A187489 -id:A187489 - cp's OEIS frontend

A161664 a(n) = Sum_{i=1..n} (i - d(i)), where d(n) is the number of divisors of n (A000005).

0, 0, 1, 2, 5, 7, 12, 16, 22, 28, 37, 43, 54, 64, 75, 86, 101, 113, 130, 144, 161, 179, 200, 216, 238, 260, 283, 305, 332, 354, 383, 409, 438, 468, 499, 526, 561, 595, 630, 662, 701, 735, 776, 814, 853, 895, 940, 978, 1024, 1068, 1115, 1161, 1212, 1258, 1309

Offset: 1

Enoch Haga, Jun 15 2009

Partial Sums of A049820. - Omar E. Pol, Jun 18 2009
The original definition was: Safe periods for the emergence of cicada species on prime number cycles.
See Table 9 in reference, page 75, which together with the chart on page 73 (see link) provide a mathematical basis for the emergence of cicada species on prime number cycles.
Also the number of 2-element nondividing subsets of {1, ..., n}.  The a(6)=7 subsets of {1,2,3,4,5,6} with two elements where no element divides the other are: {2,3}, {2,5}, {3,4}, {3,5}, {4,5}, {4,6}, {5,6}. - Alois P. Heinz, Mar 08 2011
Sum of the number of proper nondivisors of all positive integers <= n. - Omar E. Pol, Feb 13 2014

			a(8) in A000217 minus a(8) in A006218 = a(7) above (28-16=12).
Referring to the chart referenced, when n-th year = 7 there are 16 x-markers.
These represent unsafe periods for cicada emergence: 28-16=12 safe periods.
The percent of safe periods for the entire 7 years is 12/28=~42.86%; for year 7 alone the calculation is 5/7 = 71.43%, a relatively good time to emerge.

Enoch Haga, Eratosthenes goes bugs! Exploring Prime Numbers, 2007, pp 71-80; first publication 1994.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Baker, Are there Genuine Mathematical Explanations of Physical Phenomena?, Mind 114 (454) (2005) 223-238.
Enoch Haga, Prime Safe Periods
G. F. Webb, The prime number periodical Cicada problem, Discr. Cont. Dyn. Syst. 1 (3) (2001) 387.
Wildforests, Cicada, visited Dec. 2012. - From _N. J. A. Sloane_, Dec 25 2012

Cf. A000005, A000217, A049820, A006218, A051014.

Column 2 of triangle A187489 or of A355145.

with(numtheory): A161664:=n->add(i-tau(i), i=1..n): seq(A161664(n), n=1..100); # Wesley Ivan Hurt, Jul 15 2014
# second Maple program:
a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+n-numtheory[tau](n))
    end:
seq(a(n), n=1..55);  # Alois P. Heinz, Jun 24 2022

a[n_] := n*(n+1)/2 - Sum[ DivisorSigma[0, k], {k, n}]; Table[a[n], {n, 55}] (* Jean-François Alcover, Nov 07 2011 *)

from math import isqrt
def A161664(n): return (lambda m: n*(n+1)//2+m*m-2*sum(n//k for k in range(1, m+1)))(isqrt(n)) # Chai Wah Wu, Oct 08 2021

a(n) = A000217(n) - A006218(n).

For n>1: a(n) = Sum_{h=1..n} Sum_{m=1..1 + 2*floor(n/2 - 1/2)} Sum_{k=1 + floor(h/(m + 1))..floor(h/m - 1/m)} 1 (from Granvik at A368592). - Bill McEachen, Apr 01 2025

Simplified definition, offset corrected and partially edited by Omar E. Pol, Jun 18 2009

New name from Wesley Ivan Hurt, Jul 15 2014

A068063 Maximum cardinality of a nondividing subset of {1, 2, ..., n}.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8

Offset: 0

David Wasserman, Feb 15 2002

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

			a(65) = 8 because 8 is the maximal cardinality of a nondividing subset of {1, 2, ..., 65}.  Two different subsets have cardinality 8:
{36,40,48,49,53,61,64,65}, {30,44,45,49,50,59,64,65}.

Eric Weisstein's World of Mathematics, Nondividing Set.

Cf. A051014, A187489.

a(41)-a(65) from Alois P. Heinz, Mar 10 2011

A051014 Number of nondividing sets on {1,2,...,n}.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 14, 21, 27, 38, 52, 73, 90, 123, 159, 211, 263, 344, 413, 535, 658, 832, 1026, 1276, 1499, 1846, 2226, 2708, 3229, 3912, 4592, 5541, 6495, 7795, 9207, 10908, 12547, 14852, 17358, 20493, 23709, 27744, 31921, 37250, 43013, 49936, 57319, 66318

Offset: 0

Eric W. Weisstein

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

			a(5) = 11 because there are 11 nondividing subsets of {1,2,3,4,5}: {}, {1}, {2}, {3}, {4}, {5}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}.
a(7) = 21: {}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {2,3}, {2,5}, {2,7}, {3,4}, {3,5}, {3,7}, {4,5}, {4,6}, {4,7}, {5,6}, {5,7}, {6,7}, {4,6,7}.

Eric Weisstein's World of Mathematics, Nondividing Set

Row sums of A187489. Cf. A068063.

sums:= proc(s) option remember; local i, m;
          m:= max(s[]);
         `if`(m<1, {}, {m, seq([i,i+m][], i=sums(s minus {m}))})
       end:
b:= proc(i,s) option remember; local j, ok, t, si;
      if i<2 then 1
    else si:= s union {i};
         ok:= true;
         for j in sums(si) while ok do
           for t in si while ok do
             if irem(j, t)=0 and t<>j then ok:= false fi
           od
         od;
         b(i-1,s) +`if`(ok, b(i-1, si), 0)
      fi
    end:
a:= n-> `if`(n=0, 1, 1+b(n, {})):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 08 2011

sums[s_] := sums[s] = Module[{m=Max[s]},
If[m<1, {},
  Join[{m},
  Sequence@@Table[{i,i+m}, {i,sums[DeleteCases[s,m]]}]]]
];
b[i_,s_] := b[i,s] = Module[{ ok,si,sij,sik},
If[ i<2, 1, si = Union[s,{i}];
ok = True;
For[j=1, j <= Length[sums[si]] && ok, j++,
  sij = sums[si][[j]];
  For[k=1, k <= Length[si] && ok, k++,
    If[Divisible[sij,sik=si[[k]]]&&sij!=sik,ok=False]]];
    b[i-1, s] + If[ok, b[i-1,si],0]
  ]
];
a[n_] := a[n] = If[n==0, 1, 1+b[n, {}]];
Table[ Print[ a[n] ]; a[n], {n,0,47}]
(* Jean-François Alcover, Oct 10 2012, after Alois P. Heinz *)

More terms from David Wasserman, Feb 15 2002

a(41)-a(47) from Alois P. Heinz, Mar 08 2011

A187490 Number of 3-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 2, 6, 12, 22, 31, 49, 70, 99, 128, 176, 216, 284, 343, 423, 515, 633, 722, 860, 1007, 1173, 1333, 1552, 1729, 1989, 2223, 2502, 2809, 3138, 3416, 3819, 4226, 4658, 5049, 5570, 6016, 6601, 7146, 7719, 8371, 9100, 9686, 10461, 11208, 12039

Offset: 7

Alois P. Heinz, Mar 10 2011

nonn

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

			a(7) = 1 because there is one 3-element nondividing subset of {1,2,3,4,5,6,7}: {4,6,7}.
a(9) = 6: {4,6,7}, {4,6,9}, {5,6,8}, {5,8,9}, {6,7,9}, {6,8,9}.

Alois P. Heinz, Table of n, a(n) for n = 7..200
Eric Weisstein's World of Mathematics, Nondividing Set.

Column 3 of triangle A187489. Cf. A068063.

A187491 Number of 4-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 2, 3, 6, 10, 21, 32, 49, 65, 101, 150, 224, 305, 413, 525, 707, 908, 1174, 1479, 1871, 2269, 2826, 3396, 4138, 4967, 5991, 6917, 8244, 9673, 11328, 12958, 15091, 17112, 19771, 22468, 25485, 28870, 32861, 36298, 40969, 45615, 51015

Offset: 10

Alois P. Heinz, Mar 10 2011

nonn

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

			a(10) = 1 because there is one 4-element nondividing subset of {1,2,...,10}: {6,7,9,10}.

Eric Weisstein's World of Mathematics, Nondividing Set

Column 4 of triangle A187489. Cf. A068063.

A187492 Number of 5-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

2, 4, 6, 11, 15, 24, 50, 83, 127, 209, 310, 431, 679, 921, 1229, 1624, 2145, 2770, 3752, 4866, 6141, 7753, 9679, 12005, 15027, 18134, 22045, 26368, 31712, 37763, 45569, 53810, 63393, 73560, 86496, 100071, 117234, 134623, 155465, 176876

Offset: 21

Alois P. Heinz, Mar 10 2011

nonn

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

			a(21) = 2 because there are two 5-element nondividing subsets of {1,2,...,21}: {12,16,18,19,21}, {12,14,18,20,21}.

Eric Weisstein's World of Mathematics, Nondividing Set

Column 5 of triangle A187489. Cf. A068063.

A187493 Number of 6-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 3, 4, 7, 15, 27, 45, 55, 85, 133, 199, 262, 378, 534, 803, 999, 1319, 1742, 2309, 3007, 4020, 5166, 6565, 7950, 10380, 12882, 16533, 19664, 24099, 30912, 37550, 44092, 54465, 65117, 79616, 94144, 111780, 132592, 159228, 187506, 219949, 256514

Offset: 31

Alois P. Heinz, Mar 10 2011

nonn

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

			a(31) = 1 because there is one 6-element nondividing subset of {1,2,...,31}: {16,19,23,24,28,31}.

Eric Weisstein's World of Mathematics, Nondividing Set

Column 6 of triangle A187489. Cf. A068063.

A187494 Number of 7-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 9, 12, 21, 27, 41, 46, 74, 99, 137, 153, 203, 307, 414, 464, 612, 788, 1126, 1292, 1645, 2039, 2614, 3291, 4120, 5127, 6356, 7180, 9786

Offset: 43

Alois P. Heinz, Mar 10 2011

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

			a(43) = 1 because there is one 7-element nondividing subset of {1,2,...,43}: {24,28,31,35,36,40,43}.

Eric Weisstein's World of Mathematics, Nondividing Set

Column 7 of triangle A187489. Cf. A068063.

A187550 Number of 8-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 12, 15, 19, 24, 35, 46, 60, 83, 102, 135

Offset: 65

Alois P. Heinz, Mar 11 2011

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

			a(65) = 2 because there are two 8-element nondividing subsets of {1,2,...,65}: {36,40,48,49,53,61,64,65}, {30,44,45,49,50,59,64,65}.

Eric Weisstein's World of Mathematics, Nondividing Set

Column 8 of triangle A187489. Cf. A068063.

cp's OEIS Frontend

A161664 a(n) = Sum_{i=1..n} (i - d(i)), where d(n) is the number of divisors of n (A000005).

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A068063 Maximum cardinality of a nondividing subset of {1, 2, ..., n}.

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A051014 Number of nondividing sets on {1,2,...,n}.

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A187490 Number of 3-element nondividing subsets of {1, 2, ..., n}.

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A187491 Number of 4-element nondividing subsets of {1, 2, ..., n}.

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A187492 Number of 5-element nondividing subsets of {1, 2, ..., n}.

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A187493 Number of 6-element nondividing subsets of {1, 2, ..., n}.

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A187494 Number of 7-element nondividing subsets of {1, 2, ..., n}.

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A187550 Number of 8-element nondividing subsets of {1, 2, ..., n}.

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