A340179 -id:A340179 - cp's OEIS frontend

A341061 Numbers k such that A340179(k) is prime.

5, 28, 44, 51, 58, 61, 63, 90, 93, 108, 129, 136, 145, 148, 186, 208, 234, 235, 241, 247, 262, 272, 277, 278, 300, 306, 310, 314, 316, 321, 329, 335, 379, 384, 386, 414, 428, 446, 448, 449, 475, 480, 492, 514, 535, 537, 546, 548, 572, 580, 599, 609, 611, 616, 618, 626, 660, 670, 673, 680, 683

Offset: 1

J. M. Bergot and Robert Israel, Feb 04 2021

nonn

			a(3) = 44 is a term because A340179(44) = 211 is prime.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A340179, A341059, A341060.

f:= proc(n) local C, s, c;
  C:=select(t -> igcd(t, n) = 1, [$1..n-1]);
  s:= convert(C, `+`);
  add(s mod c, c = C)
end proc:
select(t -> isprime(f(t)), [$1..1000]);

A341059 Numbers k such that A340179(k) is a square.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 12, 15, 17, 27, 48, 92, 6219, 24310

Offset: 1

J. M. Bergot and Robert Israel, Feb 04 2021

			a(6) = 8 is a term because A340179(8) = 4 = 2^2.

Cf. A340179, A341060, A341061.

f:= proc(n) local C, s, c;
  C:=select(t -> igcd(t, n) = 1, [$1..n-1]);
  s:= convert(C, `+`);
  add(s mod c, c = C)
end proc:
select(t -> issqr(f(t)), [$1..7000]);

A341060 Numbers k such that A340179(k) is a multiple of k.

Original entry on oeis.org

1, 2, 10, 16, 30, 74, 81, 97, 489, 525, 607, 1861, 4439, 26051

Offset: 1

J. M. Bergot and Robert Israel, Feb 04 2021

			a(4) = 16 is a term because A340179(16) = 32 = 2*16.

Cf. A340179, A341059, A341061.

f:= proc(n) local C, s, c;
  C:=select(t -> igcd(t, n) = 1, [$1..n-1]);
  s:= convert(C, `+`);
  add(s mod c, c = C)
end proc:
select(t -> f(t) mod t = 0, [$1..5000]);

A340179[n_] := Total@Mod[#2, #1]& @@ {#, Total@#}& @ Select[Range[n], GCD[#, n] == 1&];
Reap[For[k = 1, k <= 80000, k++, If[Divisible[A340179[k], k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, May 16 2023, after Michael De Vlieger in A340179 *)

A343883 Numbers k such that A023896(k) is a multiple of A340179(k).

Original entry on oeis.org

3, 4, 6, 8, 10, 16, 30, 54

Offset: 1

J. M. Bergot and Robert Israel, May 02 2021

Conjecture: there are only 8 terms.

			a(6) = 16 is a term because A023896(16) = 64 is a multiple of A340179(16) = 32.

Cf. A023896, A340179.

filter:= proc(n) local S,s,t;
  S:= select(t -> igcd(t,n)=1, [$1..n-1]);
  s:= nops(S)*n/2;
  s mod add(s mod t,t=S) = 0;
end proc:
select(filter, [$3..1000]);

A340180 a(n) = Sum_{x in C(n)} (sigma(n) mod x), where C(n) is the set of numbers < n coprime to n, and sigma = A000203.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 7, 1, 16, 4, 16, 9, 30, 23, 26, 24, 46, 19, 60, 30, 60, 52, 84, 43, 132, 77, 105, 62, 137, 51, 166, 88, 183, 139, 182, 117, 247, 186, 239, 158, 283, 99, 327, 194, 259, 284, 373, 176, 462, 234, 442, 294, 491, 235, 508, 294, 514, 430, 585, 259, 671, 519, 546, 408, 749, 323, 798

Offset: 1

J. M. Bergot and Robert Israel, Dec 30 2020

			For n=8, sigma(8) = 15 and C(8) = {1,3,5,7} so a(8) = (15 mod 1) + (15 mod 3) + (15 mod 5) + (15 mod 7) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A340179, A337189 (n | a(n)).

f := proc(n) local C,s,c;
  s:= numtheory:-sigma(n);
  C:=select(t -> igcd(t,n) = 1, [$1..n-1]);
  add(s mod c, c=C)
end proc:
map(f, [$1..100]);

Table[Sum[Mod[DivisorSigma[1, n], k] Floor[1/GCD[k, n]], {k, n - 1}], {n, 80}] (* Wesley Ivan Hurt, Jan 30 2021 *)

a(n) = my(s=sigma(n)); sum(k=1, n, if (gcd(k, n)==1, s % k)); \\ Michel Marcus, Jan 31 2021

A340976 Sum_{1 < k < n} sigma(n) mod k, where sigma = A000203.

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 7, 8, 18, 11, 16, 27, 30, 30, 40, 47, 46, 75, 60, 72, 101, 93, 84, 109, 146, 148, 167, 142, 137, 180, 166, 197, 254, 282, 283, 301, 247, 333, 367, 347, 283, 389, 327, 367, 475, 501, 373, 591, 517, 562, 621, 597, 491, 615, 699, 637, 810, 839, 585, 783, 671, 964, 1024

Offset: 1

M. F. Hasler, Feb 01 2021

nonn

Motivated by A340180 and several other sequences that use the sum over a subset of the indices.
Is there an efficient formula for a(n)? That might answer the following questions:
1) Is a(63) = a(2^6-1) = 1024 = 2^10 just a coincidence?
2) Are there are further terms of the form 2^k, i.e., a(n) in A000079? What can be said about these n?
3) Are there other fixed points a(n) = n as for n = 7, 8?
4) What is the frequency of odd vs. even terms? a(n) is odd for consecutive indices 21..22, 35..49, 51..56, 58..61, 64..65, 68..69, 73..79, ...: Are there patterns or simple subsequence(s) of such runs of length 2 or larger?

Cf. A000203, A340179, A340180.

Table[Sum[Mod[DivisorSigma[1,n],k],{k,2,n-1}],{n,1,138}] (* Metin Sariyar, Feb 02 2021 *)

apply( {A340976(n,s=sigma(n))=sum(k=1,n-1,s%k)}, [1..66]) \\ M. F. Hasler, Feb 01 2021

T(n) = n*(n+1)/2;
S(n) = my(s=sqrtint(n)); sum(k=1, s, T(n\k) + k*(n\k)) - s*T(s); \\ A024916
g(a,b) = my(s=0); while(a <= b, my(t=b\a); my(u=b\t); s += t*(T(u) - T(a-1)); a = u+1); s;
a(n) = (n-1)*sigma(n) - S(sigma(n)) + g(n, sigma(n)); \\ Daniel Suteu, Feb 02 2021

a(n) = (n-1)*sigma(n) - A024916(sigma(n)) + Sum_{k=n..sigma(n)} k*floor(sigma(n)/k). - Daniel Suteu, Feb 02 2021

cp's OEIS Frontend

A341061 Numbers k such that A340179(k) is prime.

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A341059 Numbers k such that A340179(k) is a square.

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A341060 Numbers k such that A340179(k) is a multiple of k.

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Mathematica

A343883 Numbers k such that A023896(k) is a multiple of A340179(k).

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A340180 a(n) = Sum_{x in C(n)} (sigma(n) mod x), where C(n) is the set of numbers < n coprime to n, and sigma = A000203.

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A340976 Sum_{1 < k < n} sigma(n) mod k, where sigma = A000203.

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