A061398 -id:A061398 - cp's OEIS frontend

A061351 Between p and the next prime either there are no numbers or there is a single squarefree number.

2, 5, 29, 41, 101, 137, 281, 461, 569, 617, 641, 821, 857, 1229, 1289, 1301, 1481, 1697, 1721, 1877, 2081, 2129, 2237, 2309, 2381, 2657, 2729, 2801, 3389, 3461, 3557, 3917, 3929, 4001, 4217, 4241, 4421, 4637, 4721, 5009, 5441, 5477, 5501, 5657, 6089

Offset: 0

Labos Elemer, Jun 07 2001

nonn

Apart from the initial 2, the lesser of twin primes {p, p+2} such that the middle term p+1 is squarefree: intersection[{A014574(n)},{A005117(n)}].

			Between 29 and 31 the only composite is 30, a squarefree number. If next(p)-p>2, a nonsquarefree integer always arises between them.

Harry J. Smith, Table of n, a(n) for n=0,...,1000

Cf. A005117, A013929, A000040, A001359, A014574, A061398, A061399.

Join[{2},Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==2&&SquareFreeQ[#[[1]]+1]&][[;;,1]]] (* Harvey P. Dale, Jan 26 2025 *)

{ n=0; p=3; f="b061351.txt"; write(f, "0 2"); forprime (q=5, 355723, if (q-p == 2, if (issquarefree(p+1), write(f, n++, " ", p))); p=q ) } \\ Harry J. Smith, Jul 21 2009

A376343 Positions of twos in the run-compressed (A037201) first differences (A001223) of the primes (A000040).

Original entry on oeis.org

2, 4, 6, 9, 12, 15, 18, 24, 26, 31, 33, 37, 39, 41, 44, 47, 50, 53, 57, 62, 73, 75, 81, 90, 95, 99, 102, 105, 108, 127, 129, 131, 135, 139, 156, 158, 161, 163, 167, 173, 182, 187, 190, 193, 196, 205, 210, 214, 216, 232, 235, 241, 244, 247, 254, 263, 265, 270

Offset: 1

Gus Wiseman, Sep 26 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, we can remove 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 prime numbers (A000040) is:
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ...
with first differences (A001223):
  1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, ...
with run-compression (A037201):
  1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, ...
with twos at (A376343):
  2, 4, 6, 9, 12, 15, 18, 24, 26, 31, 33, 37, 39, 41, 44, 47, 50, 53, 57, 62, 73, ...

Positions of 2's in A037201.
The repeats were at positions A064113 before being omitted.
A variation for squarefree numbers is A376342.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
A333254 lists run-lengths of differences between consecutive primes.
Cf. A007921, A030173, A053289, A061398, A373817, A373822, A373947, A376305, A376308.

Join@@Position[First/@Split[Differences[Select[Range[100],PrimeQ]]],2]

For just the odd primes we have a(n) - 1.

A376521 Sorted positions of first appearances in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

Original entry on oeis.org

1, 2, 3, 8, 22, 28, 32, 42, 91, 141, 172, 198, 242, 259, 341, 400, 556, 692, 1119, 1737, 1779, 2072, 2101, 2913, 3126, 3204, 3246, 3457, 3598, 4294, 4383, 7596, 7651, 8284, 11986, 13729, 14220, 15101, 16273, 18217, 22303, 29523, 30243, 32236, 32808, 32820

Offset: 1

Gus Wiseman, Sep 26 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, we can remove 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 prime numbers (A000040) is:
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ...
with first differences (A001223):
  1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, ...
with run-compression (A037201):
  1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, ...
with first appearances at (A376521):
  1, 2, 3, 8, 22, 28, 32, 42, 91, 141, 172, 198, 242, 259, 341, 400, 556, 692, 1119, ...

These are the sorted positions of first appearances in A037201.
For positions of twos instead of first appearances we have A376343.
The unsorted version is A376520.
A000040 lists the prime numbers, differences A001223.
A003242 counts compressed compositions, ranks A333489.
A333254 lists run-lengths of differences between consecutive primes.
A373948 encodes compression using compositions in standard order.
Cf. A007921, A030173, A061398, A064113, A373817, A373822, A373947, A376305, A376308, A376312.

q=First/@Split[Differences[Select[Range[1000],PrimeQ]]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A377701 Number of non-perfect-powers x in the range 2^n < x < 2^(n+1).

Original entry on oeis.org

0, 1, 3, 6, 13, 29, 59, 121, 248, 501, 1008, 2024, 4064, 8150, 16323, 32686, 65418, 130906, 261913, 523966, 1048123, 2096517, 4193412, 8387355, 16775449, 33551945, 67105359, 134212792, 268428497, 536861096, 1073727974, 2147464110, 4294939718, 8589895659

Offset: 0

Gus Wiseman, Nov 05 2024

nonn

Non-perfect-powers (A007916) are numbers without a proper integer root.

Also the number of non-perfect-powers with n bits.

			The non-perfect-powers in each range (rows):
   .
   3
   5  6  7
  10 11 12 13 14 15
  17 18 19 20 21 22 23 24 26 28 29 30 31
Their binary expansions (columns):
  .  11  101  1010  10001
         110  1011  10010
         111  1100  10011
              1101  10100
              1110  10101
              1111  10110
                    10111
                    11000
                    11010
                    11100
                    11101
                    11110
                    11111

The union of all numbers counted is A007916.
For squarefree numbers we have A077643.
For prime-powers we have A244508.
For primes instead of powers of 2 we have A377433, ones A029707.
For perfect-powers we have A377467, for primes A377432, zeros A377436.
A000225(n) counts the interval from A000051(n) to A000225(n+1).
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.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf. A000015, A013597, A014210, A014234, A023055, A045542, A052410, A061398, A304521, A377434, A377435, A377702.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[Length[Select[Range[2^n+1, 2^(n+1)-1],radQ]],{n,0,15}]

from sympy import mobius, integer_nthroot
def A377701(n):
    def f(x): return int(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return f((1<Chai Wah Wu, Nov 06 2024

a(n) = 2^n-1-A377467(n). - Pontus von Brömssen, Nov 06 2024

Offset corrected by, and a(16)-a(33) from Pontus von Brömssen, Nov 06 2024

A376679 Number of strict integer factorizations of n into nonsquarefree factors > 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 08 2024

nonn

			The a(3456) = 28 factorizations are:
  (4*8*9*12)  (4*9*96)    (36*96)   (3456)
              (8*9*48)    (4*864)
              (4*12*72)   (48*72)
              (4*16*54)   (54*64)
              (4*18*48)   (8*432)
              (4*24*36)   (9*384)
              (4*27*32)   (12*288)
              (4*8*108)   (16*216)
              (8*12*36)   (18*192)
              (8*16*27)   (24*144)
              (8*18*24)   (27*128)
              (9*12*32)   (32*108)
              (9*16*24)
              (12*16*18)

Dominic McCarty, Table of n, a(n) for n = 1..10000

Positions of zeros are A005117 (squarefree numbers), complement A013929.
For squarefree instead of nonsquarefree we have A050326, non-strict A050320.
For prime-powers we have A050361, non-strict A000688.
For nonprime numbers we have A050372, non-strict A050370.
The version for partitions is A256012, non-strict A114374.
For perfect-powers we have A323090, non-strict A294068.
The non-strict version is A376657.
Nonsquarefree numbers:
- A078147 (first differences)
- A376593 (second differences)
- A376594 (inflections and undulations)
- A376595 (nonzero curvature)
A000040 lists the prime numbers, differences A001223.
A001055 counts integer factorizations, strict A045778.
A005117 lists squarefree numbers, differences A076259.
A317829 counts factorizations of superprimorials, strict A337069.
Cf. A008480, A053797, A053806, A061398, A089259, A120992, A303707, A322452, A375707, A376312.

function nextNonSquareFree(val){val+=1;for(let i=2;i*i<=val;i+=1){if(val%i==0&&val%(i*i)==0){return val}}return nextNonSquareFree(val)}function strictFactorCount(val,maxFactor){if(val==1){return 1}let sum=0;while(maxFactorDominic McCarty, Oct 19 2024

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],UnsameQ@@#&&NoneTrue[#,SquareFreeQ]&]],{n,100}] (* corrected by Gus Wiseman, Jun 27 2025 *)

A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

Original entry on oeis.org

0, 4, 0, 17, 12, 16, 18, 20, 104, 0, 68, 40, 0, 89, 199, 110, 60, 127, 68, 72, 151, 161, 172, 278, 297, 0, 104, 108, 112, 849, 128, 403, 0, 579, 150, 461, 322, 164, 680, 351, 180, 561, 192, 196, 198, 819, 648, 449, 228, 232, 470, 240, 1472, 508, 521, 532, 270

Offset: 1

Gus Wiseman, Dec 09 2024

nonn

			The nonsquarefree numbers between prime(24) = 89 and prime(25) = 97 are {90, 92, 96}, so a(24) = 278.

For prime instead of nonsquarefree we have A001043.
For composite instead of nonsquarefree we have A054265.
Zeros are A068361.
A000040 lists the primes, differences A001223, seconds A036263.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers up to prime(n), restriction of A013928.
A120327 gives the least nonsquarefree number >= n.
A378086 counts nonsquarefree numbers up to prime(n), restriction of A057627.
For squarefree numbers (A005117, differences A076259) between primes:
- length is A061398, zeros A068360
- min is A112926, differences A378037
- max is A112925, differences A378038
- sum is A373197
For nonsquarefree numbers (A013929, differences A078147) between primes:
- length is A061399
- min is A377783 (differences A377784), union A378040
- max is A378032 (differences A378034), restriction of A378033 (differences A378036)
- sum is A378618 (this)
Cf. A007674, A045882, A053806, A067535, A072284, A073247, A179211, A224363, A373198, A377049, A377431.

Table[Total[Select[Range[Prime[n],Prime[n+1]],!SquareFreeQ[#]&]],{n,100}]

A378619 Distance between n and the greatest squarefree number <= n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 12 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30000

Positions of 0 are A005117.
Positions of first appearances are A020755 - 1.
Positions of 1 are A053806.
Subtracting each term from n gives A070321.
The opposite version is A081221.
Restriction to the primes is A240473, opposite A240474.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A007674, A013928, A053797, A072284, A073247, A076259, A280892.
Cf. A007947, A067535, A081217, A112925.

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

A378619(n) = forstep(k=n,1,-1,if(issquarefree(k), return(n-k))); \\ Antti Karttunen, Jan 29 2025

from itertools import count
from sympy import factorint
def A378619(n): return n-next(m for m in count(n,-1) if max(factorint(m).values(),default=0)<=1) # Chai Wah Wu, Dec 14 2024

a(n) = n - A070321(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A376164 Maximum of the n-th maximal run of nonsquarefree numbers (increasing by 1 at a time).

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 32, 36, 40, 45, 50, 52, 54, 56, 60, 64, 68, 72, 76, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 117, 121, 126, 128, 132, 136, 140, 144, 148, 150, 153, 156, 160, 162, 164, 169, 172, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208

Offset: 1

Gus Wiseman, Sep 15 2024

nonn

			The maximal runs of nonsquarefree numbers begin:
       4
     8   9
      12
      16
      18
      20
    24  25
    27  28
      32
      36
      40
    44  45
  48  49  50

For length instead of maximum we have A053797 (firsts A373199).
For lengths of anti-runs we have A373409 (firsts A373573).
For sum instead of maximum we have A373414, anti A373412.
For minimum instead of maximum we have A053806, anti A373410.
For anti-runs instead of runs we have A068781.
For squarefree instead of nonsquarefree we have A373415, anti A007674.
For nonprime instead of nonsquarefree we have A006093 with 2 removed.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147, sums A329472.
A061398 counts squarefree numbers between primes, nonsquarefree A061399.
A120992 gives squarefree run-lengths, anti A373127 (firsts A373128).
A373413 adds up each maximal run of squarefree numbers, min A072284.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A000040, A020754, A045882, A049094, A054265, A077641, A101836, A143658, A294242, A375709.

Max/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1==#2&]//Most

A376657 Number of integer factorizations of n into nonsquarefree factors > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 4, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 07 2024

nonn

			The a(n) factorizations for n = 16, 64, 72, 144, 192, 256, 288:
  (16)   (64)     (72)    (144)    (192)     (256)      (288)
  (4*4)  (8*8)    (8*9)   (4*36)   (4*48)    (4*64)     (4*72)
         (4*16)   (4*18)  (8*18)   (8*24)    (8*32)     (8*36)
         (4*4*4)          (9*16)   (12*16)   (16*16)    (9*32)
                          (12*12)  (4*4*12)  (4*8*8)    (12*24)
                          (4*4*9)            (4*4*16)   (16*18)
                                             (4*4*4*4)  (4*8*9)
                                                        (4*4*18)

For prime-powers we have A000688.
Positions of zeros are A005117 (squarefree numbers), complement A013929.
For squarefree instead of nonsquarefree we have A050320, strict A050326.
For nonprime numbers we have A050370.
The version for partitions is A114374.
For perfect-powers we have A294068.
For non-perfect-powers we have A303707.
For non-prime-powers we have A322452.
The strict case is A376679.
Nonsquarefree numbers:
- A078147 (first differences)
- A376593 (second differences)
- A376594 (inflections and undulations)
- A376595 (nonzero curvature)
A000040 lists the prime numbers, differences A001223.
A001055 counts integer factorizations, strict A045778.
A005117 lists squarefree numbers, differences A076259.
A317829 counts factorizations of superprimorials, strict A337069.
Cf. A008480, A053797, A053806, A061398, A089259, A120992, A124010, A182853, A373198, A375707, A376312.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],NoneTrue[SquareFreeQ]]],{n,100}]

A378620 Lesser prime index of twin primes with nonsquarefree mean.

Original entry on oeis.org

2, 5, 7, 17, 20, 28, 35, 41, 43, 45, 49, 52, 57, 64, 69, 81, 83, 98, 109, 120, 140, 144, 152, 171, 173, 176, 178, 182, 190, 206, 215, 225, 230, 236, 253, 256, 262, 277, 286, 294, 296, 302, 307, 315, 318, 323, 336, 346, 373, 377, 390, 395, 405, 428, 430, 444

Offset: 1

Gus Wiseman, Dec 10 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

This is a subset of A029707 (twin prime indices). The other twin primes are A068361, so A029707 is the disjoint union of A068361 and A378620.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

The lesser of twin primes is A001359, index A029707 (complement A049579).
The greater of twin primes is A006512, index A107770 (complement appears to be A168543).
A subset of A029707 (twin prime lesser indices).
Prime indices of the primes listed by A061368.
Indices of twin primes with squarefree mean are A068361.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 finds the first position of a prime gap of 2n.
A046933 counts composite numbers between primes.
A120327 gives the least nonsquarefree number >= n.
Cf. A007674, A072284, A068360.
Cf. A057627, A061398, A061399, A067535, A070321, A071403, A112925.
Cf. A000720, A025584, A067774, A122535, A155752, A179067.

Select[Range[100],Prime[#]+2==Prime[#+1]&&!SquareFreeQ[Prime[#]+1]&]
PrimePi/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&!SquareFreeQ[Mean[#]]&][[;;,1]] (* Harvey P. Dale, Jul 13 2025 *)

prime(a(n)) = A061368(n).

cp's OEIS Frontend

A061351 Between p and the next prime either there are no numbers or there is a single squarefree number.

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A376343 Positions of twos in the run-compressed (A037201) first differences (A001223) of the primes (A000040).

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A376521 Sorted positions of first appearances in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

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A377701 Number of non-perfect-powers x in the range 2^n < x < 2^(n+1).

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A376679 Number of strict integer factorizations of n into nonsquarefree factors > 1.

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A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

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A378619 Distance between n and the greatest squarefree number <= n.

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A376164 Maximum of the n-th maximal run of nonsquarefree numbers (increasing by 1 at a time).

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