A371448
Numbers such that (1) the product of prime indices is squarefree, and (2) the binary indices of prime indices cover an initial interval of positive integers.
Original entry on oeis.org
1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 26, 30, 32, 33, 34, 40, 47, 48, 51, 52, 55, 60, 64, 66, 68, 80, 85, 86, 94, 96, 102, 104, 110, 120, 123, 127, 128, 132, 136, 141, 143, 160, 165, 170, 172, 187, 188, 192, 204, 205, 208, 215, 220, 221, 226, 240, 246
Offset: 1
The terms together with their binary indices of prime indices begin:
1: {}
2: {{1}}
4: {{1},{1}}
5: {{1,2}}
6: {{1},{2}}
8: {{1},{1},{1}}
10: {{1},{1,2}}
12: {{1},{1},{2}}
15: {{2},{1,2}}
16: {{1},{1},{1},{1}}
17: {{1,2,3}}
20: {{1},{1},{1,2}}
24: {{1},{1},{1},{2}}
26: {{1},{2,3}}
30: {{1},{2},{1,2}}
32: {{1},{1},{1},{1},{1}}
33: {{2},{1,3}}
34: {{1},{1,2,3}}
40: {{1},{1},{1},{1,2}}
47: {{1,2,3,4}}
48: {{1},{1},{1},{1},{2}}
51: {{2},{1,2,3}}
The connected components of this multiset system are counted by
A371451.
A000009 counts partitions covering initial interval, compositions
A107429.
A011782 counts multisets covering an initial interval.
A070939 gives length of binary expansion.
A131689 counts patterns by number of distinct parts.
-
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000], SquareFreeQ[Times@@prix[#]]&&normQ[Join@@bpe/@prix[#]]&]
A371453
Numbers whose binary indices are all squarefree semiprimes.
Original entry on oeis.org
32, 512, 544, 8192, 8224, 8704, 8736, 16384, 16416, 16896, 16928, 24576, 24608, 25088, 25120, 1048576, 1048608, 1049088, 1049120, 1056768, 1056800, 1057280, 1057312, 1064960, 1064992, 1065472, 1065504, 1073152, 1073184, 1073664, 1073696, 2097152, 2097184
Offset: 1
The terms together with their binary expansions and binary indices begin:
32: 100000 ~ {6}
512: 1000000000 ~ {10}
544: 1000100000 ~ {6,10}
8192: 10000000000000 ~ {14}
8224: 10000000100000 ~ {6,14}
8704: 10001000000000 ~ {10,14}
8736: 10001000100000 ~ {6,10,14}
16384: 100000000000000 ~ {15}
16416: 100000000100000 ~ {6,15}
16896: 100001000000000 ~ {10,15}
16928: 100001000100000 ~ {6,10,15}
24576: 110000000000000 ~ {14,15}
24608: 110000000100000 ~ {6,14,15}
25088: 110001000000000 ~ {10,14,15}
25120: 110001000100000 ~ {6,10,14,15}
1048576: 100000000000000000000 ~ {21}
Partitions of this type are counted by
A002100, squarefree case of
A101048.
For primes instead of squarefree semiprimes we get
A326782.
Allowing any squarefree numbers gives
A368533.
This is the squarefree case of
A371454.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
-
M:= 26: # for terms < 2^M
P:= select(isprime, [$2..(M+1)/2]): nP:= nops(P):
S:= select(`<`,{seq(seq(P[i]*P[j],i=1..j-1),j=1..nP)},M+1):
R:= map(proc(s) local i; add(2^(i-1),i=s) end proc, combinat:-powerset(S) minus {{}}):
sort(convert(R,list)); # Robert Israel, Apr 04 2024
-
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sqfsemi[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[10000],And@@sqfsemi/@bix[#]&]
-
def A371453(n): return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1')
-
from math import isqrt
from sympy import primepi, primerange
def A371453(n):
def f(x,n): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
def A006881(n):
m, k = n, f(n,n)
while m != k:
m, k = k, f(k,n)
return m
return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # Chai Wah Wu, Aug 16 2024
A371449
Numbers whose prime indices are not powers of 2.
Original entry on oeis.org
1, 5, 11, 13, 17, 23, 25, 29, 31, 37, 41, 43, 47, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 211, 215
Offset: 1
The terms together with their prime indices begin:
1: {} 85: {3,7} 169: {6,6} 253: {5,9}
5: {3} 89: {24} 173: {40} 257: {55}
11: {5} 97: {25} 179: {41} 263: {56}
13: {6} 101: {26} 181: {42} 269: {57}
17: {7} 103: {27} 185: {3,12} 271: {58}
23: {9} 107: {28} 187: {5,7} 275: {3,3,5}
25: {3,3} 109: {29} 191: {43} 277: {59}
29: {10} 113: {30} 193: {44} 281: {60}
31: {11} 115: {3,9} 197: {45} 283: {61}
37: {12} 121: {5,5} 199: {46} 289: {7,7}
41: {13} 125: {3,3,3} 205: {3,13} 293: {62}
43: {14} 127: {31} 211: {47} 295: {3,17}
47: {15} 137: {33} 215: {3,14} 299: {6,9}
55: {3,5} 139: {34} 221: {6,7} 305: {3,18}
59: {17} 143: {5,6} 223: {48} 307: {63}
61: {18} 145: {3,10} 227: {49} 313: {65}
65: {3,6} 149: {35} 229: {50} 317: {66}
67: {19} 151: {36} 233: {51} 319: {5,10}
71: {20} 155: {3,11} 235: {3,15} 325: {3,3,6}
73: {21} 157: {37} 239: {52} 331: {67}
79: {22} 163: {38} 241: {53} 335: {3,19}
83: {23} 167: {39} 251: {54} 337: {68}
Partitions of this type are counted by
A101417.
For binary indices instead of prime indices we have
A326781.
For primes instead of powers of 2 we have
A320628.
A070939 gives length of binary expansion.
A371450
MM-number of the set-system with BII-number n.
Original entry on oeis.org
1, 3, 5, 15, 13, 39, 65, 195, 11, 33, 55, 165, 143, 429, 715, 2145, 29, 87, 145, 435, 377, 1131, 1885, 5655, 319, 957, 1595, 4785, 4147, 12441, 20735, 62205, 47, 141, 235, 705, 611, 1833, 3055, 9165, 517, 1551, 2585, 7755, 6721, 20163, 33605, 100815, 1363, 4089
Offset: 0
The set-system with BII-number 30 is {{2},{1,2},{3},{1,3}} with MM-number prime(3) * prime(6) * prime(5) * prime(10) = 20735.
The terms together with their prime indices and binary indices of prime indices begin:
1 -> {} -> {}
3 -> {2} -> {{1}}
5 -> {3} -> {{2}}
15 -> {2,3} -> {{1},{2}}
13 -> {6} -> {{1,2}}
39 -> {2,6} -> {{1},{1,2}}
65 -> {3,6} -> {{2},{1,2}}
195 -> {2,3,6} -> {{1},{2},{1,2}}
11 -> {5} -> {{3}}
33 -> {2,5} -> {{1},{3}}
55 -> {3,5} -> {{2},{3}}
165 -> {2,3,5} -> {{1},{2},{3}}
143 -> {5,6} -> {{1,2},{3}}
429 -> {2,5,6} -> {{1},{1,2},{3}}
715 -> {3,5,6} -> {{2},{1,2},{3}}
2145 -> {2,3,5,6} -> {{1},{2},{1,2},{3}}
A019565 gives Heinz number of binary indices.
A070939 gives length of binary expansion.
Cf.
A000720,
A003963,
A087086,
A096111,
A275024,
A302242,
A302505,
A302521,
A326782,
A329557,
A329630,
A368109.
-
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Times@@Prime/@(Times@@Prime/@#&/@bix/@bix[n]),{n,0,30}]
A371454
Numbers whose binary indices are all semiprimes.
Original entry on oeis.org
8, 32, 40, 256, 264, 288, 296, 512, 520, 544, 552, 768, 776, 800, 808, 8192, 8200, 8224, 8232, 8448, 8456, 8480, 8488, 8704, 8712, 8736, 8744, 8960, 8968, 8992, 9000, 16384, 16392, 16416, 16424, 16640, 16648, 16672, 16680, 16896, 16904, 16928, 16936, 17152
Offset: 1
The terms together with their binary expansions and binary indices begin:
8: 1000 ~ {4}
32: 100000 ~ {6}
40: 101000 ~ {4,6}
256: 100000000 ~ {9}
264: 100001000 ~ {4,9}
288: 100100000 ~ {6,9}
296: 100101000 ~ {4,6,9}
512: 1000000000 ~ {10}
520: 1000001000 ~ {4,10}
544: 1000100000 ~ {6,10}
552: 1000101000 ~ {4,6,10}
768: 1100000000 ~ {9,10}
776: 1100001000 ~ {4,9,10}
800: 1100100000 ~ {6,9,10}
808: 1100101000 ~ {4,6,9,10}
Partitions of this type are counted by
A101048, squarefree case
A002100.
For primes instead of semiprimes we get
A326782.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
-
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
semi[n_]:=PrimeOmega[n]==2;
Select[Range[10000],And@@semi/@bix[#]&]
-
from math import isqrt
from sympy import primepi, primerange
def A371454(n):
def f(x,n): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
def A001358(n):
m, k = n, f(n,n)
while m != k:
m, k = k, f(k,n)
return m
return sum(1<<A001358(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # Chai Wah Wu, Aug 16 2024
A371290
Numbers whose product of binary indices is a prime power > 1.
Original entry on oeis.org
1, 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 64, 65, 128, 129, 130, 131, 136, 137, 138, 139, 256, 257, 260, 261, 1024, 1025, 4096, 4097, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897, 32898, 32899, 32904, 32905, 32906, 32907, 65536, 65537, 262144
Offset: 1
The terms together with their binary expansions and binary indices begin:
1: 1 ~ {1}
2: 10 ~ {2}
3: 11 ~ {1,2}
4: 100 ~ {3}
5: 101 ~ {1,3}
8: 1000 ~ {4}
9: 1001 ~ {1,4}
10: 1010 ~ {2,4}
11: 1011 ~ {1,2,4}
16: 10000 ~ {5}
17: 10001 ~ {1,5}
64: 1000000 ~ {7}
65: 1000001 ~ {1,7}
128: 10000000 ~ {8}
129: 10000001 ~ {1,8}
130: 10000010 ~ {2,8}
131: 10000011 ~ {1,2,8}
136: 10001000 ~ {4,8}
137: 10001001 ~ {1,4,8}
138: 10001010 ~ {2,4,8}
139: 10001011 ~ {1,2,4,8}
256: 100000000 ~ {9}
257: 100000001 ~ {1,9}
260: 100000100 ~ {3,9}
261: 100000101 ~ {1,3,9}
1024: 10000000000 ~ {11}
1025: 10000000001 ~ {1,11}
4096: 1000000000000 ~ {13}
4097: 1000000000001 ~ {1,13}
32768: 1000000000000000 ~ {16}
For squarefree numbers instead of prime powers we have
A371289.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
-
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],#==1||PrimePowerQ[Times@@bpe[#]]&]
A342475
Prime numbers whose binary expansion contains only prime powers of 2 and the zeroth power.
Original entry on oeis.org
5, 13, 37, 41, 137, 173, 2053, 2081, 2089, 2213, 2221, 8233, 8237, 8329, 8353, 10253, 10273, 10369, 131113, 131213, 133121, 133153, 133157, 133253, 133261, 139273, 139297, 139301, 139309, 139393, 139397, 139429, 141353, 141481, 524429, 524453, 526373, 526381, 526501
Offset: 1
5 = 2^2 + 2^0 is a term.
7 = 2^2 + 2^1 + 2^0 is not a term, because the exponent 1 is not a prime.
11 = 2^3 + 2^1 + 2^0 is not a term, because the exponent 1 is not a prime.
13 = 2^3 + 2^2 + 2^0 is a term.
-
Select[Array[1 + Total@ MapIndexed[#1*2^Prime[#2] & @@ {#1, First[#2]} &, Reverse@ IntegerDigits[#, 2]] &, 140], PrimeQ] (* Michael De Vlieger, Mar 13 2021 *)
-
isok(p) = if (isprime(p), my(b=Vecrev(binary(p))); sum(i=1, #b, b[i]*((i!=1) && !isprime(i-1))) == 0); \\ Michel Marcus, Apr 22 2021
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