cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Juri-Stepan Gerasimov, Jul 11 2011

Keywords

Comments

The even perfect numbers (A000396) are a subsequence.

Links

Crossrefs

Cf. A000396, A145551, A187680, A191906

Programs

  • Maple
    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

A191905 Composite deficient numbers k such that (product of proper divisors of k) mod (sum of proper divisors of k) is a prime number.

Original entry on oeis.org

4, 9, 10, 25, 33, 39, 49, 57, 91, 93, 98, 105, 111, 119, 121, 145, 155, 169, 183, 185, 187, 189, 201, 205, 209, 215, 225, 235, 237, 242, 245, 265, 289, 291, 299, 305, 327, 335, 351, 355, 361, 371, 403, 413, 415, 417, 425, 427, 437, 469, 471, 475, 485, 493, 497, 515, 527, 529, 535, 543, 549, 553
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 19 2011

Keywords

Links

Crossrefs

Cf. A001065, A005100, A007956, A125493, A191906.

Programs

  • Maple
    isA191905 := proc(n) if not isA125493(n) then false; else isprime( A191906(n)) ; end if; end proc:
for n from 3 to 710 do if isA191905(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Jun 27 2011
  • Mathematica
    fQ[n_]:=Module[{pd=Most[Divisors[n]]},!PerfectNumberQ[n]&&CompositeQ[n] && DivisorSigma[ 1,n]<2n&& PrimeQ[Mod[Times@@pd,Total[pd]]]] Select[Range[2,600],fQ] (* Harvey P. Dale, Jul 14 2024 *)

Extensions

Corrected by R. J. Mathar, Jun 27 2011

A192034 Least k such that (product of proper divisors of k) mod (sum of proper divisors of k) equals n.

Original entry on oeis.org

2, 8, 4, 9, 14, 25, 15, 49, 22, 18, 21, 57, 45, 169, 34, 69, 38, 205, 143, 119, 46, 87, 217, 93, 130, 133, 58, 323, 62, 111, 160, 553, 319, 63, 74, 129, 30, 305, 82, 75, 86, 36, 68, 335, 48, 159, 301, 355, 369, 171, 106, 177
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Jun 21 2011

Keywords

Comments

Greedy inverse of A191906.

Examples

			a(0)=2 because A007956(2) mod A001065(2) = 1 mod 1 = 0, and 2 is the smallest number for which this is the case;
a(1)=8 because A007956(8) mod A001065(8) = 8 mod 7 = 1, and 8 is the smallest number for which this is the case;
a(2)=4 because A007956(4) mod A001065(4) = 2 mod 3 = 2, and 4 is the smallest number for which this is the case.

Crossrefs

Cf. A001065, A007956, A191906.

Programs

  • Maple
    A192034 := proc(n) local k ; for k from 2 do if A191906(k) = n then return k ; end if; end do: end proc: # R. J. Mathar, Jul 01 2011
  • Mathematica
    ds[n_]:=Module[{divs=Most[Divisors[n]]},Mod[Times@@divs,Total[divs]]]; Join[ {2},Transpose[Table[SelectFirst[Table[{n,ds[n]},{n,2,2000}],#[[2]] == i&],{i,60}]][[1]]] (* Harvey P. Dale, Apr 11 2015 *)

Extensions

Corrected by R. J. Mathar, Jul 01 2011
Example section corrected by Jon E. Schoenfield, Feb 24 2019

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

Views

Author

Juri-Stepan Gerasimov, Jun 21 2011

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A001065, A007956, A187680, A191906.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

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

A247145 Composite numbers such that the product of the number's proper divisors is divisible by the sum of the number's proper divisors.

Original entry on oeis.org

6, 12, 24, 28, 40, 42, 56, 60, 90, 120, 140, 153, 216, 234, 270, 290, 360, 440, 496, 522, 568, 585, 588, 672, 708, 819, 924, 984, 992, 1001, 1170, 1316, 1320, 1365, 1431, 1780, 2016, 2184, 2295, 2296, 2299, 2464, 2466, 2655, 2832, 3100, 3344, 3420, 3627, 3724, 3948, 4320, 4336, 4416, 4680
Offset: 1

Views

Author

David Consiglio, Jr., Nov 20 2014

Keywords

Comments

Equal to the indices of the zero terms that correspond to composite numbers in A191906.

Examples

			12 is on the list because the proper divisors of 12 are [1,2,3,4,6]. The product of these numbers is 144. Their sum is 16. 144 is divisible by 16.

Links

Crossrefs

Cf. A145551.

Programs

  • Maple
    filter:= proc(n)
       local d,p,s;
     if isprime(n) then return false fi;
     d:= numtheory:-divisors(n) minus {n};
     convert(d,`*`) mod convert(d,`+`) = 0;
end proc:
select(filter, [$2..10000]); # Robert Israel, Dec 16 2014
  • Mathematica
    a247145[n_Integer] :=
Select[Select[Range[n], CompositeQ[#] &],
Divisible[Times @@ Most@Divisors[#], Plus @@ Most@Divisors[#]] &]; a247145[4680] (* Michael De Vlieger, Dec 15 2014 *)
fQ[n_Integer] := Block[{d = Most@Divisors@n}, Mod[Times @@ d, Plus @@ d] == 0]; Select[Range@4680, ! PrimeQ@# && fQ@# &] (* Michael De Vlieger, Dec 19 2014, suggested by Robert G. Wilson v *)
  • PARI
    forcomposite(n=1,10^3,d=divisors(n);p=prod(i=1,#d-1,d[i]);if(!(p%(sigma(n)-n)),print1(n,", "))) \\ Derek Orr, Nov 27 2014
  • Python
    from functools import reduce
from operator import mul
def divs(n):
    for i in range(1, int(n / 2 + 1)):
        if n % i == 0:
            yield i
    yield n
g = []
for a in range(2, 100):
    q = list(divs(a))[0:-1]
    if reduce(mul, q, 1) % sum(q) == 0 and len(q) != 1:
        g.append(a)
print(g)

Extensions

More terms from Derek Orr, Dec 03 2014
Showing 1-5 of 5 results.

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