A218446 -id:A218446 - cp's OEIS frontend

A218444 a(n) = Sum_{k>=0} floor(n/(5*k + 1)).

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18, 20, 21, 23, 24, 25, 27, 29, 30, 32, 33, 35, 36, 37, 38, 40, 42, 44, 46, 47, 48, 51, 52, 53, 54, 55, 57, 60, 61, 63, 64, 66, 67, 70, 71, 72, 74, 76, 77, 79, 81, 83, 84, 85, 86, 88, 90, 92, 94, 96, 97, 101, 102, 103, 104, 105

Offset: 0

Benoit Cloitre, Oct 28 2012

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Partial sums of A001876.
Cf. A218444, A218445, A218446.
Cf. A001620, A256779.

a[n_] := Sum[ Floor[n/(5*k+1)], {k, 0, Ceiling[n/5]}]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 22 2013 *)

A218444[n]:=sum(floor(n/(5*k+1)),k,0,n)$
makelist(A218444[n],n,0,80); /* Martin Ettl, Oct 29 2012 */

PARI
```
a(n)=sum(k=0,n,(n\(5*k+1)))
```

a(n) = Sum_{k>=0} floor(n/(5*k + 1)).

a(n) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,5) - (1 - gamma)/5 = A256779 - (1 - A001620)/5 = 0.651363... (Smith and Subbarao, 1981). - Amiram Eldar, Apr 20 2025

A218445 a(n) = Sum_{k>=0} floor(n/(5*k + 2)).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 8, 8, 10, 10, 11, 12, 13, 13, 14, 15, 17, 17, 19, 19, 20, 21, 23, 23, 24, 24, 26, 26, 28, 29, 31, 32, 33, 33, 34, 34, 37, 37, 39, 39, 40, 41, 43, 44, 45, 46, 48, 48, 50, 50, 52, 53, 54, 54, 56, 56, 58, 59, 61, 61, 63, 64, 66, 66, 68, 68, 71, 71, 73, 73, 74, 76, 77, 77, 78

Offset: 0

Benoit Cloitre, Oct 28 2012

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Partial sums of A001877.
Cf. A218444, A218446, A218447.
Cf. A001620, A256780.

Table[Sum[Floor[n/(5k+2)],{k,0,n}],{n,0,80}] (* Harvey P. Dale, Dec 08 2022 *)

A218445[n]:=sum(floor(n/(5*k+2)),k,0,n)$
makelist(A218445[n],n,0,80); /* Martin Ettl, Oct 29 2012 */

PARI
```
a(n)=sum(k=0,n\5,(n\(5*k+2)))
```

a(n) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,5) - (1 - gamma)/5 = A256780 - (1 - A001620)/5 = 0.105832... (Smith and Subbarao, 1981). - Amiram Eldar, Apr 20 2025

A218447 a(n) = Sum_{k>=0} floor(n/(5*k + 4)).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 9, 9, 11, 11, 11, 12, 14, 15, 15, 15, 16, 16, 17, 17, 19, 19, 20, 21, 22, 22, 23, 23, 25, 26, 26, 26, 28, 29, 29, 29, 30, 30, 32, 32, 34, 35, 36, 37, 38, 38, 38, 39, 41, 41, 41, 41, 43, 44, 45, 45, 48, 48, 49, 49, 51, 51, 52, 53, 54

Offset: 0

Benoit Cloitre, Oct 28 2012

Robert Israel, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Partial sums of A001899.
Cf. A218444, A218445, A218446.
Cf. A001620, A256849.

g:= n -> nops(select(t -> t mod 5 = 4, numtheory:-divisors(n))):
g(0):= 0:
ListTools:-PartialSums(map(g, [$0..100])); # Robert Israel, Apr 29 2021

A218447[n]:=sum(floor(n/(5*k+4)),k,0,n)$
makelist(A218447[n],n,0,80); /* Martin Ettl, Oct 20 2012 */

PARI
```
a(n)=sum(k=0,n,(n\(5*k+4)))
```

a(n) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(4,5) - (1 - gamma)/5 = A256849 - (1 - A001620)/5 = -0.213442... (Smith and Subbarao, 1981). - Amiram Eldar, Apr 20 2025

A218466 Least k > n for which phi(k - n) = phi(k + n) or 0 if no such k exists.

Original entry on oeis.org

5, 10, 27, 17, 25, 54, 23, 34, 61, 47, 55, 108, 47, 46, 139, 68, 58, 122, 71, 85, 144, 95, 115, 207, 101, 94, 183, 92, 145, 278, 104, 136, 177, 116, 175, 244, 161, 142, 306, 149, 184, 283, 191, 187, 410, 230, 235, 267, 146, 202, 299, 188, 157, 366, 275, 184

Offset: 1

Irina Gerasimova, Mar 26 2013

nonn

Is there an upper bound for a(n) for a given n? - Michael B. Porter, Apr 06 2013

			a(3) = 27 since phi(27 - 3) = phi(24) = 8 and phi(27 + 3) = phi(30) = 8, and 27 is the smallest number greater than 3 for which the two are equal.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000010, A217768, A223091.

/* will not terminate if k does not exist */
a218446(n) = {local(k); k = n + 1; while(eulerphi(k - n) <> eulerphi(k + n), k = k + 1); k} \\ Michael B. Porter, Mar 30 2013

Extended by R. J. Mathar, Mar 27 2013

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A218444 a(n) = Sum_{k>=0} floor(n/(5*k + 1)).

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A218445 a(n) = Sum_{k>=0} floor(n/(5*k + 2)).

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A218447 a(n) = Sum_{k>=0} floor(n/(5*k + 4)).

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A218466 Least k > n for which phi(k - n) = phi(k + n) or 0 if no such k exists.

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