A187680 -id:A187680 - cp's OEIS frontend

A346956 Numbers k such that A000203(k) and A007955(k) are both divisible by A187680(k).

4, 9, 14, 16, 25, 38, 42, 49, 51, 55, 62, 64, 70, 81, 86, 92, 96, 117, 121, 130, 134, 138, 140, 158, 159, 161, 168, 169, 182, 206, 209, 234, 254, 256, 266, 267, 278, 282, 284, 289, 302, 322, 326, 351, 361, 376, 390, 398, 408, 410, 422, 426, 434, 446, 477, 508, 529, 532, 534, 542, 551, 566, 590

Offset: 1

J. M. Bergot and Robert Israel, Aug 08 2021

nonn

Numbers k such that both the sum s and product p of the divisors of k are divisible by (p mod s).

			a(3) = 14 is a term because A000203(14) = 1+2+7+14 = 24, A007955(14) = 1*2*7*14 = 196, A187680(14) = 196 mod 24 = 4, and both 24 and 196 are divisible by 4.

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

Cf. A000203, A007955, A187680.

Includes A188061.

filter:= proc(n) local d,s,p,r;
  d:= numtheory:-divisors(n);
  s:= convert(d,`+`);
  p:= convert(d,`*`);
  r:= p mod s;
  r <> 0 and p mod r = 0 and s mod r = 0
end proc:
select(filter, [$1..1000]);

okQ[n_] := Module[{d, s, p, m},
  d = Divisors[n];
  s = Total[d];
  p = Times @@ d;
  m = Mod[p, s];
  If[m == 0, False, Divisible[s, m] && Divisible[p, m]]];
Select[Range[1000], okQ] (* Jean-François Alcover, May 16 2023 *)

isok(k) = my(d=divisors(k), s=vecsum(d), p=vecprod(d), m=p % s); (m>0) && !(s%m) && !(p%m); \\ Michel Marcus, Aug 09 2021

A192853 Places n such that the two remainders A187680(n) and A191906(n) are both zero.

Original entry on oeis.org

6, 28, 120, 270, 496, 672, 924, 1320, 3948, 7980, 8128, 10920, 12690, 15456, 18018, 25296, 27930, 29190, 30240, 30294, 32760, 35640, 52080, 55692, 61770, 69936, 76986, 83160, 83580, 86814, 106950, 127218, 130200, 131040, 141360, 155610

Offset: 1

Juri-Stepan Gerasimov, Jul 11 2011

nonn

The even perfect numbers (A000396) are a subsequence.

Robert Israel, Table of n, a(n) for n = 1..214

Cf. A000396, A145551, A187680, A191906

filter:= proc(n) local Q,p,s;
  Q:= numtheory:-divisors(n) minus {n};
  p:= convert(Q,`*`); s:= convert(Q,`+`);
  p mod s = 0 and (p * n ) mod (s + n ) = 0
end proc:
select(filter, [$2..2*10^5]); # Robert Israel, Apr 22 2025

A191906 The remainder of (product of proper divisors of n) mod (sum of proper divisors of n).

Original entry on oeis.org

0, 0, 2, 0, 0, 0, 1, 3, 2, 0, 0, 0, 4, 6, 4, 0, 9, 0, 4, 10, 8, 0, 0, 5, 10, 1, 0, 0, 36, 0, 1, 3, 14, 9, 41, 0, 16, 5, 0, 0, 0, 0, 16, 12, 20, 0, 44, 7, 6, 9, 36, 0, 54, 4, 0, 11, 26, 0, 0, 0, 28, 33, 8, 8, 66, 0, 42, 15, 10, 0, 81, 0, 34, 39, 16, 1, 72, 0, 10, 9, 38, 0, 84, 16, 40, 21

Offset: 2

Juri-Stepan Gerasimov, Jun 19 2011

			a(2) = 1 mod 1 = 0;
a(3) = 1 mod 1 = 0;
a(4) = 2 mod 3 = 2.

Antti Karttunen, Table of n, a(n) for n = 2..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Cf. A001065, A007956, A187680.

A007956 := n -> mul(i, i=op(numtheory[divisors](n) minus {1, n}));
A001065 := proc(n) numtheory[sigma](n)-n ; end proc:
A191906 := proc(n) A007956(n) mod A001065(n) ; end proc:
seq(A191906(n),n=2..90) ; # R. J. Mathar, Jun 25 2011

Table[With[{pd=Most[Divisors[n]]},Mod[Times@@pd,Total[pd]]],{n,2,90}] (* Harvey P. Dale, Nov 24 2021 *)

A191906(n) = { my(m=1,s=0); fordiv(n, d, if(dAntti Karttunen, Jul 11 2019

a(n) = A007956(n) mod A001065(n).

A188061 Numbers k such that (product of divisors of k) == 1 (mod sum of divisors of k).

Original entry on oeis.org

4, 9, 16, 25, 49, 55, 64, 81, 121, 161, 169, 209, 256, 289, 351, 361, 529, 551, 625, 649, 729, 841, 961, 1024, 1079, 1189, 1369, 1407, 1443, 1681, 1849, 2015, 2209, 2289, 2401, 2809, 2849, 2915, 2975, 3401, 3481, 3721, 3857, 4096, 4489, 4599, 4887, 5041, 5329, 6049, 6241, 6319, 6561, 6889, 6993, 7921, 8569, 9409, 9701

Offset: 1

Juri-Stepan Gerasimov, Mar 20 2011, Jun 18 2011

nonn

This sequence includes every number of the form p^(2n), where p is a prime. Other semiprime members include 55, 161, 209, 551, 649, 1079, 1189, 3401, 6049, 6319, 9701. Are there infinitely many nonsquare semiprimes in the sequence? Is there some simpler property of primes p and q that puts pq in this sequence?

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A000203, A007955, A187680.

mptQ[n_]:=Module[{dn=Divisors[n]},Mod[Times@@dn,Total[dn]]==1]; Join[{1},Select[Range[10000],mptQ]]  (* Harvey P. Dale, Mar 28 2011 *)

proddiv(n)=local(t);t=numdiv(n);if(t%2==0,n^(t\2),sqrtint(n)^t)
for(n=1,10000,if(Mod(proddiv(n),sigma(n))==1,print1(n",")))

from gmpy2 import powmod, is_square, isqrt
from sympy import divisor_sigma
A188061_list = [n for n in range(1,10**4) if powmod(isqrt(n) if is_square(n) else n, int(divisor_sigma(n,0))//(1 if is_square(n) else 2), int(divisor_sigma(n,1))) == 1] # Chai Wah Wu, Mar 10 2016

A187680(a(n)) = 1.

More terms from Franklin T. Adams-Watters, Mar 21 2011

A187711 Integers k which equal (product of divisors of k) mod (sum of divisors of k).

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 17, 19, 20, 23, 29, 31, 33, 37, 40, 41, 43, 47, 53, 59, 61, 67, 71, 73, 76, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 136, 137, 139, 145, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 207, 211, 223, 227, 229, 233, 239, 241, 251, 257, 261, 263, 269, 271, 277

Offset: 1

Juri-Stepan Gerasimov, Mar 17 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000203, A007955, A187680.

isA187711 := proc(n) is(A187680(n) = n) end proc:
for n from 2 to 300 do if isA187711(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Mar 17 2011

Select[Range[300], Mod[#^(DivisorSigma[0, #]/2), DivisorSigma[1, #]] == # &] (* G. C. Greubel, Nov 05 2018 *)

{ k : k = A187680(k) }.

A187712 Composite numbers k such that k = (product of divisors of k) mod (sum of divisors of k).

Original entry on oeis.org

10, 20, 33, 40, 76, 136, 145, 207, 261, 385, 464, 528, 588, 897, 931, 1441, 1519, 1611, 1816, 1989, 2016, 2205, 2241, 2353, 3280, 3504, 3724, 3808, 4067, 4320, 4864, 5696, 6256, 7201, 7345, 8036, 10688, 10936, 11376, 13000, 16840, 17101, 18625, 19359, 19504, 19840

Offset: 1

Juri-Stepan Gerasimov, Mar 17 2011

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000203, A002808, A007955, A187680, A187711.

Select[Range[20000], CompositeQ[#] && PowerMod[#, DivisorSigma[0, #]/2, DivisorSigma[1, #]] == # &] (* Amiram Eldar, Mar 22 2024 *)

is1(n) = my(f = factor(n), s = sigma(f), d = numdiv(f)); if(d%2, Mod(sqrtint(n), s)^d, Mod(n, s)^(d/2)) == n;
is(n) = n > 1 && !isprime(n) && is1(n); \\ Amiram Eldar, Mar 22 2024

A187711 INTERSECT A002808.

More terms from Amiram Eldar, Mar 22 2024

A192035 Numbers k with equal remainders of (product of divisors of k) mod (sum of divisors of k) and (product of proper divisors of k) mod (sum of proper divisors of k).

Original entry on oeis.org

6, 14, 28, 51, 120, 260, 270, 496, 672, 679, 752, 924, 1260, 1320, 1540, 1960, 2055, 2262, 2651, 3808, 3948, 4381, 6413, 6435, 6944, 7900, 7980, 8010, 8128, 9809, 9945, 10242, 10920, 12690, 15456, 16830, 18018, 21728, 21970, 22320, 25296, 27930, 29190, 29792

Offset: 1

Juri-Stepan Gerasimov, Jun 21 2011

nonn

The even perfect numbers (A000396) are a subsequence.

The deficient numbers (A005100) in the sequence are 14, 51, 679, 752, 2055, 2651, 4381, 6413, 9809, 9945, 21970, ... - Juri-Stepan Gerasimov, Jul 07 2011

			14 is in this sequence because (1*2*7*14) mod (1+2+7+14) = 196 mod 24 = 4 and (1*2*7) mod (1+2+7) = 14 mod 10 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..600

Cf. A001065, A007956, A187680, A191906.

erQ[n_]:=Module[{divs=Divisors[n],ds=DivisorSigma[1,n]},Mod[ Times@@ divs,ds] == Mod[ Times@@Most[divs],ds-n]]; Select[Range[2,30000],erQ] (* Harvey P. Dale, Jun 13 2015 *)
Select[Range[2, 30000], Mod[(p = #^(DivisorSigma[0, #]/2)), (s = DivisorSigma[1, #])] == Mod[p/#, s - #] &] (* Amiram Eldar, Jul 21 2019 *)

{ k : A187680(k) = A191906(k) }.

Values from a(4) onwards from R. J. Mathar, Jul 05 2011

A191868 Numbers n such that (product of divisors of 2n) mod (sum of divisors of 2n) is equal to number of divisors of 2n.

Original entry on oeis.org

1, 4, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 448, 457, 463, 487, 499, 523, 541

Offset: 1

Juri-Stepan Gerasimov, Jun 18 2011

Cf. A000005, A000203, A007955, A016777, A187680.

is(n)=my(t=2*n,P=if(issquare(t,&t), t^numdiv(t^2),t^(numdiv(t)/2))); P%sigma(2*n)==numdiv(2*n) \\ Charles R Greathouse IV, Jun 19 2011

More terms from Charles R Greathouse IV, Jun 19 2011

cp's OEIS Frontend

A346956 Numbers k such that A000203(k) and A007955(k) are both divisible by A187680(k).

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A192853 Places n such that the two remainders A187680(n) and A191906(n) are both zero.

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A191906 The remainder of (product of proper divisors of n) mod (sum of proper divisors of n).

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A188061 Numbers k such that (product of divisors of k) == 1 (mod sum of divisors of k).

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A187711 Integers k which equal (product of divisors of k) mod (sum of divisors of k).

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Maple

Mathematica

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A187712 Composite numbers k such that k = (product of divisors of k) mod (sum of divisors of k).

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Mathematica

PARI

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A192035 Numbers k with equal remainders of (product of divisors of k) mod (sum of divisors of k) and (product of proper divisors of k) mod (sum of proper divisors of k).

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