A126590 -id:A126590 - cp's OEIS frontend

A126592 Sum of numbers less than or equal to n which are multiples of 3 or 5.

0, 0, 3, 3, 8, 14, 14, 14, 23, 33, 33, 45, 45, 45, 60, 60, 60, 78, 78, 98, 119, 119, 119, 143, 168, 168, 195, 195, 195, 225, 225, 225, 258, 258, 293, 329, 329, 329, 368, 408, 408, 450, 450, 450, 495, 495, 495, 543, 543, 593, 644, 644, 644, 698, 753, 753, 810, 810

Offset: 1

Zak Seidov, Mar 13 2007

Sum of numbers m <= n such that (m mod 3) * (m mod 5) = 0.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Project Euler, Problem 1: Multiples of 3 and 5
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1).

Cf. A126073, A126590.

[(3*Floor(n/3)*(1 + Floor(n/3)) + 5*Floor(n/5)*(1 + Floor(n/5)) - 15*Floor(n/15)*(1 + Floor(n/15)))/2: n in [1..30]]; // G. C. Greubel, Mar 06 2018

an[n_, d_] := d * Floor[n/d]; sn[n_, d_] := (an[n, d] * (an[n, d] + d))/(2 * d); Table[sn[n, 3] + sn[n, 5] - sn[n, 15], {n, 1000}]
Accumulate[Table[If[Divisible[n, 3] || Divisible[n, 5], n, 0], {n, 60}]] (* Harvey P. Dale, Jun 09 2016 *)
Accumulate[Table[n Boole[GCD[n, 15] > 1], {n, 50}]] (* Alonso del Arte, Dec 23 2018 *)

{b(n,x)=floor(n/x)*(1 + floor(n/x))};
for(n=1,30, print1((3*b(n,3) + 5*b(n,5) - 15*b(n,15))/2, ", ")) \\ G. C. Greubel, Mar 06 2018

(for (n <- 2 to 50) yield if ((n % 3) * (n % 5) == 0) { n } else { 0 }).scanLeft(0)( + ) // Alonso del Arte, Dec 23 2018

an(n, d) = d * floor(n/d), sn(n, d) = (an(n, d) * (an(n, d) + d))/(2*d), a(n) = sn(n, 3) + sn(n, 5) - sn(n, 15).

A126073 Sum of numbers <= n which are multiples of 3 or 5 but not 15.

Original entry on oeis.org

0, 0, 3, 3, 8, 14, 14, 14, 23, 33, 33, 45, 45, 45, 45, 45, 45, 63, 63, 83, 104, 104, 104, 128, 153, 153, 180, 180, 180, 180, 180, 180, 213, 213, 248, 284, 284, 284, 323, 363, 363, 405, 405, 405, 405, 405, 405, 453, 453, 503, 554, 554, 554, 608, 663, 663, 720, 720

Offset: 1

Zak Seidov, Mar 13 2007

nonn

Sum of numbers m<=n such that mod(m,3)*mod(m,5)=0 and mod(m,15)>0.
First differences (fd) are
0,3,0,5,6,0,0,9,10,0,12,0,0,0,0,
0,18,0,20,21,0,0,24,25,0,27,0,0,0,0,
0,33,0,35,36,0,0,39,40,0,42,0,0,0,0,...
fd(1..15)={0,3,0,5,6,0,0,9,10,0,12,0,0,0,0}; for n>15
fd(n)=fd(n-15)+15 if fd(n-15)>0, fd(n)=0 otherwise.

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

Cf. A126590, A126592.

an[n_,d_]:=d*Floor[n/d];sn[n_,d_]:=(an[n,d]*(an[n,d] + d))/(2*d); Table[sn[n,3]+sn[n,5]-2*sn[n,15],{n,1000}]
Accumulate[Table[If[Mod[n,3]Mod[n,5]==0&&Mod[n,15]>0,n,0],{n,60}]] (* Harvey P. Dale, Jul 19 2025 *)

an[n,d]=d*Floor[n/d];sn[n,d]=(an[n,d]*(an[n,d] + d))/(2*d); a(n)=sn[n,3]+sn[n,5]-2*sn[n,15].

A335650 Numbers that are multiples of 2,3,5, or 7 but not multiples of the product of any combination of 2,3,5, and 7.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 16, 22, 25, 26, 27, 32, 33, 34, 38, 39, 44, 46, 49, 51, 52, 55, 57, 58, 62, 64, 65, 68, 69, 74, 76, 77, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 104, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 125, 128, 129, 133, 134, 136, 141, 142

Offset: 1

Peter Andrew, Jun 15 2020

nonn

			4 is a term because 4 = 2 * 2;
77 is a term because 77 = 7 * 11;
6 is not a term because 6 = 2 * 3;
21 is not a term because 21 = 3 * 7;
30 is not a term because 30 = 2 * 3 * 5;
210 is not a term because 210 = 2 * 3 * 5 * 7.

Cf. A126590.

a335650 = [x | x <- [0..], (gcd x 210) `elem` [2,3,5,7]]

q:= n-> is(igcd(n, 210) in {2,3,5,7}):
select(q, [$0..200])[];  # Alois P. Heinz, Jun 16 2020

Select[Range[150], Count[IntegerExponent[#, {2, 3, 5, 7}], 0] == 3 &] (* Amiram Eldar, Jun 16 2020 *)

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A126592 Sum of numbers less than or equal to n which are multiples of 3 or 5.

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A126073 Sum of numbers <= n which are multiples of 3 or 5 but not 15.

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A335650 Numbers that are multiples of 2,3,5, or 7 but not multiples of the product of any combination of 2,3,5, and 7.

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