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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A321970 Numbers k such that 7^k ends with k.

Original entry on oeis.org

3, 43, 343, 2343, 72343, 172343, 5172343, 65172343, 565172343, 1565172343, 11565172343, 511565172343, 5511565172343, 65511565172343, 265511565172343, 1265511565172343, 31265511565172343, 331265511565172343, 3331265511565172343, 43331265511565172343
Offset: 1

Views

Author

Ivan Stoykov, Nov 26 2018

Keywords

Comments

Leftmost digit of a(n) is A133617(n-1) for n <= 30. - Alois P. Heinz, Nov 26 2018

Examples

			7^3 = 343, and it ends with 3, so 3 is a term.

Links

Crossrefs

Cf. A133617.
Sequence A064541 is similar, but uses the smallest single-digit prime as a base, unlike this one, which uses the largest single-digit prime as a base.

Programs

  • Mathematica
    a[1] = 3; a[n_] := a[n] = For[ida = IntegerDigits[a[n-1]]; k = 1, True, k++, idk = IntegerDigits[k]; pm = PowerMod[7, an = FromDigits[Join[idk, ida]], 10^IntegerLength[an]]; If[pm == an, Return[an]]]; Array[a, 20] (* after Jean-François Alcover in A064541 *)

A067869 Numbers n such that n and 2^n end with the same 5 digits.

Original entry on oeis.org

48736, 148736, 248736, 348736, 448736, 548736, 648736, 748736, 848736, 948736, 1048736, 1148736, 1248736, 1348736, 1448736, 1548736, 1648736, 1748736, 1848736, 1948736, 2048736, 2148736, 2248736, 2348736, 2448736, 2548736
Offset: 1

Views

Author

Benoit Cloitre, Mar 07 2002

Keywords

Links

Crossrefs

Cf. A064541.
Subsequence of A067844, A067845, A067846, and A067847.

Programs

  • PARI
    isok(n) = (2^n - n) % 100000 == 0; \\ Michel Marcus, Nov 23 2013

Formula

a(n) = 48736+10^5(n-1).
a(n) = 2*a(n-1)-a(n-2). G.f.: x*(48736+51264*x)/(1-x)^2. - Colin Barker, Jun 05 2012

A067865 Numbers n such that n and 2^n end with the same two digits.

Original entry on oeis.org

36, 136, 236, 336, 436, 536, 636, 736, 836, 936, 1036, 1136, 1236, 1336, 1436, 1536, 1636, 1736, 1836, 1936, 2036, 2136, 2236, 2336, 2436, 2536, 2636, 2736, 2836, 2936, 3036, 3136, 3236, 3336, 3436, 3536, 3636, 3736, 3836, 3936, 4036, 4136, 4236, 4336
Offset: 1

Views

Author

Benoit Cloitre, Mar 07 2002

Keywords

Comments

2^36=68719476736 hence 36 is in the sequence.

Links

Crossrefs

Cf. A064541.
Subsequence of A067844.

Programs

  • PARI
    isok(n) = (2^n - n) % 100 == 0; \\ Michel Marcus, Nov 23 2013

Formula

a(n) = 36+100(n-1).
a(n) = 2*a(n-1)-a(n-2). G.f.: 4*x*(9+16*x)/(1-x)^2. [Colin Barker, Dec 01 2012]

A067866 Numbers n such that n and 2^n end with the same three digits.

Original entry on oeis.org

736, 1736, 2736, 3736, 4736, 5736, 6736, 7736, 8736, 9736, 10736, 11736, 12736, 13736, 14736, 15736, 16736, 17736, 18736, 19736, 20736, 21736, 22736, 23736, 24736, 25736, 26736, 27736, 28736, 29736, 30736, 31736, 32736, 33736, 34736
Offset: 1

Views

Author

Benoit Cloitre, Mar 07 2002

Keywords

Links

Crossrefs

Cf. A064541.
Subsequence of A067844 and A067845.

Programs

  • PARI
    isok(n) = (2^n - n) % 1000 == 0; \\ Michel Marcus, Nov 23 2013

Formula

a(n) = 736 + 1000(n-1).
a(n) = 2*a(n-1)-a(n-2). G.f.: 8*x*(92+33*x)/(1-x)^2. [Colin Barker, Dec 01 2012]

A067867 Numbers n such that n and 2^n end with the same 4 digits.

Original entry on oeis.org

8736, 18736, 28736, 38736, 48736, 58736, 68736, 78736, 88736, 98736, 108736, 118736, 128736, 138736, 148736, 158736, 168736, 178736, 188736, 198736, 208736, 218736, 228736, 238736, 248736, 258736, 268736, 278736, 288736, 298736, 308736
Offset: 1

Views

Author

Benoit Cloitre, Mar 07 2002

Keywords

Links

Crossrefs

Cf. A064541.
Subsequence of A067844, A067845 and A067846.

Programs

  • PARI
    isok(n) = (2^n - n) % 10000 == 0; \\ Michel Marcus, Nov 23 2013

Formula

a(n) = 8736 + 10^4(n-1).
a(n) = 2*a(n-1)-a(n-2). G.f.: 16*x*(546+79*x)/(1-x)^2. [Colin Barker, Dec 01 2012]

A351410 Numbers m such that the decimal representation of 8^m ends in m.

Original entry on oeis.org

56, 856, 5856, 25856, 225856, 5225856, 95225856, 895225856, 6895225856, 16895225856, 416895225856, 5416895225856, 35416895225856, 7035416895225856, 77035416895225856, 577035416895225856, 1577035416895225856, 21577035416895225856, 521577035416895225856, 1521577035416895225856, 81521577035416895225856
Offset: 1

Views

Author

Bernard Schott, Feb 10 2022

Keywords

Comments

The Crux Mathematicorum link calls these numbers "expomorphic" relative to "base" b, with here b = 8.
Under that definition, the term after a(13) = 35416895225856 is not "035416895225856" or "35416895225856" but a(14) = 7035416895225856.
Conjecture: if k(n) is "expomorphic" relative to "base" b, then the next one in the sequence, k(n+1), consists of the last n+1 digits of b^k(n).
This conjecture is true. See A133618. - David A. Corneth, Feb 10 2022

Examples

			8^56 = 374144419156711147060143317175368453031918731001856, so 56 is a term.
8^856 = ...5856 ends in 856, so 856 is another term.

Links

  • Charles W. Trigg, Problem 559, Crux Mathematicorum, pp. 192-194, Vol. 7, Jun. 1981.

Crossrefs

Cf. A064541, A183613, A288845, A306570, A290788, A321970, this sequence, A306686, A289138.
Cf. A003226 (automorphic numbers), A033819 (trimorphic numbers).
Cf. A133618 (leading digits).

Extensions

a(7)-a(8) from Michel Marcus, Feb 10 2022
More terms from David A. Corneth, Feb 10 2022
Previous Showing 11-16 of 16 results.

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