A104080 -id:A104080 - cp's OEIS frontend

A145667 a(n) = number of components of the graph P(n,2) (defined in Comments).

0, 1, 1, 2, 1, 2, 4, 11, 13, 19, 29, 43, 107, 169, 350, 603, 1134, 2070, 3803, 7502, 13989, 26495, 50826, 97369, 185827, 357307, 690577, 1332382, 2565110, 4958962, 9594425, 18569626, 36009794

Offset: 1

W. Edwin Clark, Mar 17 2009

Let H(n,b) be the Hamming graph whose vertices are the sequences of length n over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(n,b) be the subgraph of H(n,b) induced by the set of vertices which are base b representations of primes with n digits (not allowing leading 0 digits).

Cf. A104080, A014234, A145668, A145669, A145670, A145671, A145672, A145673, A145674, A158576, A158577, A158578, A158579.

a(18)-a(31) from Max Alekseyev, May 12 2011

a(32)-a(33) from Max Alekseyev, Dec 23 2024

A145668 a(n) = size of the n-th term in S(2) (defined in Comments).

Original entry on oeis.org

2, 2, 1, 1, 5, 3, 4, 9, 2, 1, 1, 7, 4, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 12, 1, 20, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 29, 19, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 3, 1, 75, 2, 19, 4, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 23, 1, 82, 76, 1, 1, 3, 1, 1, 3, 3, 4, 2, 3, 3, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 9, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

W. Edwin Clark, Mar 17 2009

Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.

Max Alekseyev, Table of n, a(n) for n = 1..100000

Cf. A104080, A014234, A145667, A145669, A145670, A145671, A145672, A145673, A145674, A158576, A158577, A158578, A158579.

More terms from Max Alekseyev, May 12 2011

A372517 Least k such that the k-th prime number has exactly n ones in its binary expansion.

Original entry on oeis.org

1, 2, 4, 9, 11, 64, 31, 76, 167, 309, 502, 801, 1028, 7281, 6363, 12079, 12251, 43237, 43390, 146605, 291640, 1046198, 951351, 2063216, 3957778, 11134645, 14198321, 28186247, 54387475, 249939829, 105097565, 393248783, 751545789, 1391572698, 2182112798, 8242984130

Offset: 1

Gus Wiseman, May 12 2024

In other words, the a(n)-th prime is the least with binary weight n. The sorted version is A372686.

			The primes A000040(a(n)) together with their binary expansions and binary indices begin:
        2:                     10 ~ {2}
        3:                     11 ~ {1,2}
        7:                    111 ~ {1,2,3}
       23:                  10111 ~ {1,2,3,5}
       31:                  11111 ~ {1,2,3,4,5}
      311:              100110111 ~ {1,2,3,5,6,9}
      127:                1111111 ~ {1,2,3,4,5,6,7}
      383:              101111111 ~ {1,2,3,4,5,6,7,9}
      991:             1111011111 ~ {1,2,3,4,5,7,8,9,10}
     2039:            11111110111 ~ {1,2,3,5,6,7,8,9,10,11}
     3583:           110111111111 ~ {1,2,3,4,5,6,7,8,9,11,12}
     6143:          1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}
     8191:          1111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13}
    73727:      10001111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,17}
    63487:       1111011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13,14,15,16}

Positions firsts of first appearances in A014499.
Taking primes gives A061712.
Counting zeros (weight) gives A372474, firsts of A035103.
For binary length we have A372684 (take primes A104080), firsts of A035100.
The sorted version is A372686, taking primes A372685.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A029837 gives greatest binary index, least A001511.
A030190 gives binary expansion, reversed A030308.
A048793 lists binary indices, reverse A272020, sum A029931.
A372471 lists binary indices of primes.
Cf. A000040, A066195, A069010, A070939, A071814, A211997, A372429, A372516.

spsm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
j=DigitCount[#,2,1]&/@Select[Range[1000],PrimeQ];
Table[Position[j,k][[1,1]],{k,spsm[j]}]

a(n) = my(k=1, p=2); while(hammingweight(p) !=n, p = nextprime(p+1); k++); k; \\ Michel Marcus, May 13 2024

from itertools import count
from sympy import isprime, primepi
from sympy.utilities.iterables import multiset_permutations
def A372517(n):
    for l in count(n-1):
        m = 1<Chai Wah Wu, May 13 2024

A000040(a(n)) = A061712(n).

a(32)-a(36) from Pontus von Brömssen, May 13 2024

A373126 Difference between 2^n and the greatest squarefree number <= 2^n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1

Offset: 0

Gus Wiseman, May 29 2024

nonn

			The greatest squarefree number <= 2^21 is 2097149, and 2^21 = 2097152, so a(21) = 3.

For prime instead of squarefree we have A013603, opposite A092131.
For primes instead of powers of 2: A240474, A240473, A112926, A112925.
Difference between 2^n and A372889.
The opposite is A373125, delta of A372683.
A005117 lists squarefree numbers, first differences A076259.
A053797 gives lengths of gaps between squarefree numbers.
A061398 counts squarefree numbers between primes (exclusive).
A070939 or (preferably) A029837 gives length of binary expansion.
A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
A143658 counts squarefree numbers up to 2^n.
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
For primes between powers of 2:
- sum A293697 (except initial terms)
- length A036378
- min A104080 or A014210, indices A372684 (firsts of A035100)
- max A014234
Cf. A010036, A029931, A046933, A049093-A049096, A077641, A372540, A373197.

Table[2^n-NestWhile[#-1&,2^n,!SquareFreeQ[#]&],{n,0,100}]

a(n) = 2^n-A372889(n). - R. J. Mathar, May 31 2024

A145669 a(n) = smallest member of the n-th term in S(2) (defined in Comments).

Original entry on oeis.org

2, 5, 11, 13, 17, 37, 41, 67, 73, 107, 127, 131, 149, 173, 191, 193, 211, 223, 233, 239, 241, 251, 257, 263, 277, 281, 337, 349, 353, 373, 419, 431, 443, 491, 509, 521, 541, 547, 557, 613, 653, 661, 683, 701, 709, 719, 733, 761, 769, 787, 853, 877, 907, 1019, 1031, 1091, 1093, 1153, 1163, 1187, 1193, 1201, 1259, 1381, 1433, 1451, 1453, 1553, 1597, 1637, 1657, 1709, 1721, 1753, 1759, 1777, 1783, 1811, 1889, 1907, 1931, 1973, 2027

Offset: 1

W. Edwin Clark, Mar 17 2009

Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.

Max Alekseyev, Table of n, a(n) for n = 1..100000

Cf. A104080, A014234, A145667, A145668, A145670, A145671, A145672, A145673, A145674, A158576, A158577, A158578, A158579.

More terms from Max Alekseyev, May 12 2011

A145670 a(n) = largest member of the n-th term in S(2) (defined in Comments).

Original entry on oeis.org

3, 7, 11, 13, 31, 61, 59, 113, 89, 107, 127, 227, 181, 173, 191, 229, 211, 223, 233, 239, 241, 251, 257, 479, 277, 503, 337, 349, 353, 373, 419, 431, 443, 491, 509, 619, 1021, 953, 557, 613, 653, 661, 683, 701, 709, 751, 733, 761, 773, 787, 853, 877, 971, 1019, 2029, 1123, 1879, 1409, 1163, 1699, 1193, 1201, 1259, 1381, 1433, 1451, 1453, 1553, 1597, 1637, 1913, 1709, 1979, 1753, 1759, 1777, 2039, 1811, 2017, 1907, 1931, 1973, 2027

Offset: 1

W. Edwin Clark, Mar 17 2009

Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.

Max Alekseyev, Table of n, a(n) for n = 1..100000

Cf. A104080, A014234, A145667, A145668, A145669, A145671, A145672, A145673, A145674, A158576, A158577, A158578, A158579.

More terms from Max Alekseyev, May 12 2011

A373123 Sum of all squarefree numbers from 2^(n-1) to 2^n - 1.

Original entry on oeis.org

1, 5, 18, 63, 218, 891, 3676, 15137, 60580, 238672, 953501, 3826167, 15308186, 61204878, 244709252, 979285522, 3917052950, 15664274802, 62663847447, 250662444349, 1002632090376, 4010544455838, 16042042419476, 64168305037147, 256675237863576

Offset: 1

Gus Wiseman, May 27 2024

nonn

			This is the sequence of row sums of A005117 treated as a triangle with row-lengths A077643:
   1
   2   3
   5   6   7
  10  11  13  14  15
  17  19  21  22  23  26  29  30  31
  33  34  35  37  38  39  41  42  43  46  47  51  53  55  57  58  59  61  62

Counting all numbers (not just squarefree) gives A010036.
For the sectioning of A005117:
Row-lengths are A077643, partial sums A143658.
First column is A372683, delta A373125, indices A372540, firsts of A372475.
Last column is A372889, delta A373126, indices A143658, diffs A077643.
For primes instead of powers of two:
- sum A373197
- length A373198 = A061398 - 1
- maxima A112925, opposite A112926
For prime instead of squarefree:
- sum A293697 (except initial terms)
- length A036378
- min A104080 or A014210, indices A372684 (firsts of A035100)
- max A014234, delta A013603
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers, first differences A076259.
A030190 gives binary expansion, reversed A030308.
A070939 or (preferably) A029837 gives length of binary expansion.
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
Cf. A029931, A048793, A049093, A049094, A059015, A069010, A077641.

Table[Total[Select[Range[2^(n-1),2^n-1],SquareFreeQ]],{n,10}]

a(n) = my(s=0); forsquarefree(i=2^(n-1), 2^n-1, s+=i[1]); s; \\ Michel Marcus, May 29 2024

A372516 Number of ones minus number of zeros in the binary expansion of the n-th prime number.

Original entry on oeis.org

0, 2, 1, 3, 2, 2, -1, 1, 3, 3, 5, 0, 0, 2, 4, 2, 4, 4, -1, 1, -1, 3, 1, 1, -1, 1, 3, 3, 3, 1, 7, -2, -2, 0, 0, 2, 2, 0, 2, 2, 2, 2, 6, -2, 0, 2, 2, 6, 2, 2, 2, 6, 2, 6, -5, -1, -1, 1, -1, -1, 1, -1, 1, 3, 1, 3, 1, -1, 3, 3, -1, 3, 5, 3, 5, 7, -1, 1, -1, 1, 1

Offset: 1

Gus Wiseman, May 13 2024

sign

Absolute value is A177718.

			The binary expansion of 83 is (1,0,1,0,0,1,1), and 83 is the 23rd prime, so a(23) = 4 - 3 = 1.

The sum instead of difference is A035100, firsts A372684 (primes A104080).
The negative version is A037861(A000040(n)).
Restriction of A145037 to the primes.
The unsigned version is A177718.
- Positions of zeros are A177796, indices of the primes A066196.
- Positions of positive terms are indices of the primes A095070.
- Positions of negative terms are indices of the primes A095071.
- Positions of negative ones are A372539, indices of the primes A095072.
- Positions of ones are A372538, indices of the primes A095073.
- Positions of nonnegative terms are indices of the primes A095074.
- Positions of nonpositive terms are indices of the primes A095075.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A030190 gives binary expansion, reversed A030308.
A035103 counts zeros in binary expansion of primes, firsts A372474.
A048793 lists binary indices, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A101211 lists run-lengths in binary expansion, row-lengths A069010.
A372471 lists the binary indices of each prime.
Cf. A000043, A003714, A005940, A059305, A061712, A066195, A071814, A211997, A372429, A372517, A372686.

Table[DigitCount[Prime[n],2,1]-DigitCount[Prime[n],2,0],{n,100}]
DigitCount[#,2,1]-DigitCount[#,2,0]&/@Prime[Range[100]] (* Harvey P. Dale, May 09 2025 *)

a(n) = A000120(A000040(n)) - A080791(A000040(n)).
a(n) = A014499(n) - A035103(n).
a(n) = A145037(A000040(n))

A203074 a(0)=1; for n > 0, a(n) = next prime after 2^(n-1).

Original entry on oeis.org

1, 2, 3, 5, 11, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659

Offset: 0

Frank M Jackson and N. J. A. Sloane, Dec 28 2011

nonn

Equals {1} union A014210. Unlike A014210, every positive integer can be written in one or more ways as a sum of terms of this sequence. See A203075, A203076.

a(n)*2^(n-1) = A133814(n-1) for n > 1 and a(n)*2^(n-1) for n > O is a subsequence of primitive practical numbers (A267124). - Frank M Jackson, Dec 29 2024

			a(5) = 17, since this is the next prime after 2^(5-1) = 2^4 = 16.

M. F. Hasler & Bill McEachen, Table of n, a(n) for n = 0..1300 (missing lines n = 1159..1165 from Bill McEachen)
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A013632, A013597, A014210, A104080, A203075, A203076.

[1] cat [NextPrime(2^(n-1)): n in [1..40]]; // Vincenzo Librandi, Feb 23 2018

nextprime[n_Integer] := (k=n+1;While[!PrimeQ[k], k++];k); aprime[m_Integer] := (If[m==0, 1, nextprime[2^(m-1)]]); Table[aprime[l], {l,0,100}]
nxt[{n_,a_}]:={n+1,NextPrime[2^n]}; NestList[nxt,{0,1},40][[All,2]] (* Harvey P. Dale, Oct 10 2017 *)

a(n)=if(n,nextprime(2^n/2+1),1) \\ Charles R Greathouse IV

A203074(n)=nextprime(2^(n-1)+1)-!n  \\ M. F. Hasler, Mar 15 2012

A203074(n) = 2^(n-1) + A013597(n-1), for n > 0. - M. F. Hasler, Mar 15 2012

a(n) = A104080(n-1) for n > 2. - Georg Fischer, Oct 23 2018

A104082 Smallest prime >= 4^n.

Original entry on oeis.org

2, 5, 17, 67, 257, 1031, 4099, 16411, 65537, 262147, 1048583, 4194319, 16777259, 67108879, 268435459, 1073741827, 4294967311, 17179869209, 68719476767, 274877906951, 1099511627791, 4398046511119, 17592186044423, 70368744177679, 281474976710677, 1125899906842679

Offset: 0

Cino Hilliard, Mar 03 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A104080 (for 2^n), A104081 (for 3^n).

Cf. A014210.

[NextPrime(4^n-1): n in [0..50]]; // Jinyuan Wang, Nov 09 2018

NextPrime[4^Range[0, 23]] (* Vladimir Joseph Stephan Orlovsky, Mar 13 2011 *)

g(n,b=4) = for(x=0,n,print1(nextprime(b^x)","))

a(n) = A104080(2n). - Jinyuan Wang, Nov 09 2018

cp's OEIS Frontend

A145667 a(n) = number of components of the graph P(n,2) (defined in Comments).

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A145668 a(n) = size of the n-th term in S(2) (defined in Comments).

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A372517 Least k such that the k-th prime number has exactly n ones in its binary expansion.

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A373126 Difference between 2^n and the greatest squarefree number <= 2^n.

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A145669 a(n) = smallest member of the n-th term in S(2) (defined in Comments).

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A145670 a(n) = largest member of the n-th term in S(2) (defined in Comments).

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A373123 Sum of all squarefree numbers from 2^(n-1) to 2^n - 1.

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Mathematica

PARI

A372516 Number of ones minus number of zeros in the binary expansion of the n-th prime number.

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Mathematica

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A203074 a(0)=1; for n > 0, a(n) = next prime after 2^(n-1).

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