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 1-8 of 8 results.

A304096 Number of Lucas numbers larger than 3 (4, 7, 11, 18, ...) that divide n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1
Offset: 1

Views

Author

Antti Karttunen, May 13 2018

Keywords

Comments

a(n) is the number of the divisors d of n that are of the form d = A000045(k-1) + A000045(k+1), for k >= 3.

Examples

			The divisors of 4 are 1, 2 and 4. Of these only 4 is a Lucas number larger than 3, thus a(4) = 1.
The divisors of 28 are 1, 2, 4, 7, 14 and 28. Of these 4 and 7 are Lucas numbers (A000032) larger than 3, thus a(28) = 2.

Links

Crossrefs

Cf. A000032, A000045, A000204, A093540, A102460, A304092, A304094, A304095, A304104.
Cf. also A005086, A300837.

Programs

  • PARI
    A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304096(n) = sumdiv(n,d,(d>3)*A102460(d));

Formula

a(n) = Sum_{d|n, d>3} A102460(d).
a(n) = A304094(n) - A079978(n) - 1.
a(n) = A304092(n) - A059841(n) - A079978(n) - 1.
a(n) = A007949(A304104(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 - 4/3 = 0.629524... . - Amiram Eldar, Dec 31 2023

A304091 a(n) is the number of the proper divisors of n that are Lucas numbers (A000032, with 2 included).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 3, 2, 3, 1, 3, 1, 3, 3, 3, 1, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 2, 5, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 2, 4, 2, 3, 1, 4, 1, 2, 3, 3, 1, 4, 1, 3, 2, 3, 1, 5, 1, 2, 2, 3, 3, 3, 1, 3, 2, 2, 1, 5, 1, 2, 3, 4, 1, 4, 2, 3, 2, 3, 1, 4, 1, 3, 3, 3, 1, 3, 1, 3, 3
Offset: 1

Views

Author

Antti Karttunen, May 13 2018

Keywords

Links

Crossrefs

Cf. A000032, A093540, A102460, A304092, A304093.
Cf. also A293435, A300836.

Programs

  • PARI
    A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304091(n) = sumdiv(n,d,(dA102460(d));

Formula

a(n) = Sum_{d|n, dA102460(d).
a(n) = A304092(n) - A102460(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2 + A093540 = 2.462858... . - Amiram Eldar, Jul 05 2025

A304094 Number of Lucas numbers (A000204: 1, 3, 4, 7, 11, ... excluding 2) that divide n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 13 2018

Keywords

Links

Crossrefs

Cf. A000204, A093540, A304092, A304093, A304096.
Cf. also A005086, A300837.

Programs

  • PARI
    isA000204(n) = { my(u1=1,u2=3,old_u1); if(n<=2,(n%2),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304094(n) = sumdiv(n,d,isA000204(d));

Formula

a(n) = A304092(n) - A059841(n).
a(n) = A304096(n) + A079978(n) + 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 = 1.962858... . - Amiram Eldar, Dec 31 2023

A356062 a(n) is the smallest integer that has exactly n Lucas divisors (A000032).

Original entry on oeis.org

1, 2, 4, 12, 36, 252, 2772, 52668, 1211364, 35129556, 1089016236, 44649665676, 2098534286772, 417608323067628, 88115356167269508, 24760415083002731748, 7948093241643876891108, 4140956578896459860267268
Offset: 1

Views

Author

Bernard Schott, Jul 25 2022

Keywords

Comments

The new Lucas numbers that appear at each step are in A356063.

Examples

			36 is divisible by 1, 2, 3, 4, 18, which are all Lucas numbers, and no integer < 36 has 5 divisors that are Lucas numbers, hence a(5) = 36.

Crossrefs

Cf. A000032, A304092, A356063.
Similar sequences: A087997 (palindromes), A129655 (Fibonacci), A333456 (Niven).

Extensions

a(10) from Amiram Eldar, Jul 25 2022
a(11)-a(18) from David A. Corneth, Jul 27 2022

A356123 Least Lucas number with n Lucas divisors.

Original entry on oeis.org

1, 2, 4, 18, 1364, 1860498, 2537720636, 6440026026380244498, 8784200221406821330636, 77162173529763648886126034136172445632164498, 4365101043708483494615466932242949707161871659736799144058331102381689400753867700636
Offset: 1

Views

Author

Michel Marcus, Jul 27 2022

Keywords

Links

Crossrefs

Cf. A000032, A102460, A304092, A356062, A356122, A356666.
Cf. A076985 (similar for Fibonacci numbers).

Programs

  • PARI
    L(n)=fibonacci(n+1)+fibonacci(n-1); \\ A000032
isld(n) = { my(u1=1, u2=3, old_u1); if(n<=2, sign(n), while(n>u2, old_u1=u1; u1=u2; u2=old_u1+u2); (u2==n)); }; \\ A102460
nbld(n) = sumdiv(n, d, isld(d)); \\ A304092
a(n) = my(k=0); while(nbld(L(k)) != n, k++); L(k);
  • PARI
    countLd(n) = my(c=0,x=2,y=1); while(x <= n, if(n%x == 0, c++); [x,y]=[y,x+y]); c;
a(n) = if(n==1, return(1)); my(x=2,y=1); while(1, if(countLd(x) == n, return(x)); [x,y]=[y,x+y]); \\ Daniel Suteu, Aug 24 2022

Formula

a(n) = A000032(A356666(n)). - Daniel Suteu, Aug 24 2022

Extensions

a(11) from Daniel Suteu, Aug 06 2022

A356122 Number of Lucas divisors of the n-th Lucas number.

Original entry on oeis.org

2, 1, 2, 3, 2, 2, 4, 2, 2, 4, 3, 2, 4, 2, 3, 5, 2, 2, 5, 2, 3, 5, 3, 2, 4, 3, 3, 5, 3, 2, 6, 2, 2, 5, 3, 4, 5, 2, 3, 5, 3, 2, 6, 2, 3, 7, 3, 2, 4, 3, 4, 5, 3, 2, 6, 4, 3, 5, 3, 2, 6, 2, 3, 7, 2, 4, 6, 2, 3, 5, 5, 2, 5, 2, 3, 7, 3, 4, 6, 2, 3, 6, 3, 2, 6, 4, 3, 5, 3, 2, 8, 4, 3, 5, 3, 4, 4, 2, 4, 7
Offset: 0

Views

Author

Michel Marcus, Jul 27 2022

Keywords

Links

Crossrefs

Cf. A000032, A102460, A304092, A356062, A356123.
Cf. A076984 (similar for Fibonacci numbers).

Programs

  • PARI
    L(n)=fibonacci(n+1)+fibonacci(n-1); \\ A000032
isld(n) = { my(u1=1, u2=3, old_u1); if(n<=2, sign(n), while(n>u2, old_u1=u1; u1=u2; u2=old_u1+u2); (u2==n)); }; \\ A102460
nbld(n) = sumdiv(n, d, isld(d)); \\ A304092
a(n) = nbld(L(n));
  • PARI
    a(n) = if(n==1, return(1)); my(k=fibonacci(n+1)+fibonacci(n-1),c=0,x=2,y=1); while(x <= k, if(k%x == 0, c++); [x,y]=[y,x+y]); c; \\ Daniel Suteu, Aug 25 2022

Formula

a(n) = A304092(A000032(n)).

A356666 Smallest m such that the m-th Lucas number has exactly n divisors that are also Lucas numbers.

Original entry on oeis.org

1, 0, 3, 6, 15, 30, 45, 90, 105, 210, 405, 810, 315, 630, 3645, 2025, 945, 1890, 1575, 3150, 2835, 5670, 36450, 25025, 3465, 6930, 101250, 11025, 22050, 51030, 14175, 28350, 10395, 20790, 2952450, 175175, 17325, 34650, 1937102445, 625625, 31185, 62370, 127575, 255150
Offset: 1

Views

Author

Michel Marcus, Aug 22 2022

Keywords

Comments

Further terms <= 51030: a(28) = 11025, a(29) = 22050, a(30) = 51030, a(31) = 14175, a(32) = 28350, a(33) = 10395, a(34) = 20790, a(37) = 17325, a(38) = 34650, a(41) = 31185, a(49) = 45045. - Daniel Suteu, Aug 24 2022

Links

Crossrefs

Cf. A000032, A038547, A102460, A304092, A356123.
Cf. A105802 (similar for Fibonacci).

Programs

  • PARI
    L(n)=fibonacci(n+1)+fibonacci(n-1); \\ A000032
isld(n) = { my(u1=1, u2=3, old_u1); if(n<=2, sign(n), while(n>u2, old_u1=u1; u1=u2; u2=old_u1+u2); (u2==n)); }; \\ A102460
nbld(n) = sumdiv(n, d, isld(d)); \\ A304092
a(n) = my(k=0); while(nbld(L(k)) != n, k++); k;
  • PARI
    countLd(n) = my(c=0,x=2,y=1); while(x<=n, if(n%x==0, c++); [x,y]=[y,x+y]); c;
a(n) = if(n==1, return(1)); my(k=0,x=2,y=1); while(1, if(countLd(x) == n, return(k)); [x,y,k]=[y,x+y,k+1]); \\ Daniel Suteu, Aug 24 2022

Formula

A000032(a(n)) = A356123(n).

Extensions

a(12)-a(26) from Daniel Suteu, Aug 24 2022
More terms from Daniel Suteu and David A. Corneth, Sep 04 2022

A356063 a(n) is the new Lucas divisor that appears at the step A356062(n).

Original entry on oeis.org

1, 2, 4, 3, 18, 7, 11, 76, 322, 29, 1364, 123, 47, 199, 24476, 843, 5778, 521
Offset: 1

Views

Author

Bernard Schott, Jul 25 2022

Keywords

Comments

The sequence is not monotonic.
Conjecture: the sequence is well defined, i.e., it is not possible that two new Lucas divisors arrive while one disappears for some step in A356062.

Examples

			a(1) = 1 because the smallest integer that has only one Lucas divisor is 1 since 1 is the smallest Lucas number in A000032.
A356062(6) = 252 and the set of the six Lucas divisors of 252 is {1, 2, 3, 4, 7, 18}. Then, A356062(7) = 2772 and the set of the seven Lucas divisors of 2772 is {1, 2, 3, 4, 7, 11, 18}. The new Lucas divisor that appears in this set is 11, hence a(7) = 11.

Crossrefs

Cf. A000032, A304092, A349100, A356062.
Showing 1-8 of 8 results.

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

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