A023894 -id:A023894 - cp's OEIS frontend

A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23

Offset: 1

Gus Wiseman, Dec 14 2018

			24 does not belong to the sequence because there are integer partitions of 24 containing no 1's or prime powers, namely: (24), (18,6), (14,10), (12,12), (12,6,6), (6,6,6,6).

Cf. A000607, A002095, A023893, A023894, A064573, A078135, A101417, A246655, A320322, A322452, A322454, A322547.

nn=100;
ser=Product[If[n==1||PrimePowerQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A322547 Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 71, 79

Offset: 1

Gus Wiseman, Dec 14 2018

			48 does not belong to the sequence because there are integer partitions of 48 containing no 1's, squarefree numbers, or prime powers, namely: (48), (36,12), (28,20), (24,24), (24,12,12), (18,18,12), (12,12,12,12).

Cf. A000607, A002095, A005117, A023893, A023894, A064573, A078135, A101417, A246655, A322546.

nn=100;
ser=Product[If[PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

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)).

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.

A079412 Number of ways to write n as sum of prime powers p^e such that e>0 and p does not divide n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 1, 3, 1, 11, 1, 18, 3, 7, 5, 43, 2, 65, 5, 24, 10, 137, 4, 115, 17, 84, 16, 379, 3, 519, 42, 152, 47, 317, 12, 1267, 73, 334, 41, 2213, 9, 2897, 107, 344, 174, 4871, 32, 3733, 100, 1369, 245, 10218, 51, 4037, 235, 2607, 554, 20586, 23, 25792, 795

Offset: 1

Reinhard Zumkeller, Jan 07 2003

nonn

a(p) = A023894(p) - 1 for p prime.

			13 = 11+2 = 3^2+2^2 = 3^2+2+2 = 2^3+5 = 2^3+3+2 = 7+2^2+2 = 7+3+3 = 7+2+2+2 = 5+5+3 = 5+2^2+2^2 = 5+2^2+2+2 = 5+3+3+2 = 5+2+2+2+2 = 2^2+2^2+3+2 = 2^2+3+3+3 = 2^2+3+2+2+2 = 3+3+3+2+2 = 3+2+2+2+2+2, therefore a(13)=18, (A023894(13)=19, A079413(13)=3);
14 = 11+3 = 3^2+5 = 5+3+3+3, therefore a(14)=3, (A023894(14)=23, A079413(14)=2).

Cf. A079413, A023894, A051613, A000961.

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.

A281616 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, 3, 3, 7, 8, 15, 18, 28, 36, 53, 66, 91, 117, 156, 195, 254, 318, 407, 503, 630, 777, 965, 1176, 1439, 1750, 2124, 2559, 3078, 3692, 4417, 5257, 6246, 7405, 8753, 10314, 12127, 14233, 16668, 19464, 22687, 26406, 30662, 35539, 41109, 47495, 54767, 63044, 72454, 83167, 95305, 109054, 124607, 142209, 162076, 184464

Offset: 1

Ilya Gutkovskiy, Jan 25 2017

nonn

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

Convolution of A001222 and A023894.

			a(9) = 18 because we have [9], [7, 2], [5, 4], [5, 2, 2], [4, 3, 2], [3, 3, 3], [3, 2, 2, 2] and 1 + 2 + 2 + 3 + 3 + 3 + 4 = 18.

Index entries for related partition-counting sequences

Cf. A001222, A023894, A084993, A246655.

nmax = 57; 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)).

A308558 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 into powers of k > 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 1, 4, 2, 2, 2, 1, 6, 3, 2, 2, 2, 1, 6, 3, 2, 2, 2, 2, 1, 10, 3, 3, 2, 2, 2, 2, 1, 10, 5, 3, 2, 2, 2, 2, 2, 1, 14, 5, 3, 3, 2, 2, 2, 2, 2, 1, 14, 5, 3, 3, 2, 2, 2, 2, 2, 2, 1, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 20, 7, 4, 3, 3, 2

Offset: 1

Gus Wiseman, Jun 07 2019

			Triangle begins:
  1
  1  2
  1  2  2
  1  4  2  2
  1  4  2  2  2
  1  6  3  2  2  2
  1  6  3  2  2  2  2
  1 10  3  3  2  2  2  2
  1 10  5  3  2  2  2  2  2
  1 14  5  3  3  2  2  2  2  2
  1 14  5  3  3  2  2  2  2  2  2
  1 20  7  4  3  3  2  2  2  2  2  2
  1 20  7  4  3  3  2  2  2  2  2  2  2
Row n = 6 counts the following partitions:
  (111111)  (42)      (33)      (411)     (51)      (6)
            (222)     (3111)    (111111)  (111111)  (111111)
            (411)     (111111)
            (2211)
            (21111)
            (111111)

Same as A102430 except for the k = 1 column.
Row sums are A102431(n) + 1.
Column k = 2 is A018819.
Column k = 3 is A062051.
Cf. A000961, A001597, A007916, A008284, A023894, A052410, A101417, A112344, A323053, A323054.

Table[If[k==1,1,Length[Select[IntegerPartitions[n],And@@(IntegerQ[Log[k,#]]&/@#)&]]],{n,10},{k,n}]

A339242 Number of partitions of n into prime power parts (1 excluded) where every part appears at least 2 times.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 2, 1, 3, 0, 5, 1, 6, 3, 9, 2, 12, 4, 15, 8, 21, 8, 28, 13, 34, 20, 45, 23, 59, 34, 73, 47, 92, 57, 119, 78, 145, 103, 182, 128, 229, 166, 277, 213, 344, 265, 427, 334, 513, 420, 629, 517, 771, 641, 923, 794, 1120, 967, 1355, 1182, 1618, 1447, 1946, 1745

Offset: 0

Ilya Gutkovskiy, Nov 28 2020

nonn

			a(12) = 5 because we have [4, 4, 4], [4, 4, 2, 2], [3, 3, 3, 3], [3, 3, 2, 2, 2] and [2, 2, 2, 2, 2, 2].

Index entries for sequences related to partitions

Cf. A007690, A023894, A161077, A246655, A339218, A339241.

nmax = 65; CoefficientList[Series[Product[1 + Boole[PrimePowerQ[k]] x^(2 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{p prime, k>=1} (1 + x^(2*p^k) / (1 - x^(p^k))).

A352165 Number of partitions of n into odd prime powers (1 included).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 26, 31, 37, 44, 52, 61, 71, 83, 97, 112, 130, 150, 173, 199, 228, 261, 298, 340, 386, 439, 497, 563, 637, 718, 809, 910, 1023, 1147, 1286, 1439, 1608, 1796, 2003, 2231, 2483, 2761, 3065, 3401, 3770, 4175, 4619

Offset: 0

Ilya Gutkovskiy, Mar 06 2022

nonn

Cf. A000961, A023893, A023894, A061345, A099773, A134345, A280151, A352166.

nmax = 55; CoefficientList[Series[Product[1/(1 - Boole[(PrimePowerQ[k] || k == 1) && OddQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

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

cp's OEIS Frontend

A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

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A322547 Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.

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

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

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A079412 Number of ways to write n as sum of prime powers p^e such that e>0 and p does not divide n.

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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.

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A281616 Expansion of Sum_{p prime, i>=1} x^(p^i)/(1 - x^(p^i)) / Product_{p prime, j>=1} (1 - x^(p^j)).

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A308558 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 into powers of k > 0.

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A339242 Number of partitions of n into prime power parts (1 excluded) where every part appears at least 2 times.

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