A372689
Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a prime number.
Original entry on oeis.org
2, 3, 4, 6, 9, 11, 12, 16, 18, 23, 26, 29, 33, 38, 41, 43, 44, 48, 50, 55, 58, 61, 64, 69, 71, 72, 74, 79, 81, 86, 89, 91, 92, 96, 101, 103, 104, 106, 111, 113, 118, 121, 131, 132, 134, 137, 142, 144, 149, 151, 152, 154, 159, 163, 164, 166, 169, 174, 176, 181
Offset: 1
The terms together with their binary expansions and binary indices begin:
2: 10 ~ {2}
3: 11 ~ {1,2}
4: 100 ~ {3}
6: 110 ~ {2,3}
9: 1001 ~ {1,4}
11: 1011 ~ {1,2,4}
12: 1100 ~ {3,4}
16: 10000 ~ {5}
18: 10010 ~ {2,5}
23: 10111 ~ {1,2,3,5}
26: 11010 ~ {2,4,5}
29: 11101 ~ {1,3,4,5}
33: 100001 ~ {1,6}
38: 100110 ~ {2,3,6}
41: 101001 ~ {1,4,6}
43: 101011 ~ {1,2,4,6}
44: 101100 ~ {3,4,6}
48: 110000 ~ {5,6}
50: 110010 ~ {2,5,6}
55: 110111 ~ {1,2,3,5,6}
58: 111010 ~ {2,4,5,6}
61: 111101 ~ {1,3,4,5,6}
Numbers k such that
A029931(k) is prime.
Union of prime-indexed rows of
A118462.
For prime indices instead of binary indices we have
A316091.
A372687 counts strict partitions of prime binary rank, counted by
A372851.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices
A372886.
Binary indices:
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:
-
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
-
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
-
Select[Range[100],PrimeQ[Total[First /@ Position[Reverse[IntegerDigits[Prime[#],2]],1]]]&]
A316092
Heinz numbers of integer partitions of prime numbers into prime parts.
Original entry on oeis.org
3, 5, 11, 15, 17, 31, 33, 41, 45, 59, 67, 83, 93, 109, 127, 153, 157, 177, 179, 191, 211, 241, 275, 277, 283, 297, 327, 331, 353, 367, 369, 375, 401, 405, 425, 431, 459, 461, 509, 537, 547, 563, 587, 599, 603, 605, 617, 709, 739, 773, 775, 797, 825, 831, 837
Offset: 1
Sequence of integer partitions of prime numbers into prime parts together with their Heinz numbers begins:
3: (2)
5: (3)
11: (5)
15: (2,3)
17: (7)
31: (11)
33: (2,5)
41: (13)
45: (2,2,3)
Cf.
A000041,
A000607,
A056239,
A056768,
A076610,
A100118,
A112798,
A215366,
A296150,
A300383,
A316091.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[900],And[PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]
Showing 1-4 of 4 results.
Comments