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.

A127163 Integers whose aliquot sequences terminate by encountering the prime 3. Also known as the prime family 3.

Original entry on oeis.org

3, 4, 9, 12, 15, 16, 26, 30, 33, 42, 45, 46, 52, 54, 66, 72, 78, 86, 87, 90, 102, 105, 114, 121, 123, 126, 135, 144, 165, 166, 174, 186, 198, 207, 212, 243, 246, 247, 249, 258, 259, 270
Offset: 1

Views

Author

Ant King, Jan 07 2007

Keywords

Comments

This sequence is complete only as far as the last term given, for the eventual fate of the aliquot sequence generated by 276 is not (yet) known

Examples

			a(5)=15 because the fifth integer whose aliquot sequence terminates by encountering the prime 3 as a member of its trajectory is 15. The complete aliquot sequence generated by iterating the proper divisors of 15 is 15->9->4->3->1->0

References

  • Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.

Links

Crossrefs

Cf. A080907, A127161, A127162, A127164, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

Programs

  • Mathematica
    s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[275], MemberQ[Trajectory[ # ], 3] &]

Formula

Define s(i)=sigma(i)-i=A000203(i)-i. Then if the aliquot sequence obtained by repeatedly applying the mapping i->s(i) terminates by encountering the prime 3 as a member of its trajectory, i is included in this sequence

A127161 Integers whose aliquot sequences terminate by encountering a prime number.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75
Offset: 1

Views

Author

Ant King, Jan 06 2007

Keywords

Comments

This sequence is the same as A080907 from A080907's second term onwards.

Examples

			a(10)=12 because the tenth integer whose aliquot sequence terminates by encountering a prime as a member of its trajectory is 12. The complete aliquot sequence generated by iterating the proper divisors of 12 is 12->16->15->9->4->3->1->0

References

  • Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.

Links

Crossrefs

Cf. A080907, A127162, A127163, A127164, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

Programs

  • Mathematica
    s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[2, 275], Last[Trajectory[ # ]] == 0 &]

Formula

Define s(i)=sigma(i)-i=A000203(i)-i. Then if the aliquot sequence obtained by repeatedly iterating s contains a prime as a member of its trajectory, i is included in this sequence

A127162 Composite numbers whose aliquot sequences terminate by encountering a prime number.

Original entry on oeis.org

4, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 98, 99
Offset: 1

Views

Author

Ant King, Jan 06 2007

Keywords

Examples

			a(5)=12 because the fifth composite number whose aliquot sequence terminates by encountering a prime as a member of its trajectory is 12. The complete aliquot sequence generated by iterating the proper divisors of 12 is 12->16->15->9->4->3->1->0

References

  • Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.

Links

Crossrefs

Cf. A080907, A127161, A127163, A127164, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

Programs

  • Mathematica
    s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[2, 275], ! PrimeQ[ # ] && Last[Trajectory[ # ]] == 0 &]

Formula

Define s(i)=sigma(i)-i=A000203(i)-i. Then if i is composite and the aliquot sequence obtained by repeatedly applying the mapping i->s(i) contains a prime as a member of its trajectory, i is included in this sequence.

A127652 Integers whose unitary aliquot sequences are longer than their ordinary aliquot sequences.

Original entry on oeis.org

25, 28, 36, 40, 50, 68, 70, 74, 94, 95, 98, 116, 119, 134, 142, 143, 154, 162, 170, 175, 182, 189, 190, 200, 220, 226, 242, 245, 262, 273
Offset: 1

Views

Author

Ant King, Jan 24 2007

Keywords

Comments

Here the length of an aliquot sequence is defined to be the length of the transient part of its trajectory + the length of its terminal cycle.

Examples

			a(5)=50 because the fifth integer whose unitary aliquot sequence is longer than its ordinary aliquot sequence is 50.

References

  • Riele, H. J. J. te; Unitary Aliquot Sequences. MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
  • Riele, H. J. J. te; Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.

Links

Crossrefs

Cf. A127161, A127162, A127163, A127164, A063769, A063990, A097032, A098007, A097010, A127653, A098185, A127654, A063991, A127655, A097037, A097036.

Programs

  • Mathematica
    UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;g[n_] := If[n > 0, sstar[n], 0];UnitaryTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];s[n_]:=DivisorSigma[1,n]-n;h[n_] := If[n > 0, s[n], 0];OrdinaryTrajectory[n_] := Most[NestWhileList[h, n, UnsameQ, All]];Select[Range[275],Length[UnitaryTrajectory[ # ]]>Length[OrdinaryTrajectory[ # ]] &]

Formula

Sequence gives those values of n for which A097032(n)>A098007(n).

A358266 Numbers k such that the aliquot sequence of 2^k ends with the prime 7.

Original entry on oeis.org

3, 10, 12, 141, 278, 387, 421
Offset: 1

Views

Author

Jean Luc Garambois, Nov 06 2022

Keywords

Examples

			a(4)=141 because the aliquot sequence that starts with the integer 2^141 ends with the prime number 7 and there are only three smaller powers of 2 that do the same: 2^3, 2^10 and 2^12.

Links

Crossrefs

Cf. A127164, A358239.

Programs

  • PARI
    f(n) = if (n==1, return(2)); my(list = List(), s=2^n); for (i=1, oo, s = sigma(s) - s; if (#select(x->(x==s), list), return(0)); if (isprime(s), return (s)); listput(list, s); );
isok(m) = f(m) == 7;

Formula

Define s(i) = sigma(i) - i = A000203(i) - i. Then k is a term of this sequence if the aliquot sequence obtained by repeatedly applying the mapping i->s(i) taking as initial value 2^k terminates in the prime 7.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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