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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A079665 Triangular array read by rows: row s contains integers of the form (2^s+1)/(2^r+1) in order of increasing r <= s-1.

Original entry on oeis.org

3, 11, 13, 43, 171, 57, 205, 683, 241, 2731, 3277, 10923, 3641, 993, 43691, 52429, 4033, 174763, 61681, 699051, 233017, 16257, 838861, 2796203, 65281, 11184811, 1016801, 13421773, 44739243, 14913081, 261633, 15790321, 178956971, 214748365
Offset: 1

Views

Author

Jose R. Brox (tautocrona(AT)terra.es), Jan 25 2003

Keywords

Comments

Conjecture: (b^s+1)/(b^r+1) is an integer if and only if: 1) r

Examples

			s=2: no terms
s=3: 3
s=4: no terms
s=5: 11
s=6: 13
s=7: 43
s=8: no terms
s=9: 171, 57
...

Crossrefs

Cf. A079672, A079581, A079673, A370425.

Programs

  • PARI
    for(x=2,30, for(y=1,x-1,if(Mod(2^x+1,2^y+1),0,print1((2^x+1)\(2^y+1)","))))

Extensions

Definition corrected by Max Alekseyev, Feb 18 2024

A281728 Johnson's non-Wieferich numbers of the first kind.

Original entry on oeis.org

3, 11, 13, 43, 241, 683, 2731, 43691, 61681, 174763, 2796203, 15790321, 715827883, 4278255361, 2932031007403, 4363953127297
Offset: 1

Views

Author

Felix Fröhlich, Jan 28 2017

Keywords

Comments

This is the case a = 2 of primes p such that p-1 has the a-adic expansion bb...b00...0bb...b00...0_a, where b = a-1 with each of the t blocks of digits b or 0 having length k and additionally q_a == (a^k + 1)/(t + 1)*k =/= 0 (mod p), where q_a denotes the Fermat quotient to base a (cf. Johnson, 1977).
These are prime numbers of the form (2^m + 1)/(2^n + 1). Note that if m,n > 0, then 2^n + 1 divides 2^m + 1 if and only if m/n is odd. - Thomas Ordowski, Feb 17 2024

Examples

			(2^49+1)/(2^7+1) = 4363953127297 = 111111100000001111111000000011111110000001.

Links

Crossrefs

Cf. A370425 (integers of the form (2^m+1)/(2^n+1)).
Showing 1-2 of 2 results.

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

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