A080671 -id:A080671 - cp's OEIS frontend

A007775 Numbers not divisible by 2, 3 or 5.

1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209

Offset: 1

N. J. A. Sloane

Also numbers n such that the sum of the 4th powers of the first n positive integers is divisible by n, or A000538(n) = n*(n+1)(2*n+1)(3*n^2+3*n-1)/30 is divisible by n. - Alexander Adamchuk, Jan 04 2007
Also the 7-rough numbers: positive integers that have no prime factors less than 7. - Michael B. Porter, Oct 09 2009
a(n) mod 3 has period 8, repeating [1,1,2,1,2,1,2,2] = (n mod 2) + floor(((n-1) mod 8)/7) - floor(((n-2) mod 8)/7) + 1. floor(a(n)/3) is the set of numbers k such that k is congruent to {0,2,3,4,5,6,7,9} mod 10 = floor((5*n-2)/4)-floor((n mod 8)/6). - Gary Detlefs, Jan 08 2012
Numbers k such that C(k+3,3)==1 (mod k) and C(k+5,5)==1 (mod k). - Gary Detlefs, Sep 15 2013
a(n) mod 30 has period 8 repeating [1, 7, 11, 13, 17, 19, 23, 29]. The mean of these 8 numbers is 120/8 = 15. (a(n)-15) mod 30 has period 8 repeating [-14, -8, -4, -2, 2, 4, 8, 14]. One half of the absolute value produces the symmetric sequence [7, 4, 2, 1, 1, 2, 4, 7] = A061501(((n-1) + 16) mod 8). - Gary Detlefs, Sep 24 2013
a(n) are exactly those positive integers m such that the sequence b(n) = n*(n + m)*(n + 2*m)*(n + 3*m)(n + 4*m)/120 is integral. Cf. A007310. - Peter Bala, Nov 13 2015
The asymptotic density of this sequence is 4/15. - Amiram Eldar, Sep 30 2020
If a(n) + a(n+1) = 0 (mod 30), then a(n-j) + a(n+j+1) = a(n) + a(n+1) for each j in [1, n-1]. - Alexandre Herrera, Jun 27 2023

N. J. A. Sloane, Table of n, a(n) for n = 1..8000
Peter Bala, A note on A007775.
Abel Jansma, E_8 Symmetry Structures in the Ising model, Master's thesis, University of Amsterdam, 2018.
Eric Weisstein's World of Mathematics, Rough Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).
Index entries for sequences related to smooth numbers

Cf. A000538, A054403, A145011 (first differences).
For k-rough numbers with other values of k, see A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063.
Complement is A080671.
For digital root of Fibonacci numbers indexed by this sequence, see A227896.

a007775 n = a007775_list !! (n-1)
a007775_list = 1 : filter ((> 5) . a020639) [1..]
-- Reinhard Zumkeller, Jan 06 2013

I:=[1, 7, 11, 13, 17, 19, 23, 29, 31]; [n le 9 select I[n] else Self(n-1) +Self(n-8) - Self(n-9): n in [1..80]]; // G. C. Greubel, Oct 22 2018

for i from 1 to 500 do if gcd(i,30) = 1 then print(i); fi; od;
for k from 1 to 300 do if ((k^2 mod 48=1) or (k^2 mod 48=25)) and ((k^2 mod 120=1) or (k^2 mod 120=49)) then print(k) fi od. # Gary Detlefs, Dec 30 2011

Select[ Range[ 300 ], GCD[ #1, 30 ]==1& ]
Select[Range[250], Mod[#, 2]>0&&Mod[#, 3]>0&&Mod[#, 5]>0&] (* Vincenzo Librandi, Feb 08 2014 *)
a[ n_] := Quotient[ n, 8, 1] 30 + {1, 7, 11, 13, 17, 19, 23, 29}[[Mod[n, 8, 1]]]; (* Michael Somos, Jun 02 2014 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 7, 11, 13, 17, 19, 23, 29, 31}, 100] (* Mikk Heidemaa, Dec 07 2017 *)
Cases[Range@1000, x_ /; NoneTrue[Array[Prime, 3], Divisible[x, #] &]] (* Mikk Heidemaa, Dec 07 2017 *)
CoefficientList[ Series[(x^8 + 6x^7 + 4x^6 + 2x^5 + 4x^4 + 2x^3 + 4x^2 + 6x + 1)/((x - 1)^2 (x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)), {x, 0, 55}], x] (* Robert G. Wilson v, Dec 07 2017 *)

isA007775(n) = gcd(n,30)==1 \\ Michael B. Porter, Oct 09 2009

{a(n) = n\8 * 30 + [ -1, 1, 7, 11, 13, 17, 19, 23][n%8 + 1]} /* Michael Somos, Feb 05 2011 */

{a(n) = n\8 * 6 + 9 + 3 * (n+1)\2 * 2 - max(5, (n-2)%8) * 2} /* Michael Somos, Jun 02 2014 */

Vec(x*(1+6*x+4*x^2+2*x^3+4*x^4+2*x^5+4*x^6+6*x^7+x^8)/((1+x)*(x^2+1)*(x^4+1)*( x-1)^2) + O(x^100)) \\ Altug Alkan, Nov 16 2015

def A007775(n): return ((m:=n-1)<<2|1)-(m>>2&-2)+(2,0,-2,0)[m-1>>1&3] # Chai Wah Wu, Feb 02 2025

a = lambda n: ((((n-1)<< 2)-((n-1)>>2))|1) + ((((n-1)<<1)-((n-1)>> 1)) & 2)
print([a(n) for n in (1..56)]) # after Andrew Lelechenko, Peter Luschny, Jul 08 2017

A141256(a(n)) = n+1. - Reinhard Zumkeller, Jun 17 2008
From R. J. Mathar, Feb 27 2009: (Start)
a(n+8) = a(n) + 30.
a(n) = a(n-1) + a(n-8) - a(n-9).
G.f.: x*(1 + 6*x + 4*x^2 + 2*x^3 + 4*x^4 + 2*x^5 + 4*x^6 + 6*x^7 + x^8)/((1 + x)*(x^2 + 1)*(x^4 + 1)*(x-1)^2). (End)
a(n) = 4*n - 3 - 2*floor((n-1)/8) + (1 + (-1)^floor((n-2)/2))*(-1)^floor((n-2)/4), n >= 1. - Timothy Hopper, Mar 14 2010
a(1 - n) = -a(n). - Michael Somos, Feb 05 2011
Numbers k such that ((k^2 mod 48=1) or (k^2 mod 48=25)) and ((k^2 mod 120=1) or (k^2 mod 120=49)). - Gary Detlefs, Dec 30 2011
Numbers k such that k^2 mod 30 is 1 or 19. - Gary Detlefs, Dec 31 2011
a(n) = 3*(floor((5*n-2)/4) - floor((n mod 8)/6)) + (n mod 2) + floor(((n-1) mod 8)/7) - floor(((n-2) mod 8)/7) + 1. - Gary Detlefs, Jan 08 2012
a(n) = 4*n - 3 + 2*(floor((n+6)/8) - floor((n+4)/8) - floor((n+2)/8) + floor(n/8) - floor((n-1)/8)), n >= 1. From the o.g.f. given above by R. J. Mathar (with the denominator written as (1-x^8)*(1-x)), and a two-step reduction of the floor functions. Compare with Hopper's and Detlefs's formulas above. - Wolfdieter Lang, Jan 26 2012
a(n) = (6*f(n) - 3 + (-1)^f(n))/2, where f(n)= n + floor(n/4)+ floor(((n+4) mod 8)/6). - Gary Detlefs, Sep 15 2013
a(n) = 30*floor((n-1)/8) + 15 + 2*f((n-1) mod 8 + 16)*(-1)^floor(((n+3) mod 8)/4), where f(n) = (n*(n+1)/2+1) mod 10. - Gary Detlefs, Sep 24 2013
a(n) = 3*n + 6*floor(n/8) + (n mod 2) - 2*floor(((n-2) mod 8)/6) - 2*floor(((n-2) mod 8)/7) + 1. - Gary Detlefs, Jun 01 2014
a(n+1) = ((n << 2 - n >> 2) || 1) + ((n << 1 - n >> 1) && 2), where << and >> are bitwise left and right shifts, || and && are bitwise "or" and "and". - Andrew Lelechenko, Jul 08 2017
a(n) = 2*n + 2*floor(1/2 + (7*n)/8) + 2*(91 mod (2 - ((3*n)/4 + n^2/4) mod 2)) - 3 (n > 0). - Mikk Heidemaa, Dec 06 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(23 + sqrt(5) - sqrt(6*(5 + sqrt(5))))*Pi/15. - Amiram Eldar, Dec 13 2021

A354591 Numbers k that can be written as the sum of 4 divisors of k (not necessarily distinct).

Original entry on oeis.org

4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 170, 172

Offset: 1

Wesley Ivan Hurt, Aug 18 2022

nonn

All terms are even. - Robert Israel, Aug 31 2022
Is it true that a(n) = 2*A080671(n)? - Michel Marcus, Sep 01 2022 (True for n <= 10000. - N. J. A. Sloane, Sep 01 2022)
This is true.  In other words, k is in the sequence if and only if k is even and divisible by 3, 4 or 5.  Proof: the positive integer solutions of 1/a + 1/b + 1/c + 1/d = 1 can be enumerated explicitly, and each contains at least one even number and at least one divisible by 3, 4 or 5.  Of course k = k/a + k/b + k/c + k/d if and only if 1 = 1/a + 1/b + 1/c + 1/d, and this writes k as the sum of 4 divisors of k if k is divisible by a,b,c, and d.  If k is even and divisible by 3, we can use 1 = 1/3 + 1/3 + 1/6 + 1/6; if divisible by 4, 1 = 1/4 + 1/4 + 1/4 + 1/4; if even and divisible by 5, 1 = 1/2 + 1/5 + 1/5 + 1/10. - Robert Israel, Sep 01 2022
The asymptotic density of this sequence is 11/30. - Amiram Eldar, Aug 08 2023

			20 is in the sequence since 20 = 10+5+4+1 = 5+5+5+5 where each summand divides 20.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1).

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), this sequence (j=4), A355641 (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).

Cf. A080671.

F:= proc(x,S,j) option remember;
      local s,k;
      if j = 0  then return(x = 0) fi;
      if S = [] or x > j*S[-1] then return false fi;
      s:= S[-1];
      for k from 0 to min(j,floor(x/s)) do
        if procname(x-k*s, S[1..-2],j-k) then return true fi
      od;
      false
end proc:
select(t -> F(t, sort(convert(numtheory:-divisors(t),list)),4), [$1..200]); # Robert Israel, Aug 31 2022

q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[200], q[#, 4] &] (* Amiram Eldar, Aug 19 2022 *)
CoefficientList[Series[2 (2 - x + 2*x^2 - x^3 + 2*x^4 + x^6 + 2*x^8 + x^10 + 2*x^12 + x^14 + 2*x^16 - x^17 + 2*x^18 - x^19 + 2*x^20)/((x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(1 + x + x^5 + x^6 + x^7 + x^8 + x^9 + x^2 + x^4 + x^3 + x^10)*(x - 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jul 17 2025 *)

isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [4,4]); \\ Michel Marcus, Aug 19 2022

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - 2*a(n-8) + 2*a(n-9) - 2*a(n-10) + 2*a(n-11) - 2*a(n-12) + 2*a(n-13) - 2*a(n-14) + 2*a(n-15) - 2*a(n-16) + 2*a(n-17) - 2*a(n-18) + 2*a(n-19) - 2*a(n-20) + 2*a(n-21) - a(n-22). - Wesley Ivan Hurt, Jun 29 2024

G.f.: 2*x*(2 - x + 2*x^2 - x^3 + 2*x^4 + x^6 + 2*x^8 + x^10 + 2*x^12 + x^14 + 2*x^16 - x^17 + 2*x^18 - x^19 + 2*x^20)/((x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(1 + x + x^5 + x^6 + x^7 + x^8 + x^9 + x^2 + x^4 + x^3 + x^10)*(x - 1)^2). - Wesley Ivan Hurt, Jul 17 2025

A366981 Number of divisors of n in the set {3,4,5}.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Oct 30 2023

If n is even and a(n) > 0, then n can be written as the sum of 4 divisors of n (not necessarily distinct). For example, 6 = 1+2+1+2 and 12 = 3+3+3+3 but 14 cannot be written as the sum of 4 of its divisors since 14 is even, but a(14) = 0. [See discussion in A354591].

			a(12) = 2 since exactly two of its divisors are members of the set {3,4,5}.

Cf. A080671, A354591.

Table[Sum[Sum[KroneckerDelta[k, j], {j, 3, 5}] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
a[n_] := Total[Sign[IntegerExponent[n, {3, 4, 5}]]]; Array[a, 100] (* Amiram Eldar, Nov 12 2023 *)

a(n) = sumdiv(n, d, (d>=3) && (d<=5)); \\ Michel Marcus, Nov 11 2023

a(n) = (valuation(n,3)>0) + (valuation(n,2)>1) + (valuation(n,5)>0); \\ Amiram Eldar, Nov 12 2023

def A366981(n): return sum(int(not n%d) for d in range(3,6)) # Chai Wah Wu, Nov 12 2023

a(n) = Sum_{d|n, d = 3, 4, or 5} 1.
a(n) = Sum_{d|n} ([d = 3] + [d = 4] + [d = 5]), where [ ] is the Iverson bracket.
a(n) = Sum_{d|n} Sum_{k=3..5} [d = k], where [ ] is the Iverson bracket.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 47/60. - Amiram Eldar, Nov 12 2023

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A007775 Numbers not divisible by 2, 3 or 5.

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A354591 Numbers k that can be written as the sum of 4 divisors of k (not necessarily distinct).

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A366981 Number of divisors of n in the set {3,4,5}.

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A335774 Numbers k such that in prime factorization of k the second smallest factor is 7.

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