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.

A358320 Least odd number m such that m*2^n is a perfect, amicable or sociable number, and -1 if no such number exists.

Original entry on oeis.org

12285, 3, 7, 779, 31, 37, 127, 651, 2927269, 93, 25329329, 7230607, 8191, 66445153, 7613527, 18431675687, 131071, 264003743, 524287, 59592560831, 949755039781
Offset: 0

Views

Author

Jean-Marc Rebert, Nov 09 2022

Keywords

Comments

For n in {1,2,4,6,12,16,18}, a(n)*2^n is a perfect number. See A090748.
For n in {0,3,5,8,10,11,13,14,15,17,19}, a(n)*2^n is an amicable number.
For n in {7,9} a(n)*2^n is a sociable number of order 28.
That is, h_k(m*2^n) = m*2^n for some k > 0, where h_{k+1}(n) = h_k(h(n)) and h(n) = A001065(n), the sum of aliquot parts of n. - Charles R Greathouse IV, Nov 09 2022
Least m such that m*2^n is in A347770. - Charles R Greathouse IV, Nov 09 2022

Examples

			a(1) = 3, because 3 is an odd number and 3 * 2^1 = 6 is a perfect number, and no lesser number has this property.

Links

Crossrefs

Cf. A347770, A000396, A001065, A002025, A259180, A262625.
Cf. A090748, A358415.

Programs

  • PARI
    sigmap(n)=if(n<=1, return(0)); sigma(n)-n
cycle(n,TT=28)=my(x=n,T=1); while(x>0&&T<=TT, x=sigmap(x); if(x==n, return(T)); T++)
a(n,TT=28)=my(p2n=2^n); forstep(m=1, +oo, 2, if(cycle(p2n*m,TT), return(m)))

Extensions

a(0), a(15)-a(20) from Jean-Marc Rebert, Nov 17 2022

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))

A358546 Least odd number m such that m mod 3 > 0 and m*3^n is an amicable number, and -1 if no such number exists.

Original entry on oeis.org

5480828320492525, 4865, 7735, 455, 131285, 849355, 11689795, 286385, 187047685, 104255, 32851039955, 2085985, 47942199242945, 189296520259, 349700961302721360788238344333849, 580068028631, 50392682631679406080371010751466781
Offset: 0

Views

Author

Jean-Marc Rebert, Nov 21 2022

Keywords

Comments

If a(n) > -1 then a(n)*3^n is the least amicable number k such that A007949(k) = n.

Examples

			a(1) = 4865, because 4865 is an odd number and 4865 % 3 > 0 and 4865 * 3 = 14595 is an amicable number, and no lesser number has this property.

Links

Crossrefs

Cf. A347770, A000396, A001065, A002025, A259180, A262625.
Cf. A090748, A358415, A358320, A358022.

Programs

  • PARI
    sigmap(k)=if(k,sigma(k)-k,0)
cycle(k, TT=2)=my(x=k, T=1); while(x>0&&T<=TT, x=sigmap(x); if(x==k, return(T)); T++)
a(n, TT=2)=my(p3n=3^n); forstep(m=1, +oo, 2, if(m%3&&cycle(p3n*m, TT)==2, return(m)))
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