A054685 -id:A054685 - 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]&]

A331847 Number of compositions (ordered partitions) of n into distinct prime powers (1 excluded).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 5, 3, 11, 10, 13, 18, 19, 52, 30, 61, 77, 114, 109, 146, 260, 318, 341, 356, 631, 666, 927, 848, 1849, 1978, 2305, 2213, 3560, 4302, 4748, 5588, 6779, 13952, 9044, 15534, 16897, 25084, 20731, 29524, 34882, 49360, 50765, 55112, 106903, 83652, 128552, 106638

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(10) = 10 because we have [8, 2], [7, 3], [5, 3, 2], [5, 2, 3], [3, 7], [3, 5, 2], [3, 2, 5], [2, 8], [2, 5, 3] and [2, 3, 5].

Index entries for sequences related to compositions

Cf. A032020, A032021, A032022, A035942, A054685, A218396, A219107, A246655, A280195, A331843, A331844, A331845, A331846.

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.

A280152 Expansion of Product_{k>=1} (1 + floor(1/omega(2k+1))x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 3, 3, 3, 3, 4, 4, 5, 4, 6, 6, 6, 7, 7, 9, 8, 9, 10, 11, 12, 11, 14, 14, 16, 15, 18, 19, 19, 21, 22, 25, 25, 27, 28, 32, 32, 34, 36, 40, 41, 42, 47, 49, 52, 53, 57, 62, 63, 67, 71, 76, 79, 82, 88, 93, 98, 100, 108, 114, 118, 124

Offset: 0

Ilya Gutkovskiy, Dec 27 2016

nonn

Number of partitions of n into distinct odd prime powers (1 excluded).

			a(16) = 3 because we have [13, 3], [11, 5], [9, 7].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for related partition-counting sequences

Cf. A001221, A054685, A061345, A246655, A280151.

nmax = 78; CoefficientList[Series[Product[1 + Floor[1/PrimeNu[2 k + 1]] x^(2 k + 1), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + floor(1/omega(2*k+1))*x^(2*k+1)).

A300584 Number of partitions of n into distinct prime power parts (not including 1) that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 2, 1, 4, 1, 5, 2, 4, 3, 8, 2, 11, 4, 6, 6, 18, 3, 13, 10, 10, 8, 35, 7, 42, 9, 20, 24, 22, 10, 72, 33, 35, 11, 102, 21, 120, 29, 26, 65, 161, 16, 117, 56, 93, 52, 246, 43, 113, 36, 139, 160, 370, 39, 422, 208, 102, 62, 216, 110, 613, 145, 305, 130, 780, 57, 878

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

			a(9) = 2 because we have [7, 2] and [5, 4].

Index entries for sequences related to partitions

Cf. A054685, A200745, A209402, A246655, A300580.

Table[SeriesCoefficient[Product[(1 + Boole[Mod[n, k] != 0 && PrimePowerQ[k]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 73}]

A328556 Expansion of Product_{p prime, k>=1} (1 - x^(p^k)).

Original entry on oeis.org

1, 0, -1, -1, -1, 0, 1, 1, 0, 0, 1, 1, 1, 0, -1, -1, -2, -1, 0, 0, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, -3, -3, -1, 1, 1, 0, -1, -1, 2, 2, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, 0, -2, -3, -1, -1, 0, 2, 0, 1, 3, 0, 1, 3, 1, -3, -2, -3, -2, 3, 2, -1, 0, -2, 1, 1, -2, -1, 1, 2, 2, 3, -1, -2, 4

Offset: 0

Ilya Gutkovskiy, Nov 01 2019

Convolution inverse of A023894.
The difference between the number of partitions of n into an even number of distinct prime power parts and the number of partitions of n into an odd number of distinct prime power parts (1 excluded).
Conjecture: the last zero (38th) occurs at n = 340.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A023894, A046675, A054685, A246655, A292561.

N:= 100: # for a(0)..a(N)
R:= 1:
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  for k from 1 to floor(log[p](N)) do
    R:= series(R*(1-x^(p^k)),x,N+1)
  od;
od:
seq(coeff(R,x,j),j=0..N); # Robert Israel, Nov 03 2019

nmax = 90; CoefficientList[Series[Product[(1 - Boole[PrimePowerQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[Boole[PrimePowerQ[d]] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 90}]

G.f.: Product_{k>=1} (1 - x^A246655(k)).

A079413 Number of ways to write n as sum of powers p^e of distinct primes p such that e>0 and p does not divide n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 2, 1, 3, 1, 3, 2, 3, 3, 3, 2, 5, 3, 4, 3, 9, 3, 5, 4, 6, 4, 18, 3, 20, 8, 7, 8, 10, 6, 30, 9, 11, 8, 41, 5, 47, 11, 12, 13, 63, 10, 42, 13, 23, 16, 89, 13, 35, 20, 34, 28, 126, 11, 134, 35, 36, 44, 57, 15, 185, 40, 64, 19, 236, 31, 251, 64, 55, 54, 117, 24, 341

Offset: 1

Reinhard Zumkeller, Jan 07 2003

nonn

a(p) = A051613(p) - 1 for p prime.

			13 = 11+2 = 3^3+2^2 = 2^3+5, therefore a(13)=3, (A051613(13)=4, A054685(13)=6, A079412(13)=18);
14 = 11+3 = 3^2+5, therefore a(14)=2, (A051613(14)=4, A054685(14)=7, A079412(14)=3).

Cf. A079412, A054685, A051613, A023894, A000961.

A281668 Expansion of Sum_{p prime, i>=1} x^(p^i)/(1 + x^(p^i)) * Product_{p prime, j>=1} (1 + x^(p^j)).

Original entry on oeis.org

0, 1, 1, 1, 3, 2, 5, 3, 8, 7, 10, 12, 13, 20, 18, 26, 25, 36, 34, 45, 47, 59, 62, 71, 82, 91, 105, 112, 132, 143, 163, 174, 201, 220, 244, 266, 298, 327, 362, 388, 437, 470, 521, 558, 621, 671, 733, 788, 864, 938, 1011, 1100, 1182, 1295, 1379, 1501, 1606, 1753, 1861, 2017, 2158, 2335, 2493, 2672, 2871, 3078

Offset: 1

Ilya Gutkovskiy, Jan 26 2017

nonn

Total number of parts in all partitions of n into distinct prime power parts (1 excluded).

			a(10) = 7 because we have [8, 2], [7, 3], [5, 3, 2] and 2 + 2 + 3 = 7.

Index entries for related partition-counting sequences

Cf. A015723, A024938, A054685, A246655, A281616.

nmax = 66; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[i]] x^i/(1 + x^i), {i, 2, nmax}] Product[1 + Floor[1/PrimeNu[j]] x^j, {j, 2, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{p prime, i>=1} x^(p^i)/(1 + x^(p^i)) * Product_{p prime, j>=1} (1 + x^(p^j)).

A352166 Number of partitions of n into distinct odd prime powers (1 included).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 4, 5, 6, 6, 6, 7, 8, 9, 9, 10, 12, 12, 13, 14, 16, 17, 17, 19, 21, 23, 23, 25, 28, 30, 31, 33, 37, 38, 40, 43, 47, 50, 52, 55, 60, 64, 66, 70, 76, 81, 83, 89, 96, 101, 105, 110, 119, 125, 130, 138, 147, 155, 161

Offset: 0

Ilya Gutkovskiy, Mar 06 2022

nonn

Cf. A000961, A024939, A054685, A061345, A106244, A134337, A280152, A352165.

nmax = 70; CoefficientList[Series[Product[(1 + Boole[(PrimePowerQ[k] || k == 1) && OddQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} (1 + x^A061345(k)).

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

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A331847 Number of compositions (ordered partitions) of n into distinct prime powers (1 excluded).

Views

Author

Keywords

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A280152 Expansion of Product_{k>=1} (1 + floor(1/omega(2*k+1))*x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A300584 Number of partitions of n into distinct prime power parts (not including 1) that do not divide n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A328556 Expansion of Product_{p prime, k>=1} (1 - x^(p^k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A079413 Number of ways to write n as sum of powers p^e of distinct primes p such that e>0 and p does not divide n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A281668 Expansion of Sum_{p prime, i>=1} x^(p^i)/(1 + x^(p^i)) * Product_{p prime, j>=1} (1 + x^(p^j)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A352166 Number of partitions of n into distinct odd prime powers (1 included).

Views

Author

Keywords

A280152 Expansion of Product_{k>=1} (1 + floor(1/omega(2k+1))x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).