A070321 -id:A070321 - cp's OEIS frontend

A378457 Difference between n and the greatest prime power <= n, allowing 1.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 0, 1, 2, 3, 4

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Prime powers allowing 1 are listed by A000961.

			The greatest prime power <= 6 is 5, so a(6) = 1.

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we have A010051 (almost) (A179278).
Subtracting from n gives (A031218).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
Adding one gives A276781.
For nonsquarefree we have (A378033).
For non perfect power we have (A378363).
For non prime power we have A378366 (A378367).
The opposite is A378370 = A377282-1.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n, weak version A007918.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A007920, A013632, A065514, A074984, A377051, A377054, A377281, A377289, A377468, A378357, A378371.

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

a(n) = n - A031218(n).

a(n) = A276781(n) - 1.

A285719 a(1) = 1, and for n > 1, a(n) = the largest squarefree number k such that n-k is also squarefree.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 11, 13, 14, 15, 15, 17, 17, 19, 19, 21, 22, 23, 23, 23, 26, 26, 26, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 39, 41, 42, 43, 43, 43, 46, 47, 47, 47, 46, 51, 51, 53, 53, 55, 55, 57, 58, 59, 59, 61, 62, 62, 62, 65, 66, 67, 67, 69, 70, 71, 71, 73, 74, 74, 74, 77, 78, 79, 79, 79, 82, 83, 83, 85, 86, 87, 87, 89, 89, 91, 91

Offset: 1

Antti Karttunen, May 02 2017

nonn

For any n > 1 there is at least one decomposition of n as a sum of two squarefree numbers (cf. A071068 and the Mathematics Stack Exchange link). Of all pairs (x,y) of positive squarefree numbers for which x <= y and x+y = n, sequences A285718 and A285719 give the unique pair for which the difference y-x is the largest possible.

Note: a(n+1) differs from A070321(n) for the first time at n=50, with a(51) = 46, while A070321(50) = 47.

			For n=51 we see that 50 (2*5*5), 49 (7*7) and 48 (2^4 * 3) are all nonsquarefree (A013929). 47 (a prime) is squarefree, but 51 - 47 = 4 is not. On the other hand, both 46 (2*23) and 5 are squarefree numbers, thus a(51) = 46.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Mathematics Stack Exchange, Sums of square free numbers, is this conjecture equivalent to Goldbach's conjecture? (See especially the answer of Aryabhata)
K. Rogers, The Schnirelmann density of the squarefree integers, Proc. Amer. Math. Soc. 15 (1964), pp. 515-516.

Cf. A005117, A008683, A013928, A013929, A070321, A071068, A285718, A285735.

lsfn[n_]:=Module[{k=n-1},While[!SquareFreeQ[k]||!SquareFreeQ[n-k],k--];k]; Join[{1},Array[ lsfn,100,2]] (* Harvey P. Dale, Apr 27 2023 *)

from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n) == n
def a285718(n):
    if n==1: return 0
    x = 1
    while True:
        if issquarefree(x) and issquarefree(n - x):return x
        else: x+=1
def a285719(n): return n - a285718(n)
print([a285719(n) for n in range(1, 121)]) # Indranil Ghosh, May 02 2017

(define (A285719 n) (- n (A285718 n)))
(define (A285719 n) (if (= 1 n) n (let loop ((k (A013928 n))) (if (not (zero? (A008683 (- n (A005117 k))))) (A005117 k) (loop (- k 1))))))

a(n) = n - A285718(n).

A378366 Difference between n and the greatest non prime power <= n (allowing 1).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we almost have A010051 (A179278).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
For prime power we have A378457 = A276781-1 (A031218).
For nonsquarefree we have (A378033).
For non perfect power we almost have A075802 (A378363).
Subtracting from n gives (A378367).
The opposite is A378371, adding n A378372.
A000015 gives the least prime power >= n (cf. A378370 = A377282 - 1).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n, weak version A007918.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A007920, A013632, A065514, A074984, A113646, A345531, A377051, A377054, A377281, A377289, A378357.

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

a(n) = n - A378367(n).

A081218 Greatest squarefree number not exceeding n-th prime power which is not prime.

Original entry on oeis.org

1, 3, 7, 7, 15, 23, 26, 31, 47, 62, 79, 119, 123, 127, 167, 241, 255, 287, 341, 359, 511, 527, 623, 727, 839, 959, 1023, 1330, 1367, 1679, 1847, 2047, 2186, 2195, 2207, 2399, 2807, 3122, 3478, 3719, 4094, 4487, 4911, 5039, 5327, 6239, 6559, 6857, 6887, 7919

Offset: 1

Reinhard Zumkeller, Mar 10 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A025475, A070321, A081217.

f[n_] := Module[{k = n-1}, While[!SquareFreeQ[k], k--]; k]; Join[{1}, f /@ Select[Range[8000], CompositeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Mar 26 2025 *)

a(n) = A070321(A025475(n)).

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

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).

A384420 The number of exponentially squarefree prime powers (not including 1) that divide n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 28 2025

The number of terms in A384419 that are larger than 1 and divide n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A070321, A077761, A378085, A384419, A384421.

s[n_] := Module[{k = n}, While[! SquareFreeQ[k], k--]; k]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = while(!issquarefree(n), n--); n;
a(n) = vecsum(apply(s, factor(n)[, 2]));

Additive with a(p^e) = A070321(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B + C), where B is Mertens's constant (A077761), and C = Sum_{p prime, e>=2} A378085(e-1)/p^e = 0.72770645470600638249... .

A386468 The maximum exponent in the prime factorization of the largest exponentially squarefree divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 22 2025

First differs from A375428 at n = 64.
Differs from A368105 at n = 1, 36, 64, 72, 100, ... .
Except for a(1), all the terms are by definition squarefree numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A051903, A070321, A365683, A368105, A375428, A378085.

a[n_] := Module[{k = Max[FactorInteger[n][[;; , 2]]]}, While[! SquareFreeQ[k], k--]; k]; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, my(k = vecmax(factor(n)[,2])); while(!issquarefree(k), k--); k);

a(n) = A051903(A365683(n)).
a(n) = A070321(A051903(n)) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} A378085(k-1)*(1-1/zeta(k)) = 1.66055078443790141429... .

A283807 Smallest prime p such that A005117(k+1) - A005117(k-1) = n, where p = A005117(k) for some k.

Original entry on oeis.org

2, 3, 7, 47, 97, 241, 5051, 204329, 217069, 29002021, 190346677, 3568762019, 221167421, 18725346527

Offset: 2

Juri-Stepan Gerasimov, Mar 17 2017

			2 is in this sequence because A005117(2+1) - A005117(2-1) = 3 - 1 = 2, where A005117(2) = 2 is prime for k = 2.
3 is in this sequence because A005117(3+1) - A005117(3-1) = 5 - 2 = 3, where A005117(3) = 3 is prime for k = 3.
7 is in this sequence because A005117(6+1) - A005117(6-1) = 10 - 6 = 4, where A005117(6) = 7 is prime for k = 6.
47 is in this sequence because A005117(31+1) - A005117(31-1) = 51 - 46 = 5, where A005117(31) = 47 is prime for k = 31.
97 is in this sequence because A005117(61+1) - A005117(61-1) = 101 - 95 = 6, where A005117(61) = 97 is prime for k = 61.
241 is in this sequence because A005117(150+1) - A005117(150-1) = 246 - 239 = 7, where A005117(150) = 241 is prime for k = 150.
5051 is in this sequence because A005117(3071+1) - A005117(3071-1) = 5053 - 5045 = 8, where A005117(3071) = 5051 is prime for k = 3071.

Cf. A000040, A005117 (squarefree numbers), A067535, A070321.

s = Select[Range[10^6], SquareFreeQ]; Table[k = 1; While[Nand[PrimeQ@ Set[p, s[[k]]], s[[k + 1]] - s[[k - 1]] == n], k++]; p, {n, 2, 10}] (* Michael De Vlieger, Mar 18 2017 *)

a(10) from Michael De Vlieger, Mar 18 2017

a(11)-a(15) from Giovanni Resta, Mar 22 2017

cp's OEIS Frontend

A378457 Difference between n and the greatest prime power <= n, allowing 1.

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A285719 a(1) = 1, and for n > 1, a(n) = the largest squarefree number k such that n-k is also squarefree.

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A378366 Difference between n and the greatest non prime power <= n (allowing 1).

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A081218 Greatest squarefree number not exceeding n-th prime power which is not prime.

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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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A378620 Lesser prime index of twin primes with nonsquarefree mean.

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A384420 The number of exponentially squarefree prime powers (not including 1) that divide n.

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