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 1-4 of 4 results.

A283092 The smallest square referenced in A280296 (Squarefree numbers n such that 2^n - 1 is divisible by a square > 1).

Original entry on oeis.org

9, 49, 9, 9, 9, 9, 9, 49, 121, 9, 9, 961, 9, 9, 9, 9, 49, 9, 529, 9, 49, 9, 961, 9, 9, 9, 49, 9, 9, 49, 9, 9, 9, 9, 961, 9, 49, 529, 9, 9, 9, 9, 9, 1849, 9, 49, 9, 9, 49, 9, 9, 9, 9, 529, 9, 121, 49, 9, 9, 9
Offset: 1

Views

Author

Robert Price, Mar 06 2017

Keywords

Examples

			A280296(1)=6, 2^6-1 = 63 = 3*3*7, so 63 is not squarefree the square being 3*3=9.

Crossrefs

Cf. A280296, A283245.

A283245 Square root of the smallest square referenced in A280296 (Squarefree numbers n such that 2^n - 1 is divisible by a square > 1).

Original entry on oeis.org

3, 7, 3, 3, 3, 3, 3, 7, 11, 3, 3, 31, 3, 3, 3, 3, 7, 3, 23, 3, 7, 3, 31, 3, 3, 3, 7, 3, 3, 7, 3, 3, 3, 3, 31, 3, 7, 23, 3, 3, 3, 3, 3, 43, 3, 7, 3, 3, 7, 3, 3, 3, 3, 23, 3, 11, 7, 3, 3, 3
Offset: 1

Views

Author

Robert Price, Mar 06 2017

Keywords

Examples

			A280296(1)=6, 2^6-1 = 63 = 3*3*7, so 63 is not squarefree the square being 3*3=9 and the root is 3.

Crossrefs

Cf. A280296, A283092.

A372541 Least k such that the k-th squarefree number has exactly n ones in its binary expansion.

Original entry on oeis.org

1, 3, 6, 11, 20, 60, 78, 157, 314, 624, 1245, 3736, 4982, 9962, 19920, 39844, 79688, 239046, 318725, 956194, 1912371, 2549834, 5099650, 15298984, 20398664, 40797327, 81594626, 163189197, 326378284, 979135127, 1305513583, 2611027094, 5222054081, 10444108051
Offset: 0

Views

Author

Gus Wiseman, May 09 2024

Keywords

Examples

			The squarefree numbers A005117(a(n)) together with their binary expansions and binary indices begin:
       1:                   1 ~ {1}
       3:                  11 ~ {1,2}
       7:                 111 ~ {1,2,3}
      15:                1111 ~ {1,2,3,4}
      31:               11111 ~ {1,2,3,4,5}
      95:             1011111 ~ {1,2,3,4,5,7}
     127:             1111111 ~ {1,2,3,4,5,6,7}
     255:            11111111 ~ {1,2,3,4,5,6,7,8}
     511:           111111111 ~ {1,2,3,4,5,6,7,8,9}
    1023:          1111111111 ~ {1,2,3,4,5,6,7,8,9,10}
    2047:         11111111111 ~ {1,2,3,4,5,6,7,8,9,10,11}
    6143:       1011111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,13}
    8191:       1111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13}
   16383:      11111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14}
   32767:     111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
   65535:    1111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}
  131071:   11111111111111111 ~ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}

Links

Crossrefs

Positions of firsts appearances in A372433.
Counting zeros instead of ones gives A372473, firsts in A372472.
For prime instead of squarefree we have A372517, firsts of A014499.
Counting bits (length) gives A372540, firsts of A372475, runs A077643.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A145037, A097110 count ones minus zeros, for primes A372516, A177796.
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
Cf. A023416, A049093, A049094, A069010, A070939, A071403, A211997, A280296, A372474.

Programs

  • Mathematica
    nn=10000;
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
dcs=DigitCount[Select[Range[nn],SquareFreeQ],2,1];
Table[Position[dcs,i][[1,1]],{i,spnm[dcs-1]}]
  • Python
    from math import isqrt
from itertools import count
from sympy import factorint, mobius
from sympy.utilities.iterables import multiset_permutations
def A372541(n):
    if n==0: return 1
    for l in count(n):
        m = 1<Chai Wah Wu, May 10 2024

Extensions

a(23)-a(33) from Chai Wah Wu, May 10 2024

A280302 Smallest k such that (n+1)^k - n^k is divisible by a square > 1.

Original entry on oeis.org

6, 10, 4, 2, 21, 20, 3, 20, 33, 6, 20, 2, 2, 5, 21, 6, 10, 6, 6, 4, 4, 2, 7, 2, 6, 3, 10, 4, 18, 6, 2, 10, 20, 6, 57, 17, 2, 14, 42, 2, 10, 10, 6, 39, 14, 4, 10, 20, 2, 21, 20, 6, 4, 21, 6, 20, 10, 2, 5, 2, 5, 2, 20, 6, 42, 14, 2, 6, 55, 6, 3, 7, 2, 42, 3, 2
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 31 2016

Keywords

Comments

a(209) > 70.
a(n) <= p^2 - p, where p = A053670(n). - Jinyuan Wang, May 15 2020

Examples

			a(1) = 6 is because (1+1)^6 - 1^6 = 63 is divisible by 9 = 3^2.

Links

Crossrefs

Cf. A049094, A053670, A237043, A280203, A280208, A280209, A280296, A282176, A282177, A280307.
Cf. A289629, A280547, A289985.

Programs

  • PARI
    a(n) = {my(k = 1); while (issquarefree((n+1)^k - n^k), k++); k;} \\ Michel Marcus, Jan 14 2017

Extensions

More terms from Lars Blomberg, Jan 10 2017
Showing 1-4 of 4 results.

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