A085755
Number of partitions of n into a prime number of prime parts.
Original entry on oeis.org
1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 6, 8, 8, 9, 9, 12, 12, 16, 16, 19, 19, 26, 24, 31, 29, 39, 35, 50, 44, 61, 55, 74, 67, 93, 80, 111, 99, 136, 119, 166, 145, 197, 179, 239, 213, 292, 255, 342, 310, 409, 365, 492, 436, 577, 524, 682, 614, 814, 724, 947, 865, 1113, 1007, 1314
Offset: 4
a(20) = 12 because there are 12 partitions of 20 into a prime number of prime parts: 2+3+3+3+3+3+3 = 2+2+2+3+3+3+5 = 2+2+2+2+2+5+5 = 2+2+2+2+2+3+7 = 2+3+5+5+5 = 2+3+3+5+7 = 2+2+2+7+7 = 2+2+2+3+11 = 2+7+11 = 2+5+13 = 7+13 = 3+17.
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b:= proc(n, i, t) if n<0 then 0 elif n=0 then `if`(isprime(t), 1, 0) elif i=1 then `if`(irem(n,2)=0 and isprime(t +n/2), 1, 0) else b(n,i,t):= b(n -ithprime(i), i, t+1) +b(n, i-1, t) fi end: a:= proc(n) local i; for i while ithprime(i)Alois P. Heinz, Apr 30 2009
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Table[Length[Select[IntegerPartitions[n],PrimeQ[Length[#]]&&AllTrue[ #, PrimeQ]&]],{n,4,70}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 18 2016 *)
A372688
Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.
Original entry on oeis.org
0, 0, 2, 2, 1, 3, 3, 6, 3, 6, 9, 20, 13, 22, 22, 45, 47, 70, 75, 100, 107, 132, 157, 202, 229, 302, 396, 495, 536, 699, 820, 962, 1193, 1507, 1699, 2064, 2455, 2945, 3408, 4026, 4691, 5749, 6670, 7614, 9127, 10930, 12329, 14370, 16955, 19961, 22950, 26574, 30941
Offset: 0
The partition (3,2,1) has rank 2^(3-1) + 2^(2-1) + 2^(1-1) = 7, which is prime, so (3,2,1) is counted under a(6).
The a(2) = 2 through a(10) = 9 partitions:
(2) (21) (31) (221) (51) (421) (431) (441) (91)
(11) (111) (2111) (321) (2221) (521) (3321) (631)
(11111) (3111) (4111) (5111) (4221) (721)
(22111) (33111) (3331)
(211111) (42111) (7111)
(1111111) (411111) (32221)
(322111)
(3211111)
(31111111)
For all positive integers (not just prime) we get
A000041.
These partitions have Heinz numbers
A277319.
A014499 lists binary indices of prime numbers.
A372885 lists primes whose binary indices sum to a prime, indices
A372886.
Cf.
A000040,
A005940,
A023506,
A029837,
A035100,
A038499,
A096111,
A372429,
A372441,
A372471,
A372689.
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Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^#]/2]&]],{n,0,30}]
A052467
Binomial transform of {b(n)}, where b(n)=1 for prime n and b(n)=0 otherwise.
Original entry on oeis.org
0, 1, 3, 6, 11, 20, 37, 70, 134, 255, 476, 869, 1564, 2821, 5201, 9948, 19793, 40562, 84271, 174952, 359576, 728805, 1457402, 2885051, 5681277, 11185110, 22103926, 43939533, 87864092, 176447165, 354929146, 713198803, 1428312446, 2846268351
Offset: 1
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b[n_] := Boole[ PrimeQ[n]]; a[n_] := Sum[ Binomial[n, k]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 34}] // Differences (* Jean-François Alcover, Oct 25 2012 *)
Description corrected by
T. D. Noe, May 17 2003
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
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 multiplicities is
A055923, ranked by
A056166.
These partitions are ranked by
A346068.
A001221 counts constant partitions of prime length, ranked by
A053810.
A038499 counts partitions of prime length.
A101436 counts parts of prime signature that are themselves prime.
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Table[Length[Select[IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]
A363045
Number of partitions of n whose greatest part is a multiple of 3.
Original entry on oeis.org
1, 0, 0, 1, 1, 2, 4, 5, 7, 11, 14, 19, 27, 34, 45, 60, 77, 99, 130, 163, 208, 265, 333, 417, 526, 651, 810, 1004, 1237, 1519, 1869, 2278, 2780, 3382, 4101, 4958, 5995, 7210, 8669, 10398, 12444, 14859, 17730, 21086, 25057, 29718, 35186, 41584, 49100, 57842, 68075, 79991
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
end:
a:= n-> add(b(n-3*i, min(n-3*i, 3*i)), i=0..n/3):
seq(a(n), n=0..60); # Alois P. Heinz, May 14 2023
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := Sum[b[n - 3*i, Min[n - 3*i, 3*i]], {i, 0, n/3}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 23 2023, after Alois P. Heinz *)
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my(N=60, x='x+O('x^N)); Vec(sum(k=0, N, x^(3*k)/prod(j=1, 3*k, 1-x^j)))
A372885
Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.
Original entry on oeis.org
2, 3, 11, 23, 29, 41, 43, 61, 71, 79, 89, 101, 103, 113, 131, 137, 149, 151, 163, 181, 191, 197, 211, 239, 269, 271, 281, 293, 307, 331, 349, 353, 373, 383, 401, 433, 457, 491, 503, 509, 523, 541, 547, 593, 641, 683, 701, 709, 743, 751, 761, 773, 827, 863, 887
Offset: 1
The binary indices of 89 are {1,4,5,7}, with sum 17, which is prime, so 89 is in the sequence.
The terms together with their binary expansions and binary indices begin:
2: 10 ~ {2}
3: 11 ~ {1,2}
11: 1011 ~ {1,2,4}
23: 10111 ~ {1,2,3,5}
29: 11101 ~ {1,3,4,5}
41: 101001 ~ {1,4,6}
43: 101011 ~ {1,2,4,6}
61: 111101 ~ {1,3,4,5,6}
71: 1000111 ~ {1,2,3,7}
79: 1001111 ~ {1,2,3,4,7}
89: 1011001 ~ {1,4,5,7}
101: 1100101 ~ {1,3,6,7}
103: 1100111 ~ {1,2,3,6,7}
113: 1110001 ~ {1,5,6,7}
131: 10000011 ~ {1,2,8}
137: 10001001 ~ {1,4,8}
149: 10010101 ~ {1,3,5,8}
151: 10010111 ~ {1,2,3,5,8}
163: 10100011 ~ {1,2,6,8}
181: 10110101 ~ {1,3,5,6,8}
191: 10111111 ~ {1,2,3,4,5,6,8}
197: 11000101 ~ {1,3,7,8}
For prime instead of binary indices we have
A006450, prime case of
A316091.
Prime numbers p such that
A029931(p) is also prime.
The indices of these primes are
A372886.
A372687 counts strict partitions of prime binary rank, counted by
A372851.
A372688 counts partitions of prime binary rank, with Heinz numbers
A277319.
Binary indices:
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filter:= proc(p)
local L,i,t;
L:= convert(p,base,2);
isprime(add(i*L[i],i=1..nops(L)))
end proc:
select(filter, [seq(ithprime(i),i=1..200)]); # Robert Israel, Jun 19 2025
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Select[Range[100],PrimeQ[#] && PrimeQ[Total[First/@Position[Reverse[IntegerDigits[#,2]],1]]]&]
A372886
Indices of prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.
Original entry on oeis.org
1, 2, 5, 9, 10, 13, 14, 18, 20, 22, 24, 26, 27, 30, 32, 33, 35, 36, 38, 42, 43, 45, 47, 52, 57, 58, 60, 62, 63, 67, 70, 71, 74, 76, 79, 84, 88, 94, 96, 97, 99, 100, 101, 108, 116, 124, 126, 127, 132, 133, 135, 137, 144, 150, 154, 156, 160, 161, 162, 164, 172
Offset: 1
The binary indices of 89 = prime(24) are {1,4,5,7}, with sum 17, which is prime, so 24 is in the sequence.
Numbers k such that
A029931(prime(k)) is prime.
Indices of primes that belong to
A372689.
The indexed prime numbers themselves are
A372885.
Binary indices:
A372687 counts strict partitions of prime binary rank, counted by
A372851.
A372688 counts partitions of prime binary rank, with Heinz numbers
A277319.
-
filter:= proc(p)
local L,i,t;
L:= convert(p,base,2);
isprime(add(i*L[i],i=1..nops(L)))
end proc:
select(t -> filter(ithprime(t)), [$1..1000]); # Robert Israel, Jun 19 2025
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Select[Range[100],PrimeQ[Total[First /@ Position[Reverse[IntegerDigits[Prime[#],2]],1]]]&]
A372687
Number of prime numbers whose binary indices sum to n. Number of strict integer partitions y of n such that Sum_i 2^(y_i-1) is prime.
Original entry on oeis.org
0, 0, 1, 1, 1, 0, 2, 1, 2, 0, 3, 3, 1, 4, 1, 6, 5, 8, 4, 12, 8, 12, 7, 20, 8, 16, 17, 27, 19, 38, 19, 46, 33, 38, 49, 65, 47, 67, 83, 92, 94, 113, 103, 130, 146, 127, 215, 224, 176, 234, 306, 270, 357, 383, 339, 393, 537, 540, 597, 683, 576, 798, 1026, 830, 1157
Offset: 0
The a(2) = 1 through a(17) = 8 prime numbers:
2 3 5 . 17 11 19 . 257 131 73 137 97 521 4099 1031
7 13 67 41 71 263 2053 523
37 23 43 139 1033 269
29 83 193 163
53 47 149
31 101
89
79
The a(2) = 1 through a(11) = 3 strict partitions:
(2) (2,1) (3,1) . (5,1) (4,2,1) (4,3,1) . (9,1) (6,4,1)
(3,2,1) (5,2,1) (6,3,1) (8,2,1)
(7,2,1) (5,3,2,1)
For all positive integers (not just prime) we get
A000009.
Number of prime numbers p with
A029931(p) = n.
Number of times n appears in
A372429.
Number of rows of
A372471 with sum n.
These (strict) partitions have Heinz numbers
A372851.
A014499 lists binary indices of prime numbers.
A096111 gives product of binary indices.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices
A372886.
Cf.
A000040,
A005940,
A023506,
A029837,
A035100,
A071814,
A230877,
A231204,
A359359,
A372436,
A372441.
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&PrimeQ[Total[2^#]/2]&]],{n,0,30}]
A372850
Numbers whose distinct prime indices are the binary indices of some prime number.
Original entry on oeis.org
3, 6, 9, 10, 12, 18, 20, 22, 24, 27, 30, 36, 40, 42, 44, 46, 48, 50, 54, 60, 66, 70, 72, 80, 81, 84, 88, 90, 92, 96, 100, 102, 108, 114, 118, 120, 126, 130, 132, 140, 144, 150, 160, 162, 168, 176, 180, 182, 184, 192, 198, 200, 204, 216, 228, 236, 238, 240, 242
Offset: 1
The distinct prime indices of 45 are {2,3}, which are the binary indices of 6, which is not prime, so 45 is not in the sequence.
The distinct prime indices of 60 are {1,2,3}, which are the binary indices of 7, which is prime, so 60 is in the sequence.
The terms together with their prime indices begin:
3: {2}
6: {1,2}
9: {2,2}
10: {1,3}
12: {1,1,2}
18: {1,2,2}
20: {1,1,3}
22: {1,5}
24: {1,1,1,2}
27: {2,2,2}
30: {1,2,3}
36: {1,1,2,2}
40: {1,1,1,3}
42: {1,2,4}
44: {1,1,5}
46: {1,9}
48: {1,1,1,1,2}
50: {1,3,3}
54: {1,2,2,2}
60: {1,1,2,3}
66: {1,2,5}
70: {1,3,4}
For prime indices with multiplicity we have
A277319, counted by
A372688.
Partitions of this type are counted by
A372887.
A014499 lists binary indices of prime numbers.
A372885 lists primes whose binary indices sum to a prime, indices
A372886.
Cf.
A000040,
A005940,
A023506,
A029837,
A035100,
A071814,
A096111,
A372429,
A372436,
A372441,
A372471,
A372689.
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prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeQ[Total[2^(Union[prix[#]]-1)]]&]
A372851
Squarefree numbers whose prime indices are the binary indices of some prime number.
Original entry on oeis.org
3, 6, 10, 22, 30, 42, 46, 66, 70, 102, 114, 118, 130, 182, 238, 246, 266, 318, 330, 354, 370, 402, 406, 434, 442, 510, 546, 646, 654, 690, 762, 770, 798, 930, 938, 946, 962, 986, 1066, 1102, 1122, 1178, 1218, 1222, 1246, 1258, 1334, 1378, 1430, 1482, 1578
Offset: 1
The prime indices of 70 are {1,3,4}, which are the binary indices of 13, which is prime, so 70 is in the sequence.
The prime indices of 15 are {2,3}, which are the binary indices of 6, which is not prime, so 15 is not in the sequence.
The terms together with their prime indices begin:
3: {2}
6: {1,2}
10: {1,3}
22: {1,5}
30: {1,2,3}
42: {1,2,4}
46: {1,9}
66: {1,2,5}
70: {1,3,4}
102: {1,2,7}
114: {1,2,8}
118: {1,17}
130: {1,3,6}
182: {1,4,6}
238: {1,4,7}
246: {1,2,13}
266: {1,4,8}
318: {1,2,16}
330: {1,2,3,5}
354: {1,2,17}
370: {1,3,12}
402: {1,2,19}
For odd instead of prime we have
A039956.
For even instead of prime we have
A056911.
Strict partitions of this type are counted by
A372687.
Non-strict partitions of this type are counted by
A372688, ranks
A277319.
A014499 lists binary indices of prime numbers.
A372885 lists primes whose binary indices sum to a prime, indices
A372886.
Cf.
A000040,
A005940,
A025147,
A035100,
A071814,
A096111,
A096765,
A231204,
A372429,
A372471,
A372689.
Showing 1-10 of 24 results.
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