A014234 -id:A014234 - cp's OEIS frontend

A372684 Least k such that prime(k) >= 2^n.

1, 3, 5, 7, 12, 19, 32, 55, 98, 173, 310, 565, 1029, 1901, 3513, 6543, 12252, 23001, 43391, 82026, 155612, 295948, 564164, 1077872, 2063690, 3957810, 7603554, 14630844, 28192751, 54400029, 105097566, 203280222, 393615807, 762939112, 1480206280, 2874398516, 5586502349

Offset: 1

Gus Wiseman, May 30 2024

nonn

			The numbers prime(a(n)) together with their binary expansions and binary indices begin:
        2:                       10 ~ {2}
        5:                      101 ~ {1,3}
       11:                     1011 ~ {1,2,4}
       17:                    10001 ~ {1,5}
       37:                   100101 ~ {1,3,6}
       67:                  1000011 ~ {1,2,7}
      131:                 10000011 ~ {1,2,8}
      257:                100000001 ~ {1,9}
      521:               1000001001 ~ {1,4,10}
     1031:              10000000111 ~ {1,2,3,11}
     2053:             100000000101 ~ {1,3,12}
     4099:            1000000000011 ~ {1,2,13}
     8209:           10000000010001 ~ {1,5,14}
    16411:          100000000011011 ~ {1,2,4,5,15}
    32771:         1000000000000011 ~ {1,2,16}
    65537:        10000000000000001 ~ {1,17}
   131101:       100000000000011101 ~ {1,3,4,5,18}
   262147:      1000000000000000011 ~ {1,2,19}
   524309:     10000000000000010101 ~ {1,3,5,20}
  1048583:    100000000000000000111 ~ {1,2,3,21}
  2097169:   1000000000000000010001 ~ {1,5,22}
  4194319:  10000000000000000001111 ~ {1,2,3,4,23}
  8388617: 100000000000000000001001 ~ {1,4,24}

The opposite (greatest k such that prime(k) <= 2^n) is A007053.
Positions of first appearances in A035100.
The distance from prime(a(n)) to 2^n is A092131.
Counting zeros instead of all bits gives A372474, firsts of A035103.
Counting ones instead of all bits gives A372517, firsts of A014499.
For primes between powers of 2:
- sum A293697
- length A036378
- min A104080 or A014210
- max A014234, delta A013603
For squarefree numbers between powers of 2:
- sum A373123
- length A077643, run-lengths of A372475
- min A372683, delta A373125, indices A372540
- max A372889, delta A373126, indices A143658
For squarefree numbers between primes:
- sum A373197
- length A373198 = A061398 - 1
- min A000040
- max A112925, opposite A112926
Cf. A000120, A029837, A029931, A030190, A049095, A069010, A070939, A077641, A080791, A211997.

Table[PrimePi[If[n==1,2,NextPrime[2^n]]],{n,30}]

a(n) = primepi(nextprime(2^n)); \\ Michel Marcus, May 31 2024

a(n>1) = A007053(n) + 1.
a(n) = A000720(A104080(n)).
prime(a(n)) = A104080(n).
prime(a(n)) - 2^n = A092131(n).

More terms from Michel Marcus, May 31 2024

A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

4, 9, 11, 30, 327, 445, 3512, 7789, 9361, 26519413

Offset: 1

Gus Wiseman, Nov 02 2024

Perfect powers (A001597) are numbers with a proper integer root, the complement of A007916.
Is this sequence finite?
The Redmond-Sun conjecture (see A308658) implies that this sequence is finite. - Pontus von Brömssen, Nov 05 2024

			Primes 9 and 10 are 23 and 29, and the interval (24,25,26,27,28) contains two perfect powers (25,27), so 9 is in the sequence.

For powers of 2 see A013597, A014210, A014234, A188951, A244508, A377467.
For no prime-powers we have A377286, ones in A080101.
For a unique prime-power we have A377287.
For squarefree numbers see A377430, A061398, A377431, A068360, A224363.
These are the positions of terms > 1 in A377432.
For a unique perfect power we have A377434.
For no perfect powers we have A377436.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect power <= n.
A131605 lists perfect powers that are not prime-powers.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053607, A304521.
A377468 gives the least perfect power > n.
Cf. A000720, A023055, A031218, A045542, A052410, A053706, A069623, A116086, A116455, A216765, A308658, A336416, A345531, A375740, A376560, A376561, A377057.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Count[Range[Prime[#]+1, Prime[#+1]-1],_?perpowQ]>1&]

from itertools import islice
from sympy import prime
from gmpy2 import is_power, next_prime
def A377466_gen(startvalue=1): # generator of terms >= startvalue
    k = max(startvalue,1)
    p = prime(k)
    while (q:=next_prime(p)):
        c = 0
        for i in range(p+1,q):
            if is_power(i):
                c += 1
                if c>1:
                    yield k
                    break
        k += 1
        p = q
A377466_list = list(islice(A377466_gen(),9)) # Chai Wah Wu, Nov 04 2024

a(n) = A000720(A116086(n)) = A000720(A116455(n)) for n <= 10. This would hold for all n if there do not exist more than two perfect powers between any two consecutive primes, which is implied by the Redmond-Sun conjecture. - Pontus von Brömssen, Nov 05 2024

a(10) from Pontus von Brömssen, Nov 04 2024

A373125 Difference between 2^n and the least squarefree number >= 2^n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 28 2024

nonn

For prime instead of squarefree we have A092131, opposite A013603.
For primes instead of powers of 2: A240474, A240473, A112926, A112925.
Difference between 2^n and A372683(n).
The opposite is A373126, delta of A372889.
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, delta A013603
Cf. A010036, A029931, A049093-A049096, A077641, A372540, A373197, A373198.

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

a(n) = A372683(n)-2^n. - R. J. Mathar, May 31 2024

A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

4, 9, 30, 327, 3512

Offset: 1

Gus Wiseman, Oct 25 2024

Is this sequence finite? For this conjecture see A053706, A080101, A366833.

Any further terms are > 10^12. - Lucas A. Brown, Nov 08 2024

			Primes 9 and 10 are 23 and 29, and the interval (24, 25, 26, 27, 28) contains the prime-powers 25 and 27, so 9 is in the sequence.

Lucas A. Brown, Python program.

The interval from A008864(n) to A006093(n+1) has A046933 elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The corresponding primes are A053706.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
These are the positions of 2 in A080101, or 3 in A366833.
For at least one prime-power we have A377057, primes A053607.
For no prime-powers we have A377286.
For exactly one prime-power we have A377287.
For squarefree instead of prime-power see A377430, A061398, A377431, A068360.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376596, A376597, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==2&]

prime(a(n)) = A053706(n).

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A372684 Least k such that prime(k) >= 2^n.

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A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

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A373125 Difference between 2^n and the least squarefree number >= 2^n.

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A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

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

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Mathematica

Formula

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

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Extensions

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

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Comments