A373198 -id:A373198 - cp's OEIS frontend

A373197 Sum of all squarefree numbers from prime(n) to prime(n+1) - 1.

2, 3, 11, 17, 11, 42, 17, 62, 49, 59, 133, 114, 83, 89, 98, 223, 59, 254, 206, 71, 302, 161, 341, 462, 97, 203, 314, 107, 330, 824, 386, 398, 275, 856, 149, 460, 635, 494, 337, 702, 179, 1294, 191, 582, 197, 1635, 1950, 449, 227, 690, 943, 239, 983, 1013, 1036

Offset: 1

Gus Wiseman, May 29 2024

nonn

			This is the sequence of row sums of A005117 treated as a triangle with row-lengths A373198:
   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

Counting all numbers (not just squarefree) gives A371201.
For the sectioning of A005117 (squarefree between prime):
- sum is A373197 (this sequence)
- length is A373198 = A061398 - 1
- min is A000040
- max is A112925, opposite A112926
For squarefree numbers between powers of two:
- sum is A373123
- length is A077643, partial sums A143658
- min is A372683, delta A373125, indices A372540, firsts of A372475
- max is A372889, delta A373126
For primes between powers of two:
- sum is A293697 (except initial terms)
- length is A036378
- min is A104080 or A014210, indices A372684 (firsts of A035100)
- max is A014234, delta A013603
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
Cf. A046933, A049093, A049094, A076259, A077641.

Table[Total[Select[Range[Prime[n],Prime[n+1]-1],SquareFreeQ]],{n,15}]

A377431 Numbers k such that there is at least one squarefree number between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 44, 46, 47, 48, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86

Offset: 1

Gus Wiseman, Oct 29 2024

nonn

			Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains 10, which is squarefree, so 4 is in the sequence.

These are the positive positions in A061398, or terms >= 2 in A373198.
The complement (no squarefree numbers) is A068360.
For prime-power instead of squarefree we have A377057, strict version A377287.
For exactly one squarefree number we have A377430.
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composites, complement A008578.
A005117 lists the squarefree numbers, complement A013929.
A377038 gives k-differences of squarefree numbers.
Cf. A007674, A013603, A029707, A036378, A046933, A072284, A076259, A077641, A077643, A075526, A078147, A112925, A112926, A120992, A293697.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],SquareFreeQ]]>=1&]

A378086 Number of nonsquarefree numbers < prime(n).

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 5, 6, 7, 11, 11, 13, 14, 14, 16, 20, 22, 23, 25, 26, 27, 29, 31, 33, 36, 39, 39, 40, 41, 42, 49, 50, 53, 53, 57, 58, 61, 63, 64, 68, 70, 71, 74, 75, 76, 77, 81, 84, 86, 87, 88, 90, 91, 97, 99, 101, 103, 104, 107, 109, 109, 113, 119, 120, 121

Offset: 1

Gus Wiseman, Dec 04 2024

nonn

			The nonsquarefree numbers counted under each term begin:
  n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9: n=10: n=11: n=12:
  --------------------------------------------------------------
   .    .    4    4    9    12   16   18   20   28    28    36
                       8    9    12   16   18   27    27    32
                       4    8    9    12   16   25    25    28
                            4    8    9    12   24    24    27
                                 4    8    9    20    20    25
                                      4    8    18    18    24
                                           4    16    16    20
                                                12    12    18
                                                9     9     16
                                                8     8     12
                                                4     4     9
                                                            8
                                                            4

For nonprime numbers we have A014689.
Restriction of A057627 to the primes.
First-differences are A061399 (zeros A068361), squarefree A061398 (zeros A068360).
For composite instead of squarefree we have A065890.
For squarefree we have A071403, differences A373198.
Greatest is A378032 (differences A378034), restriction of A378033 (differences A378036).
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A070321 gives the greatest squarefree number up to n.
A112925 gives the greatest squarefree number between primes, differences A378038.
A112926 gives the least squarefree number between primes, differences A378037.
A120327 gives the least nonsquarefree number >= n, first-differences A378039.
A377783 gives the least nonsquarefree > prime(n), differences A377784.
Cf. A013928, A046933, A053797, A053806, A072284, A076259, A224363, A337030, A377049, A378040, A378082, A378084.

Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}]

from math import isqrt
from sympy import prime, mobius
def A378086(n): return (p:=prime(n))-sum(mobius(k)*(p//k**2) for k in range(1,isqrt(p)+1)) # Chai Wah Wu, Dec 05 2024

a(n) = A057627(prime(n)).

A372683 Least squarefree number >= 2^n.

Original entry on oeis.org

1, 2, 5, 10, 17, 33, 65, 129, 257, 514, 1027, 2049, 4097, 8193, 16385, 32770, 65537, 131073, 262145, 524289, 1048577, 2097154, 4194305, 8388609, 16777217, 33554433, 67108865, 134217730, 268435457, 536870913, 1073741826, 2147483649, 4294967297, 8589934594

Offset: 0

Gus Wiseman, May 26 2024

nonn

			The terms together with their binary expansions and binary indices begin:
       1:                    1 ~ {1}
       2:                   10 ~ {2}
       5:                  101 ~ {1,3}
      10:                 1010 ~ {2,4}
      17:                10001 ~ {1,5}
      33:               100001 ~ {1,6}
      65:              1000001 ~ {1,7}
     129:             10000001 ~ {1,8}
     257:            100000001 ~ {1,9}
     514:           1000000010 ~ {2,10}
    1027:          10000000011 ~ {1,2,11}
    2049:         100000000001 ~ {1,12}
    4097:        1000000000001 ~ {1,13}
    8193:       10000000000001 ~ {1,14}
   16385:      100000000000001 ~ {1,15}
   32770:     1000000000000010 ~ {2,16}
   65537:    10000000000000001 ~ {1,17}
  131073:   100000000000000001 ~ {1,18}
  262145:  1000000000000000001 ~ {1,19}
  524289: 10000000000000000001 ~ {1,20}

For primes instead of powers of two we have A112926, opposite A112925, sum A373197, length A373198.
Counting zeros instead of all bits gives A372473, firsts of A372472.
These are squarefree numbers at indices A372540, firsts of A372475.
Counting ones instead of all bits gives A372541, firsts of A372433.
The opposite (greatest squarefree number <= 2^n) is A372889.
The difference from 2^n is A373125.
For prime instead of squarefree we have:
- bits A372684, firsts of A035100
- zeros A372474, firsts of A035103
- ones A372517, firsts of A014499
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308, length A070939 or A029837.
A061398 counts squarefree numbers between primes (exclusive).
A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
A143658 counts squarefree numbers up to 2^n.
Cf. A029931, A048793, A049095, A049096, A059015, A069010, A076259, A077641, A211997, A230877.

Table[NestWhile[#+1&,2^n,!SquareFreeQ[#]&],{n,0,10}]

a(n) = my(k=2^n); while (!issquarefree(k), k++); k; \\ Michel Marcus, May 29 2024

from itertools import count
from sympy import factorint
def A372683(n): return next(i for i in count(1<Chai Wah Wu, Aug 26 2024

a(n) = A005117(A372540(n)).

a(n) = A067535(2^n). - R. J. Mathar, May 31 2024

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

Original entry on oeis.org

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

A373409 Length of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Original entry on oeis.org

2, 6, 2, 5, 2, 1, 6, 4, 2, 7, 1, 5, 2, 2, 1, 4, 4, 3, 6, 2, 2, 4, 7, 5, 7, 1, 1, 6, 6, 2, 3, 4, 7, 3, 3, 5, 1, 3, 1, 3, 2, 2, 3, 5, 5, 7, 1, 5, 7, 5, 1, 8, 4, 2, 5, 2, 2, 3, 3, 1, 7, 3, 4, 7, 1, 5, 2, 5, 2, 6, 7, 6, 7, 5, 1, 2, 3, 5, 6, 4, 1, 3, 5, 7, 2, 3, 2

Offset: 1

Gus Wiseman, Jun 06 2024

nonn

An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.

Conjecture: The maximum is 9, and there is no antirun of more than 9 nonsquarefree numbers. Confirmed up to 100,000,000.

			Row-lengths of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99
The first maximal antirun of length 9 is the following, shown with prime indices:
  6345: {2,2,2,3,15}
  6348: {1,1,2,9,9}
  6350: {1,3,3,31}
  6352: {1,1,1,1,78}
  6354: {1,2,2,71}
  6356: {1,1,4,49}
  6358: {1,5,7,7}
  6360: {1,1,1,2,3,16}
  6363: {2,2,4,26}

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

Positions of first appearances are A373573, sorted A373574.
Functional neighbors: A027833, A053797, A068781, A373127, A373403, A373410, A373412.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A025157, A049093, A061398, A061399, A077641, A077643, A110969, A174965, A294242, A350842, A373198.

Length/@Split[Select[Range[1000],!SquareFreeQ[#]&],#1+1!=#2&]//Most

A376305 Run-compression of the sequence of first differences of squarefree numbers.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3

Offset: 1

Gus Wiseman, Sep 20 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
The sequence of first differences (A076259) of squarefree numbers is:
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
The run-compression is A376305 (this sequence).

This is the run-compression of first differences of A005117.
For prime instead of squarefree numbers we have A037201, halved A373947.
Before compressing we had A076259, ones A375927.
For run-lengths instead of compression we have A376306.
For run-sums instead of compression we have A376307.
For prime-powers instead of squarefree numbers we have A376308.
For positions of first appearances instead of compression we have A376311.
The version for nonsquarefree numbers is A376312.
Positions of 1's are A376342.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed or anti-run compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, by compressed length A116608.
A274174 counts contiguous compositions, ranks A374249.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A274174, A373197, A373198, A375707, A375708.

First/@Split[Differences[Select[Range[100],SquareFreeQ]]]

A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

Original entry on oeis.org

1, 3, 8, 23, 29, 33, 45, 98, 153, 188, 216, 262, 281, 366, 428, 589, 737, 1182, 1830, 1878, 2190, 2224, 3076, 3301, 3384, 3426, 3643, 3792, 4521, 4611, 7969, 8027, 8687, 12541, 14356, 14861, 15782, 17005, 19025, 23282, 30801, 31544, 33607, 34201, 34214, 38589

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The unsorted version is A073051.

A run of a sequence (in this case A002808) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of composite numbers begin:
   4
   6
   8   9  10
  12
  14  15  16
  18
  20  21  22
  24  25  26  27  28
  30
  32  33  34  35  36
  38  39  40
  42
  44  45  46
  48  49  50  51  52
  54  55  56  57  58
  60
  62  63  64  65  66
  68  69  70
  72
  74  75  76  77  78
  80  81  82
  84  85  86  87  88
  90  91  92  93  94  95  96
  98  99 100
The a(n)-th rows are:
   4
   8   9  10
  24  25  26  27  28
  90  91  92  93  94  95  96
 114 115 116 117 118 119 120 121 122 123 124 125 126
 140 141 142 143 144 145 146 147 148
 200 201 202 203 204 205 206 207 208 209 210

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The unsorted version is A073051, firsts of A176246.
For squarefree runs we have the triple (1,3,5), firsts of A120992.
For prime runs we have the triple (1,2,3), firsts of A175632.
For squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373402, unsorted A373401, firsts of A027833.
For composite runs we have the triple (1,2,7), firsts of A373403.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

t=Length/@Split[Select[Range[10000],CompositeQ],#1+1==#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A373128 Least k such that the k-th maximal antirun of squarefree numbers has length n. Position of first appearance of n in A373127.

Original entry on oeis.org

1, 3, 10, 8, 19, 162, 1853, 2052, 1633, 26661, 46782, 3138650, 1080330

Offset: 1

Gus Wiseman, Jun 08 2024

An antirun of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of squarefree numbers begin:
   1
   2
   3   5
   6
   7  10
  11  13
  14
  15  17  19  21
  22
  23  26  29
  30
  31  33
  34
  35  37
The a(n)-th rows are:
    1
    3    5
   23   26   29
   15   17   19   21
   47   51   53   55   57
  483  485  487  489  491  493
For example, (23, 26, 29) is the first maximal antirun of 3 squarefree numbers, so a(3) = 10.

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

For composite instead of squarefree we have A073051.
Positions of first appearances in A373127.
The version for nonsquarefree runs is A373199, firsts of A053797.
For prime instead of squarefree we have A373401, firsts of A027833.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A120992, A174965, A251092, A373198, A373408, A373411.

t=Length/@Split[Select[Range[10000],SquareFreeQ[#]&],#1+1!=#2&]//Most;
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[Max@@#]&];
Table[Position[t,k][[1,1]],{k,spnm[t]}]

A376306 Run-lengths of the sequence of first differences of squarefree numbers.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 21 2024

nonn

			The sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
The sequence of first differences (A076259) of squarefree numbers is:
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
with runs:
  (1,1),(2),(1,1),(3),(1),(2),(1,1),(2,2,2),(1,1),(3,3),(1,1),(2),(1,1), ...
with lengths A376306 (this sequence).

Run-lengths of first differences of A005117.
Before taking run-lengths we had A076259, ones A375927.
For prime instead of squarefree numbers we have A333254.
For compression instead of run-lengths we have A376305.
For run-sums instead of run-lengths we have A376307.
For prime-powers instead of squarefree numbers we have A376309.
For positions of first appearances instead of run-lengths we have A376311.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed or anti-run compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, by compressed length A116608.
A274174 counts contiguous compositions, ranks A374249.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A373198, A375707, A376312.

Length/@Split[Differences[Select[Range[100],SquareFreeQ]]]

cp's OEIS Frontend

A373197 Sum of all squarefree numbers from prime(n) to prime(n+1) - 1.

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A377431 Numbers k such that there is at least one squarefree number between prime(k)+1 and prime(k+1)-1.

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A378086 Number of nonsquarefree numbers < prime(n).

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A372683 Least squarefree number >= 2^n.

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

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A373409 Length of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

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A376305 Run-compression of the sequence of first differences of squarefree numbers.

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A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

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A373128 Least k such that the k-th maximal antirun of squarefree numbers has length n. Position of first appearance of n in A373127.

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