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.

Previous Showing 31-33 of 33 results.

A258929 a(n) is the unique even-valued residue modulo 5^n of a number m such that m^2+1 is divisible by 5^n.

Original entry on oeis.org

2, 18, 68, 182, 1068, 1068, 32318, 280182, 280182, 3626068, 23157318, 120813568, 1097376068, 1097376068, 11109655182, 49925501068, 355101282318, 355101282318, 15613890344818, 15613890344818, 365855836217682, 2273204469030182, 2273204469030182, 49956920289342682
Offset: 1

Views

Author

Jon E. Schoenfield, Jun 15 2015

Keywords

Comments

For any positive integer n, if a number of the form m^2+1 is divisible by 5^n, then m mod 5^n must take one of two values--one even, the other odd. This sequence gives the even residue. (The odd residues are in A259266.)

Examples

			If m^2+1 is divisible by 5, then m mod 5 is either 2 or 3; the even value is 2, so a(1)=2.
If m^2+1 is divisible by 5^2, then m mod 5^2 is either 7 or 18; the even value is 18, so a(2)=18.
If m^2+1 is divisible by 5^3, then m mod 5^3 is either 57 or 68; the even value is 68, so a(3)=68.

Crossrefs

Cf. A048898, A048899, A257366, A259266.

Extensions

More terms and additional comments from Jon E. Schoenfield, Jun 23 2015

A259266 a(n) is the unique odd-valued residue modulo 5^n of a number m such that m^2+1 is divisible by 5^n.

Original entry on oeis.org

3, 7, 57, 443, 2057, 14557, 45807, 110443, 1672943, 6139557, 25670807, 123327057, 123327057, 5006139557, 19407922943, 102662389557, 407838170807, 3459595983307, 3459595983307, 79753541295807, 110981321985443, 110981321985443, 9647724486047943, 9647724486047943
Offset: 1

Views

Author

Jon E. Schoenfield, Jun 23 2015

Keywords

Comments

For any positive integer n, if a number of the form m^2+1 is divisible by 5^n, then m mod 5^n must take one of two values--one even, the other odd. This sequence gives the odd residue. (The even residues are in A258929.)

Examples

			If m^2+1 is divisible by 5, then m mod 5 is either 2 or 3; the odd value is 3, so a(1)=3.
If m^2+1 is divisible by 5^2, then m mod 5^2 is either 7 or 18; the odd value is 7, so a(2)=7.
If m^2+1 is divisible by 5^3, then m mod 5^3 is either 57 or 68; the odd value is 57, so a(3)=57.

Crossrefs

Cf. A048898, A048899, A257366, A258929.

A051276 Nonzero coefficients in one of the 5-adic expansions of sqrt(-1).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Examples

			2 + 1*5 + 2*5^2 + 1*5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + 2*5^10 + 2*5^11 + 4*5^13 + 1*5^14 + 3*5^15 + 2*5^16 + 4*5^17 + 4*5^19 + ...

References

  • Kurt Mahler, Introduction to p-adic numbers and their functions. Cambridge Tracts in Mathematics, 76. Cambridge University Press, Cambridge-New York, 1971. See pp. 35ff.

Crossrefs

Cf. A034935, A048898, A048899, A210850, A210851.

Programs

  • Maple
    R:= select(t -> padic:-ratvaluep(t,1)=2,[padic:-rootp(x^2+1,5,200)]):
subs(0=NULL,op([1,1,3],R)); # Robert Israel, Mar 04 2016
  • PARI
    sqrt(-1+O(5^100))

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com) and Jason Earls, Jun 15 2001
Name corrected by Robert Israel at the suggestion of Wolfdieter Lang, Mar 04 2016
Previous Showing 31-33 of 33 results.

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

Content is available under The OEIS End-User License Agreement.