, 156, 158, 163, 167, 169, 172, 173, 174, 178, 181, 182, 183, 185, 188, 193

Offset: 1

Gus Wiseman, Jul 25 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.

These are numbers divisible by a prime number not of the form prime(q^k) where q is a prime number and k >= 1.

			The terms together with their prime indices begin:
   13: {6}
   26: {1,6}
   29: {10}
   37: {12}
   39: {2,6}
   43: {14}
   47: {15}
   52: {1,1,6}
   58: {1,10}
   61: {18}
   65: {3,6}
   71: {20}
   73: {21}
   74: {1,12}
   78: {1,2,6}
   79: {22}
   86: {1,14}
   87: {2,10}

Heinz numbers of the partitions counted by A023893.
Allowing prime index 1 gives A356066.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A023894 counts partitions into prime-powers, strict A054685.
A034699 gives the maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
A355743 = numbers whose prime indices are prime-powers, squarefree A356065.
Cf. A076610, A085970, A106244, A302492, A302493, A302601, A330946, A354911.

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

A356066 Numbers with a prime index that is not a prime-power. Complement of A355743.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 52, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 84, 86, 87, 88, 89, 90, 91, 92, 94, 96, 98, 100, 101

Offset: 1

Gus Wiseman, Jul 31 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 begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}

The complement is A355743, counted by A023894.
The squarefree complement is A356065, counted by A054685.
Allowing prime index 1 gives A356064, complement A302492.
A000688 counts factorizations into prime-powers, strict A050361.
A001222 counts prime-power divisors.
A034699 gives the maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A076610, A085970, A106244, A302493, A302601, A330946, A354911.

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

Union of A299174 and A356064.

A294339 Number of ways to write 2^n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 12, 5, 6, 2, 19, 2, 6, 6, 32, 2, 19, 2, 19, 6, 6, 2, 56, 5, 6, 12, 19, 2, 26, 2, 79, 6, 6, 6, 71, 2, 6, 6, 56, 2, 26, 2, 19, 19, 6, 2, 169, 5, 19, 6, 19, 2, 56, 6, 56, 6, 6, 2, 101, 2, 6, 19, 203, 6, 26, 2, 19, 6, 26, 2, 237, 2, 6, 19, 19

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

			The a(6) = 6 ways are 64, 8^2, (2^3)^2, 4^3, (2^2)^3, 2^6.

Hans Havermann, Table of n, a(n) for n = 1..10000

Cf. A001055, A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294337, A294338.

f:= proc(n) option remember; local F,t,s,g,a;
  F:= ifactors(n)[2];
  g:= igcd(op(map(t -> t[2],F)));
  t:= 1;
  for s in numtheory:-divisors(g) minus {1} do
    t:= t + procname(mul(a[1]^(a[2]/s),a=F))*procname(s)
  od;
  t
end proc:
seq(f(2^n),n=1..100); # Robert Israel, Dec 01 2017

a[n_]:=1+Sum[a[n^(1/g)]*a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Table[a[2^n],{n,100}]

a(n) = A294338(2^n). - R. J. Mathar, Nov 27 2017

cp's OEIS Frontend

A064372 Additive function a(n) defined by the recursive formula a(1)=1 and a(p^k)=a(k) for any prime p.

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A356065 Squarefree numbers whose prime indices are all prime-powers.

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A356068 Number of integers ranging from 1 to n that are not prime-powers (1 is not a prime-power).

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A294337 Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

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A351982 Number of integer partitions of n into prime parts with prime multiplicities.

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A352492 Powerful numbers whose prime indices are all prime numbers.

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A294338 Number of ways to write n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.

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A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

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