A057627 -id:A057627 - cp's OEIS frontend

A378615 Number of non prime powers <= prime(n).

1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 14, 18, 21, 22, 25, 29, 34, 35, 39, 42, 43, 48, 50, 55, 62, 65, 66, 69, 70, 73, 84, 86, 91, 92, 101, 102, 107, 112, 115, 119, 124, 125, 134, 135, 138, 139, 150, 161, 164, 165, 168, 173, 174, 182, 186, 191, 196, 197, 202, 205

Offset: 1

Gus Wiseman, Dec 06 2024

nonn

			The non prime powers counted under each term:
  n=1  n=2  n=3  n=4  n=5  n=6  n=7  n=8  n=9  n=10
  -------------------------------------------------
   1    1    1    6   10   12   15   18   22   28
                  1    6   10   14   15   21   26
                       1    6   12   14   20   24
                            1   10   12   18   22
                                 6   10   15   21
                                 1    6   14   20
                                      1   12   18
                                          10   15
                                           6   14
                                           1   12
                                               10
                                                6
                                                1

Restriction of A356068 (first-differences A143731).
First-differences are A368748.
Maxima are A378616.
Other classes of numbers (instead of non prime powers):
- prime: A000027 (diffs A000012), restriction of A000720 (diffs A010051)
- squarefree: A071403 (diffs A373198), restriction of A013928 (diffs A008966)
- nonsquarefree: A378086 (diffs A061399), restriction of A057627 (diffs A107078)
- prime power: A027883 (diffs A366833), restriction of A025528 (diffs A010055)
- composite: A065890 (diffs A046933), restriction of A065855 (diffs  A005171)
A000040 lists the primes, differences A001223
A000961 and A246655 list the prime powers, differences A057820.
A024619 lists the non prime powers, differences A375735, seconds A376599.
A080101 counts prime powers between primes (exclusive), inclusive A366833.
A361102 lists the non powers of primes, differences A375708.
Cf. A027883, A053607, A304521, A343249, A345531, A377057, A377281, A377286, A377289, A377703, A377781, A378032.

Table[Length[Select[Range[Prime[n]],Not@*PrimePowerQ]],{n,100}]

from sympy import prime, primepi, integer_nthroot
def A378615(n): return int((p:=prime(n))-n-sum(primepi(integer_nthroot(p,k)[0]) for k in range(2,p.bit_length()))) # Chai Wah Wu, Dec 07 2024

a(n) = prime(n) - A027883(n). - Chai Wah Wu, Dec 08 2024

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}]

A082926 Difference between the number of primes equal to or less than n and (1 + the number of nonsquarefree numbers equal to or less than n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 1, 2, 3, 3, 2, 3, 4, 5, 5, 6, 5, 6, 6, 7, 7, 7, 6, 7, 6, 6, 7, 8, 8, 8, 7, 8, 8, 8, 7, 8, 7, 7, 8, 9, 9, 9, 8, 9, 10, 10, 9, 10, 10, 10, 10, 11, 10, 11, 11, 12, 12, 12, 12, 13, 12

Offset: 1

Anonymous, Apr 15 2003

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A000720, A057627.

nsfQ[n_] := n == 1 || ! SquareFreeQ[n]; Abs[Accumulate[Table[If[nsfQ[n], 1, 0] - If[PrimeQ[n], 1, 0], {n, 1, 100}]]] (* Amiram Eldar, Jun 13 2022 *)

a(n) = abs(sum(k=1, n, !issquarefree(k)) + 1 - primepi(n)); \\ Michel Marcus, Mar 27 2020

a(n) = abs(A057627(n) + 1 - A000720(n)). - Michel Marcus, Mar 27 2020

Name edited by Michel Marcus, Jun 14 2022

A336025 Numbers m providing record values for the proportion of nonsquarefree integers in the interval [1, m].

Original entry on oeis.org

4, 9, 20, 25, 27, 28, 172, 176

Offset: 1

Javier Múgica, Jul 05 2020

Also, numbers providing record low values for the proportion of squarefree integers.
The proportion of nonsquarefree integers approaches 1-6/Pi^2. For low values of m the proportion in [1, m] tends to be lower, since squares appear late. But values of m for which the ratio in the interval [1, m] is larger than the limit value do exist. The first such one is 28. Therefore this sequence is finite and it can be proved that 176 is indeed its last term. The proportion of nonsquarefree numbers in [1, 176] is 70/176 = 0.397727272... and that of squarefree ones is 0.6022727...
Rogers (1964) proved that if A(x) is the number of squarefree numbers not exceeding x then A(x)/x >= A(176)/176 = 53/88. Therefore 176 is the last term of this sequence. - Amiram Eldar, Mar 12 2021

			Up to m = 9 there are 3 numbers which are divisible by some square: 4, 8 and 9, for a proportion of 3/9 = 1/3. No interval [1, k] for k < 9 has a ratio as high, so 9 is in the sequence.

Kenneth Rogers, The Schnirelmann density of the squarefree integers, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516.

Cf. A005117, A013929, A173143, A057627.

s = {}; c = 0; rm = 0; Do[If[!SquareFreeQ[n], c++]; If[(r = c/n) > rm, rm = r; AppendTo[s, n]], {n, 1, 10^3}]; s (* Amiram Eldar, Mar 12 2021 *)
Module[{nn=200},DeleteDuplicates[{#[[1]],#[[2]]/#[[1]]}&/@Thread[{Range[nn],Accumulate[ Table[If[SquareFreeQ[n],0,1],{n,nn}]]}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]]//Rest (* Harvey P. Dale, Jun 15 2024 *)

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).

A243284 a(n) = the number of distinct ways of writing such products m = k^2 * j, 0 < j <= k, (j and k natural numbers) that m is in range [1,n]; Partial sums of A102354.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

a(n) = the number of distinct ways of writing such products m = k^2 * j, 0 < j <= k, (j and k natural numbers) that m is in range [1,n].
Different ways to write product for the same m are counted separately, e.g. for 64, both 8^2 * 1 and 4^2 * 4 are counted, so a(64) = a(63)+2 = 13+2 = 15.
Differs from A243283 for the first time at n=48, where a(48)=11, while A243283(48)=10. This is because 48 = 2*2*2*2*3 is the first integer which can be represented in the form k^2 * j, 0 < j <= k (namely as 48 = 4^2 * 3), even though it is not a member of A070003.

Partial sums of A102354.

Cf. A243283, A013928, A057627.

A336026 Numbers m such that the proportion of nonsquarefree numbers in the interval [1, m] is greater than the corresponding proportion for all k > m.

Original entry on oeis.org

176, 380, 388, 389, 392, 393, 1089, 1864, 1928, 1936, 1937, 1940, 2080, 2892, 2900, 2908, 2909, 2912, 3776, 5589, 5788, 5832, 5932, 5933, 7156, 7157, 11881, 11889, 12656, 12776, 13880, 13888, 14085, 14088, 14096, 14104, 14456, 14464, 14465, 39740

Offset: 1

Javier Múgica, Jul 05 2020

nonn

Also, numbers m such that the proportion of squarefree numbers in the interval [1, m] is less than the corresponding proportion for all k > m.
If the condition "greater than" were changed to "greater than or equal to" the sequence would also include the number 396, where the proportion is the same as at 1089, namely, 156/396 = 429/1089 = 39/99. There seems to be no other such coincidence. There is none up to 2*10^11.
All the terms are congruent to 0 or 1 modulo 4. If the modulus 36 is considered, the only possible residue classes are 0, 1, 9, 12, 20, 28, 29, 32 and 33. Similar restrictions hold for larger moduli. Thus, mod 900 there are only 132 possible residues, the least one being 28. Of these, more than half appear in pairs of two consecutive values.

			There are 151 nonsquarefree numbers up to m = 380, for a proportion of 151/380 ~= 0.39737. This proportion is never again reached for larger values of m, so the number 380 belongs to this list.

Javier Múgica, Table of n, a(n) for n = 1..211
Javier Múgica, C program for finding (not proving) the terms of this sequence. The computations for the proofs can be found in the following spreadsheets: The first four terms. The first 211 terms. The determination of the possible residue classes mod 36 and 900.

Cf. A013929, A057627, A336025.

A380413 Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).

Original entry on oeis.org

0, 1, 11, 14, 39, 53, 109, 179, 222, 240, 251, 319, 337, 481, 505, 508, 578, 664, 674, 738, 818, 835, 877, 905, 933, 1041, 1069, 1098, 1325, 1352, 1392, 1535, 1539, 1567, 1652, 1663, 1732, 1817, 1849, 1960, 2134, 2148, 2158, 2220, 2387, 2428, 2457, 2622, 2625

Offset: 1

Gus Wiseman, Feb 06 2025

nonn

A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061399 counts nonsquarefree integers between primes, see A068361, A061398, A068360, A377783, A378086.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers < prime(n), see A373198, A337030.
A112925 gives the greatest squarefree number between primes, least A112926.
Cf. A057627, A065890, A378032 (differences A378034), A378033 (differences A378036).
Cf. A072284, A120327, A224363, A378040, A378082, A378084.

y=Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}];
Select[Most[Union[y]],Count[y,#]==2&]

a(n) = A378086(A068361(n)) = A378086(A068361(n)+1).

cp's OEIS Frontend

A378615 Number of non prime powers <= prime(n).

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

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A082926 Difference between the number of primes equal to or less than n and (1 + the number of nonsquarefree numbers equal to or less than n).

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A336025 Numbers m providing record values for the proportion of nonsquarefree integers in the interval [1, m].

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A378620 Lesser prime index of twin primes with nonsquarefree mean.

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A243284 a(n) = the number of distinct ways of writing such products m = k^2 * j, 0 < j <= k, (j and k natural numbers) that m is in range [1,n]; Partial sums of A102354.

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A336026 Numbers m such that the proportion of nonsquarefree numbers in the interval [1, m] is greater than the corresponding proportion for all k > m.

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A380413 Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).

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