A080172 -id:A080172 - cp's OEIS frontend

A138817 Concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th even superperfect number A061652(n) and final digit of n-th perfect number A000396(n).

326, 748, 166, 748, 166, 166, 748, 748, 166, 166, 748, 748, 166, 748, 748, 748, 166, 166, 166, 748, 166, 166, 166, 166, 166, 166, 166, 748, 748, 166, 748, 748, 166, 748, 166, 166, 166, 166, 166

Offset: 1

Omar E. Pol, Apr 01 2008

Also, concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th superperfect number A019279(n) and final digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.
Also, concatenation of n-th term of A080172, A138125(n) and A094540(n).
a(1)=326. For n>1 a(n) is equal to 166 or 748, only.

L. C. Noll, Mersenne Prime Digits and Names.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A080172, A094540, A138125, A138817.

A138841 Concatenation of initial and final digit of n-th Mersenne prime A000668(n).

Original entry on oeis.org

33, 77, 31, 17, 81, 11, 57, 27, 21, 61, 17, 17, 61, 57, 17, 17, 41, 21, 11, 27, 41, 31, 21, 41, 41, 41, 81, 57, 57, 51, 77, 17, 11, 47, 81, 61, 11, 41, 91, 17, 27, 17, 31, 11, 27, 11, 31

Offset: 1

Omar E. Pol, Apr 01 2008

			a(5)=81 because the 5th Mersenne prime is 8191, A000668(5)=8191.

Cf. A000668, A073729, A080172, A135613, A138840, A138842, A138843.

a(n) = A073729(A000668(n)). - Michel Marcus, Apr 17 2018

a(40)-a(47) from Ivan Panchenko, Apr 17 2018

A080173 Final 2 digits of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 7, 31, 27, 91, 71, 87, 47, 51, 11, 27, 27, 51, 27, 87, 7, 51, 71, 91, 7, 11, 51, 91, 71, 51, 11, 71, 7, 7, 11, 47, 87, 91, 27, 11, 51, 71, 91, 71, 47, 7, 47, 71, 71, 27, 51, 11, 51

Offset: 1

Mark Dowdeswell, Feb 04 2003

Great Internet Mersenne Prime Search (GIMPS), Distributed Computing Projects
Mersenne Primes, History, Theorems and Lists

Cf. A080172, A000668, A000043.

Mod[2^MersennePrimeExponent[Range[48]]-1, 100] (* Mark Dowdeswell, Sep 16 2024 *)

Offset corrected by Arkadiusz Wesolowski, Jan 26 2012
a(39)-a(47) from Ivan Panchenko, Apr 11 2018
a(48) from Mark Dowdeswell, Sep 16 2024

A135613 Initial digit of Mersenne primes A000668.

Original entry on oeis.org

3, 7, 3, 1, 8, 1, 5, 2, 2, 6, 1, 1, 6, 5, 1, 1, 4, 2, 1, 2, 4, 3, 2, 4, 4, 4, 8, 5, 5, 5, 7, 1, 1, 4, 8, 6, 1, 4, 9, 1, 2, 1, 3, 1, 2, 1, 3, 5

Offset: 1

Omar E. Pol, Mar 01 2008

			a(4) = 1 because the 4th Mersenne prime A000668(4) is 127 and the initial digit of 127 is 1.

Landon Curt Noll, Mersenne Prime Digits.

Cf. A000030, A000668, A077648, A080172.

lst = {* the list of terms in A000043 *}; f[n_] := Block[{pn = 2^n - 1}, Quotient[pn, 10^Floor[ Log[10, pn]] ]]; f@# & /@ lst (* Robert G. Wilson v, Apr 01 2008 *)
IntegerDigits[#][[1]]&/@(2^#-1&/@MersennePrimeExponent[Range[47]]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 04 2019 *)

a(n) = A000030(A000668(n)). - Omar E. Pol, Jul 04 2019

More terms from Robert G. Wilson v, Apr 01 2008
a(40)-a(44) from David Radcliffe, Jan 21 2016
a(45)-a(47) from Ivan Panchenko, Apr 11 2018
a(48) from Amiram Eldar, Oct 16 2024

A138819 Concatenation of final digit of n-th even superperfect number A061652(n), final digit of n-th Mersenne prime A000668(n) and final digit of n-th perfect number A000396(n).

Original entry on oeis.org

236, 478, 616, 478, 616, 616, 478, 478, 616, 616, 478, 478, 616, 478, 478, 478, 616, 616, 616, 478, 616, 616, 616, 616, 616, 616, 616, 478, 478, 616, 478, 478, 616, 478, 616, 616, 616, 616, 616

Offset: 1

Omar E. Pol, Apr 05 2008

Also, concatenation of final digit of n-th superperfect number A019279(n), final digit of n-th Mersenne prime A000668(n) and final digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.
Also, concatenation of A138125(n), A080172(n) and A094540(n).
For n>1 a(n) is equal to 478 or 616, only.
Note that, for n>1: if the final digit of n-th Mersenne prime A000668(n) is 1 then the final digit of n-th even superperfect number is 6 and the final digit of n-th perfect number also is 6, otherwise the final digit of n-th even superperfect number is 4 and the final digit of n-th perfect number is 8 (see example).

			===================================================================
.................. SHORT TABLE OF FINAL DIGITS ...................
===================================================================
... Final digit of even ..... Final digit of ..... Final digit of
... superperfect number ..... Mersenne prime ..... perfect number
........ A061652 ............... A000668 ............. A000396
===================================================================
n = 1 ..... (2) ................... (3) .................. (6)
n > 1 ..... (4) ................... (7) .................. (8)
n > 1 ..... (6) ................... (1) .................. (6)

L. C. Noll, Mersenne Prime Digits and Names.
J. M. Pedersen, Known Perfect Numbers.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A080172, A094540, A138125, A138816, A138817, A138818, A138841, A138842, A138843.

A267317 a(n) = final digit of 2^n-1.

Original entry on oeis.org

0, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5, 1, 3, 7, 5

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

Decimal expansion of 25/1818.

Period 4: repeat [1, 3, 7, 5] for n > 0.

Eric Weisstein's World of Mathematics, Mersenne Number
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000225, A000689, A010688, A010703, A010879, A080172.

[0] cat &cat[[1, 3, 7, 5]^^25]; // Bruno Berselli, Jan 13 2016

A267317:=n->(2^n-1) mod 10: seq(A267317(n), n=0..150); # Wesley Ivan Hurt, Jun 15 2016

Mathematica
```
Table[Mod[2^n - 1, 10], {n, 0, 120}]
```

a(n) = if(n==0, 0, if(n%4==0, 5, if(n%4==1, 1, if(n%4==2, 3, if(n%4==3, 7))))) \\ Felix Fröhlich, Jan 19 2016

a(n) = lift(Mod(2^n-1, 10)) \\ Felix Fröhlich, Jan 19 2016

G.f.: x*(1 + 2*x + 5*x^2)/(1 - x + x^2 - x^3).
a(n) = A010879(A000225(n)).
a(n) = A000689(n) - 1.
a(n) = (1+(-1)^n)*(-1)^(n*(n-1)/2)/2 + 3*(1-(-1)^n)*(-1)^(n*(n+1)/2)/2 + 4 for n > 0, a(0) = 0. [Bruno Berselli, Jan 13 2016]
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = a(n-4) for n>4.
a(2k+2) = A010703(k), a(2k+1) = A010688(k). (End)
From Wesley Ivan Hurt, Jul 06 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 3.
a(n) = 4 + cos(n*Pi/2) - 3*sin(n*Pi/2) for n > 0. (End)
E.g.f.: -5 + cos(x) - 3*sin(x) + 4*exp(x). - Ilya Gutkovskiy, Jul 06 2016

cp's OEIS Frontend

A138817 Concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th even superperfect number A061652(n) and final digit of n-th perfect number A000396(n).

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A138841 Concatenation of initial and final digit of n-th Mersenne prime A000668(n).

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A080173 Final 2 digits of n-th Mersenne prime A000668(n).

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A135613 Initial digit of Mersenne primes A000668.

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A138819 Concatenation of final digit of n-th even superperfect number A061652(n), final digit of n-th Mersenne prime A000668(n) and final digit of n-th perfect number A000396(n).

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A267317 a(n) = final digit of 2^n-1.

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