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.

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A120317 Consecutive refactorable numbers a(n)-1, a(n) in which 7 the smallest prime divisor of a(n).

Original entry on oeis.org

6080399213078595601, 106451203123324908289, 842122675409157900289, 205035001401532317649921, 690310240598397005456401, 1125500133125681400538801, 1241419580861102113344769
Offset: 1

Views

Author

Walter Kehowski, Jun 20 2006

Keywords

Crossrefs

Cf. A033950, A036898, A114617.

Programs

  • Maple
    with(numtheory); RFC7:=[]: p:=ithprime(4): P:=[seq(ithprime(i),i=1..3)]; for w to 1 do for k from 3 to 12^4 by 2 do if andmap(z -> k mod z <> 0, P) then m:=p*k; n:=m^(p-1); t:=tau(n); n1:=n-1; t1:=tau(n1); if (n mod t = 0) and (n1 mod t1 = 0) then RFC7:=[op(RFC7),n]; print(ifactor(n)); fi fi; od od;

Formula

a(n) is the first integer of the form (7*k)^(7-1) such that both a(n) and a(n)-1 is refactorable and 7 is the smallest prime divisor of a(n).

A120318 Consecutive refactorable numbers a(n)-1, a(n) in which 11 is the smallest prime divisor of a(n).

Original entry on oeis.org

38604666779024731098340977806401, 7208577773559712596404976530284801, 695314235787112476661749457231833601, 313468146036745542621075945985861000534849
Offset: 1

Views

Author

Walter Kehowski, Jun 20 2006

Keywords

Crossrefs

Cf. A033950, A036898, A114617.

Programs

  • Maple
    with(numtheory); RFC11:=[]: p:=ithprime(5): P:=[seq(ithprime(i),i=1..4)]; for w to 1 do for k from 3 to 12^4 by 2 do if andmap(z -> k mod z <> 0, P) then m:=p*k; n:=m^(p-1); t:=tau(n); n1:=n-1; t1:=tau(n1); if (n mod t = 0) and (n1 mod t1 = 0) then RFC11:=[op(RFC11),n]; print(ifactor(n)); fi fi; od od;

Formula

a(n) is the first integer of the form (11*k)^(11-1) such that both a(n) and a(n)-1 is refactorable and 11 is the smallest prime divisor of a(n).

A120335 CRF(13): consecutive refactorable numbers (rf, rf-1 are refactorable) such that 13 is the smallest prime divisor of rf.

Original entry on oeis.org

79600343456925208350554324952070658488321, 67727051825754224132985695308485992267126791150081, 17096333467784942360991864487916588941402614691799041
Offset: 1

Views

Author

Walter Kehowski, Jun 22 2006

Keywords

Comments

The sequence has prime factorization (13*197)^12, (13*1093)^12, (13*1733)^12, (13*17*139)^12, (13*7877)^12, (13*16069)^12.

Examples

			a(1)=(13*197)^12 is the first number rf such that rf refactorable, 13 is the smallest prime of rf and rf-1 is refactorable.

Crossrefs

Cf. A033950, A036898, A114617.

Formula

a(n) = is the n-th number rf such that both rf and rf-1 are refactorable and 13 is the smallest prime divisor of rf.

A343817 Refactorable numbers (A033950) which set a record for the gap to the next refactorable number.

Original entry on oeis.org

1, 2, 24, 40, 108, 156, 296, 732, 1692, 31616, 51608, 568720, 766620, 6195132, 6938752, 17879440, 18578320, 35196584, 228694176, 475292728, 589169184, 1451254356, 3252050592, 4865544096, 6328305120, 8082626976, 8694028264, 9112984448, 30328732568, 46093418640
Offset: 1

Views

Author

Amiram Eldar, Apr 30 2021

Keywords

Comments

Since the asymptotic density of the refactorable numbers is 0 (Kennedy and Cooper, 1990), this sequence is infinite.
The corresponding record values are 1, 6, 12, 16, 20, 24, 32, 44, 92, 100, 144, 152, 180, 192, 208, 212, 236, 268, 280, 296, 336, 360, 368, 372, 384, 396, 408, 432, 488, 496, ...

Examples

			The first 8 refactorable numbers are 1, 2, 8, 9, 12, 18, 24 and 36. The gaps between them are 1, 6, 1, 3, 6, 6 and 12. The record gaps, 1, 6 and 12, occur after the refactorable numbers 1, 2 and 24, which are the first 3 terms of this sequence.

Links

Crossrefs

Cf. A033950, A114617.

Programs

  • Mathematica
    refQ[n_] := Divisible[n, DivisorSigma[0, n]]; seq = {}; m = 1; dm = 0; Do[If[refQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 2, 10^6}]; seq

A360779 Refactorable numbers gaps: differences between consecutive refactorable numbers.

Original entry on oeis.org

1, 6, 1, 3, 6, 6, 12, 4, 16, 4, 12, 8, 4, 4, 8, 8, 4, 20, 4, 4, 16, 4, 24, 4, 20, 21, 3, 4, 8, 8, 4, 24, 12, 8, 32, 16, 4, 12, 12, 4, 8, 12, 28, 17, 3, 4, 2, 18, 4, 8, 8, 4, 12, 12, 20, 24, 4, 4, 16, 16, 12, 13, 7, 4, 4, 24, 8, 12, 24, 4, 8, 12, 44, 16, 12, 4, 16, 4, 24
Offset: 1

Views

Author

Ctibor O. Zizka, Feb 20 2023

Keywords

Comments

Empirically it looks as though the consecutive refactorable numbers >= 8 with odd gaps between them always occur in triples: [8, 9, 12], [204, 225, 228], [424, 441, 444], [612, 625, 632], [1068, 1089, 1096], [1520, 1521, 1524], and so on. The sum of the gaps in the triple is divisible by 4. The middle term of a triple is an odd refactorable number, see A036896.

Examples

			a(1) = 2 - 1 = 1;
a(2) = 8 - 2 = 6;
a(3) = 9 - 8 = 1;
and so on.

Crossrefs

Cf. A033950, A036896, A036898, A114617.

Programs

  • Mathematica
    Differences[Select[Range[1000], Divisible[#, DivisorSigma[0, #]] &]] (* Amiram Eldar, Feb 20 2023 *)
  • PARI
    lista(nn) = my(v=select(x->!(x % numdiv(x)), [1..nn])); vector(#v-1, k, v[k+1]-v[k]); \\ Michel Marcus, Feb 20 2023

Formula

a(n) = A033950(n + 1) - A033950(n).
Previous Showing 11-15 of 15 results.

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