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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A216072 Aliquot open end sequences which belong to distinct families.

Original entry on oeis.org

276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, 2232, 2340, 2360, 2484, 2514, 2664, 2712, 2982, 3270, 3366, 3408, 3432, 3564, 3678, 3774, 3876, 3906, 4116, 4224, 4290, 4350, 4380, 4788, 4800, 4842
Offset: 1

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Author

V. Raman, Sep 01 2012

Keywords

Comments

These aliquot sequences are believed to grow forever without terminating in a prime or entering a cycle.
Sequence A131884 lists all the starting values of an aliquot sequence that lead to open-ending. It includes all values obtained by iterating from the starting values of this sequence. But this sequence lists only the values that are the lowest starting elements of open end aliquot sequences that are the part of different open-ending families. - V. Raman, Dec 08 2012

Links

Crossrefs

Cf. A008892, A098007, A131884, A115350.

A361421 Infinitary aliquot sequence starting at 840: a(1) = 840, a(n) = A126168(a(n-1)), for n >= 2.

Original entry on oeis.org

840, 2040, 4440, 9240, 25320, 51000, 117480, 271320, 765480, 1531320, 3721800, 5956440, 12295560, 25086840, 54141960, 108284280, 250301640, 502213560, 1007626440, 2017856760, 4039750920, 8079502200, 19596145800, 44369345400, 71495068200, 115576350360, 231152701080
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2023

Keywords

Comments

First differs from A045477 at n = 12.
840 is the least number whose infinitary aliquot sequence is not known to be finite or eventually periodic.
R. J. Mathar found that this sequence does not reach 0 or enter a cycle before the 1500th term (see A127661). This limit was extended to beyond the 3000th term (see the b-file).

Examples

			a(1) = 840 by definition.
a(2) = A126168(a(1)) = A126168(840) = 2040.
a(3) = A126168(a(2)) = A126168(2040) = 4440.

Links

Crossrefs

Cf. A008892, A045477, A126168, A127661, A293355.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
infs[n_] := If[n==1, 1, Times @@ f @@@ FactorInteger[n]] - n; infs[0] = 0;
seq[len_, init_] := NestWhileList[infs, init, UnsameQ, All, len];
seq[27, 840]
  • PARI
    s(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
lista(nmax) = {my(k = 840); for(n = 1, nmax, print1(k, ", "); if(k == 0, break); k = s(k)); }

A347769 a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 15, 13, 14, 17, 18, 21, 19, 20, 22, 23, 24, 36, 55, 25, 26, 27, 28, 29, 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 31, 32, 34, 35, 37, 38, 39, 40, 50, 43, 41, 44, 46, 47, 48, 76, 64, 63, 49, 51, 52, 53, 56, 57, 58, 59, 60, 108, 172
Offset: 0

Views

Author

Eric Chen, Sep 13 2021

Keywords

Comments

This sequence is a permutation of the nonnegative integers iff Catalan's aliquot sequence conjecture (also called Catalan-Dickson conjecture) is true.
a(563) = 276 is the smallest number whose aliquot sequence has not yet been fully determined.
As long as the aliquot sequence of 276 is not known to be finite or eventually periodic, a(563+k) = A008892(k).

Examples

			a(0) = 0, a(1) = 1;
since A001065(a(1)) = 0 has already appeared in this sequence, a(2) = 2;
since A001065(a(2)) = 1 has already appeared in this sequence, a(3) = 3;
...
a(11) = 11;
since A001065(a(11)) = 1 has already appeared in this sequence, a(12) = 12;
since A001065(a(12)) = 16 has not yet appeared in this sequence, a(13) = A001065(a(12)) = 16;
since A001065(a(13)) = 15 has not yet appeared in this sequence, a(14) = A001065(a(13)) = 15;
since A001065(a(14)) = 9 has already appeared in this sequence, a(15) = 13;
...

Links

Crossrefs

Cf. A032451.
Cf. A001065 (sum of aliquot parts).
Cf. A003023, A044050, A098007, A098008: ("length" of aliquot sequences, four versions).
Cf. A007906.
Cf. A115060 (maximum term of aliquot sequences).
Cf. A115350 (termination of the aliquot sequences).
Cf. A098009, A098010 (records of "length" of aliquot sequences).
Cf. A290141, A290142 (records of maximum term of aliquot sequences).
Cf. A000396, A063990, A122726, A206708, A347770.
Cf. A080907, A063769, A121507, A121508, A131884, A126016.
Aliquot sequences starting at various numbers: A000004 (0), A000007 (1), A033322 (2), A010722 (6), A143090 (12), A143645 (24), A010867 (28), A008885 (30), A143721 (38), A008886 (42), A143722 (48), A143723 (52), A008887 (60), A143733 (62), A143737 (68), A143741 (72), A143754 (75), A143755 (80), A143756 (81), A143757 (82), A143758 (84), A143759 (86), A143767 (87), A143846 (88), A143847 (96), A143919 (100), A008888 (138), A008889 (150), A008890 (168), A008891 (180), A203777 (220), A008892 (276), A014360 (552), A014361 (564), A074907 (570), A014362 (660), A269542 (702), A045477 (840), A014363 (966), A014364 (1074), A014365 (1134), A074906 (1521), A143930 (3630), A072891 (12496), A072890 (14316), A171103 (46758), A072892 (1264460).

Programs

  • PARI
    A347769_list(N)=print1(0, ", "); if(N>0, print1(1, ", ")); v=[0, 1]; b=1; for(n=2, N, if(setsearch(Set(v), sigma(b)-b), k=1; while(k<=n, if(!setsearch(Set(v), k), b=k; k=n+1, k++)), b=sigma(b)-b); print1(b, ", "); v=concat(v, b))

A372645 a(n) is the 2-adic valuation of the n-th term of the aliquot sequence of 276.

Original entry on oeis.org

2, 2, 3, 4, 4, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 0

Views

Author

Samuel Herts, May 08 2024

Keywords

Comments

a(n) is the exponent of prime 2 in the prime factorization of A008892(n).
An empirical observation would suggest that this sequence may be a(n) = 1 for n >= 793 since the likelihood of a parity switch becomes exponentially small.

Examples

			For n=4, the term in the aliquot sequence of 276 after 4 steps is A008892(4) = 1872 = 2^4 * 3^2 * 13 and the exponent of 2 there is a(4) = 4.
For n=30, the term in the aliquot sequence of 276 after 30 steps is A008892(30) = 23117724 = 2^2 * 3^4 * 7 * 10193 and the exponent of 2 there is a(30) = 2.

Links

Crossrefs

Cf. A007814, A008892.

Programs

  • PARI
    lista(nn) = my(v = vector(nn)); v[1] = 276; for (n=2, nn, v[n] = sigma(v[n-1]) - v[n-1];); apply(x->valuation(x, 2), v); \\ Michel Marcus, May 14 2024

Formula

a(n) = A007814(A008892(n)).
Previous Showing 21-24 of 24 results.

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