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 1-3 of 3 results.

A383428 Primitive terms in A066192: number k such that k is a term of A066192 and k/2 is not.

Original entry on oeis.org

4, 12, 56, 120, 528, 672, 992, 1456, 2160, 2208, 6720, 9024, 9120, 11904, 13104, 16256, 17472, 24800, 29568, 55104, 55552, 73440, 90816, 95040, 119040, 120960, 121024, 123648, 131040, 146688, 151680, 174720, 195072, 223104, 297600, 397440, 399616, 445536, 505344
Offset: 1

Views

Author

Amiram Eldar, Apr 27 2025

Keywords

Comments

If a(1) = 1 instead of 4, then this will be the sequence of primitive terms in A069519.
If k is a term then 2^m * k is a term in A066192 for all m >= 0.
If there is an odd term in this sequence it must be an odd perfect number (A000396). See the comments in A066192.
Except for 4, numbers k such that A091570(k) | k and k/A091570(k) is odd.

Links

Crossrefs

Subsequence of A066191 and A066192.
Cf. A000396, A069519, A091570.

Programs

  • Mathematica
    q[n_] := Module[{s = DivisorSigma[1, n/2^IntegerExponent[n, 2]] - If[OddQ[n], n, 0]}, Divisible[n, s] && OddQ[n/s]]; Select[Range[550000], # == 4 || (CompositeQ[#] && q[#]) &]
  • PARI
    isok(k) = if(k == 1 || isprime(k), 0, if(k == 4, 1, my(s = sigma(k >> valuation(k, 2)) - if(k%2, k)); !(k % s) && (k/s) % 2));

A066191 Numbers k such that the sum of the odd aliquot parts of k divides k.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 11, 12, 13, 16, 17, 19, 23, 24, 29, 31, 32, 37, 41, 43, 47, 48, 53, 56, 59, 61, 64, 67, 71, 73, 79, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 120, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 192, 193, 197, 199, 211, 223
Offset: 1

Views

Author

Robert G. Wilson v, Dec 15 2001

Keywords

Examples

			12 is in the sequence because the odd aliquot parts of 12 are {1,3} and their sum divides 12.

Links

Crossrefs

Disjoint union of A000040 and A066192.
Cf. A091570.

Programs

  • Maple
    with(numtheory):soa:=proc(n) local div,s,j: div:=convert(divisors(n),list): s:=0: for j from 1 to nops(div)-1 do if div[j] mod 2=1 then s:=s+div[j] else s:=s: fi: od: end: p:=proc(n) if type(n/soa(n),integer)=true then n else fi end: seq(p(n),n=1..240); # Emeric Deutsch, Feb 26 2005
  • Mathematica
    Do[ d = Drop[ Divisors[ n ], -1 ]; l = Length[ d ]; od = 1; k = 2; While[ k <= l, If[ OddQ[ d[ [ k ] ] ], od = od + d[ [ k ] ] ]; k++ ]; If[ IntegerQ[ n/od ], Print[ n ] ], {n, 2, 200} ]
Select[Range[2, 225], Divisible[#, DivisorSigma[1, #/2^IntegerExponent[#, 2]] - If[OddQ[#], #, 0]] &] (* Amiram Eldar, Apr 27 2025 *)
  • PARI
    { n=0; for (m=2, 10^9, d=divisors(m); s=1; for (i=2, numdiv(m) - 1, if (d[i]%2, s += d[i])); if (s > 0 && m%s == 0, write("b066191.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 05 2010
  • PARI
    isok(n) = !(n % sumdiv(n, d, d*(d%2)*(d!=n))); \\ Michel Marcus, Apr 06 2015
  • PARI
    isok(k) = if(k == 1, 0, !(k % (sigma(k >> valuation(k, 2)) - if(k%2, k)))); \\ Amiram Eldar, Apr 27 2025

Extensions

More terms from Emeric Deutsch, Feb 26 2005

A069519 Numbers k such that 1/(Sum_{d|k} (-1)^d/d) is an integer.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 56, 64, 96, 112, 120, 128, 192, 224, 240, 256, 384, 448, 480, 512, 528, 672, 768, 896, 960, 992, 1024, 1056, 1344, 1456, 1536, 1792, 1920, 1984, 2048, 2112, 2160, 2208, 2688, 2912, 3072, 3584, 3840, 3968, 4096, 4224, 4320
Offset: 1

Views

Author

Benoit Cloitre, Apr 16 2002

Keywords

Links

Crossrefs

Cf. A066192.

Programs

  • Mathematica
    q[n_] := IntegerQ[1/DivisorSum[n, (-1)^# / # &]]; Select[Range[4500], q] (* Amiram Eldar, Apr 27 2025 *)
  • PARI
    isok(k) = denominator(1/sumdiv(k, d, (-1)^d/d)) == 1; \\ Amiram Eldar, Apr 27 2025

Formula

a(1)=1, a(2)=2, a(n) = A066192(n-2) for n > 2.

Extensions

Name corrected by Amiram Eldar, Apr 27 2025
Showing 1-3 of 3 results.

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