A085970 -id:A085970 - cp's OEIS frontend

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

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]&]

A305975 Filter sequence: All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to their exponent k, while all other numbers occur in singular equivalence classes of their own.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 8, 9, 10, 2, 11, 2, 12, 13, 14, 2, 15, 3, 16, 5, 17, 2, 18, 2, 19, 20, 21, 22, 23, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 3, 32, 33, 34, 2, 35, 36, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 53, 54, 2, 55, 10, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71

Offset: 1

Antti Karttunen, Jul 02 2018

nonn

Restricted growth sequence transform of A305974.

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A305976(i) = A305976(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000961 (A246655), A085970, A305800, A305974, A305976, A305979.

up_to = 100000;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v065515 = partialsums(n -> (omega(n)<=1), up_to);
A065515(n) = v065515[n];
A085970(n) = (n - A065515(n));
A305974(n) = if(1==n,n,my(e = isprimepower(n)); if(e,-e,1+A085970(n)));
v305975 = rgs_transform(vector(up_to,n,A305974(n)));
A305975(n) = v305975[n];

a(prime) = 2, a(prime^2) = 3, a(prime^3) = 5, a(prime^4) = 10, a(prime^5) = 19.

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.

A085972 Number of numbers <= n that are primes or not prime powers.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 54, 55, 56, 57, 58, 59, 60, 61

Offset: 1

Reinhard Zumkeller, Jul 06 2003

nonn

a(n) = Max{k: A085971(k)<=n};

a(n) = n-A085501(n) = A000720(n)+n-A065515(n) = A085970(n)+A000720(n).

Accumulate[Table[If[PrimeQ[n]||(!PrimePowerQ[n]),1,0],{n,80}]]-1 (* Harvey P. Dale, Oct 13 2022 *)

from sympy import primepi, integer_nthroot
def A085972(n): return n-1-sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length())) # Chai Wah Wu, Aug 20 2024

A368749 a(n) is the smallest prime p such that there are n numbers between p and nextprime(p) which are not prime powers.

Original entry on oeis.org

2, 5, 13, 19, 31, 53, 359, 89, 241, 139, 509, 113, 839, 293, 2803, 1831, 523, 1069, 11447, 887, 3469, 1129, 1669, 4177, 39581, 2477, 24631, 2971, 16381, 4297, 124601, 5591, 1327, 8467, 22193, 9551, 79493, 30593, 62989, 19333, 410857, 16141, 436913, 15683, 1038337, 81463, 157579

Offset: 0

David James Sycamore and Michael De Vlieger, Jan 04 2024

nonn

Conjecture: the sequence is infinite (number of nonprime powers between consecutive primes is arbitrarily large; see A014963).

			a(2) = 13 because between 13 and 17 there are three composite numbers, only one of which (16) is a prime power, and no previous prime has this property.
a(5) = 53 because between 53 and 59 there are 5 composite numbers, none of which are prime powers, and no smaller prime has this property.

Cf. A000040, A000230, A000961, A014963, A024619, A080101, A085970, A093555, A368748.

p = q = 2; r = a[] = 0; Do[q = NextPrime[q]; If[a[#] == 0, a[#] = p; If[# > r, r = #]] &@ Count[Range[p, q - 1], ?(Not@*PrimePowerQ)]; p = q, {i, 2^16}]; {2}~Join~TakeWhile[Array[a, r], # > 0 &] (* Michael De Vlieger, Jan 04 2024 *)

f(p) = sum(k=p+1, nextprime(p+1)-1, !isprimepower(k));
a(n) = my(p=2); while(f(p) != n, p=nextprime(p+1)); p; \\ Michel Marcus, Jan 04 2024

More terms from Michel Marcus, Jan 04 2024

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}]

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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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A305975 Filter sequence: All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to their exponent k, while all other numbers occur in singular equivalence classes of their own.

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

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A085972 Number of numbers <= n that are primes or not prime powers.

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A368749 a(n) is the smallest prime p such that there are n numbers between p and nextprime(p) which are not prime powers.

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

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