A094540 -id:A094540 - cp's OEIS frontend

A138843 Concatenation of initial and final digits of n-th perfect number.

66, 28, 46, 88, 36, 86, 18, 28, 26, 16, 18, 18, 26, 18, 58, 18, 96, 36, 16, 48, 16, 56, 36, 96, 16, 86, 36, 18, 18, 16, 28, 18, 86, 88, 36, 16, 86, 96, 46

Offset: 1

Omar E. Pol, Apr 02 2008

Also, concatenation of A135617(n) and A094540(n).

			a(5)=36 because the 5th perfect number A000396(5) is 33550336 and the concatenation of initial and final digits of 33550336 is 36.

Cf. A000396, A073729, A135617, A094540, A138840, A138841, A138842, A138844.

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).

Original entry on oeis.org

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.

A104511 Last 3 digits of the n-th even perfect number.

Original entry on oeis.org

6, 28, 496, 128, 336, 56, 328, 128, 176, 216, 128, 128, 976, 128, 328, 528, 776, 56, 536, 528, 216, 576, 336, 656, 376, 816, 456, 528, 528, 16, 128, 328, 936, 128, 616, 976, 856, 736, 56, 128, 528, 128, 256, 256, 128, 376, 816, 176

Offset: 1

Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v, Feb 23 2005

Whether a perfect number ends in 6 or 28, the preceding digit is odd except for the two initial terms.

All terms except the first two are divisible by 8. - Iain Fox, Dec 06 2017

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.

Cf. A094540, A000396.

See A000043 for the present state of knowledge about Mersenne primes.

p=MersennePrimeExponent[Range[45]]; Mod[(PowerMod[2, p, 1000] - 1)(PowerMod[2, p - 1, 1000]), 1000] (* edited by Iain Fox, Dec 06 2017 *)

a(p) = lift(Mod((Mod(2, 1000)^p - 1)*Mod(2, 1000)^(p-1), 1000)) \\ (where p is the n-th Mersenne exponent A000043) Iain Fox, Dec 04 2017

Clarified definition and extended by Ivan Panchenko, Aug 05 2014
a(44)-a(45) from Iain Fox, Dec 04 2017
a(46)-a(47) from Ivan Panchenko, Apr 17 2018
a(48) from Iain Fox, Oct 25 2022

A135617 a(n) is the initial digit of n-th even perfect number.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 01 2008

a(n) is also the initial digit of n-th perfect number A000396(n) if there are no odd perfect numbers.

			a(5) = 3 because the 5th even perfect number is 33550336 and the initial digit of 33550336 is 3.

Omar E. Pol, Los números perfectos, (in Spanish).

Cf. A000030, A000396, A077648, A094540, A133033.

lst = {* the list of terms in A000043 *}; f[n_] := Block[{pn = (2^n - 1) (2^(n - 1))}, Quotient[pn, 10^Floor[ Log[10, pn]] ]]; f@# & /@ lst (* Robert G. Wilson v, Apr 01 2008 *)

More terms from Robert G. Wilson v, Apr 01 2008
Definition clarified by Omar E. Pol, Apr 14 2018
a(40)-a(47) from Ivan Panchenko, Apr 16 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.

A138839 Concatenation of initial and final digits of n-th perfect number, divided by 2.

Original entry on oeis.org

33, 14, 23, 44, 18, 43, 9, 14, 13, 8, 9, 9, 13, 9, 29, 9, 48, 18, 8, 24, 8, 28, 18, 48, 8, 43, 18, 9, 9, 8, 14, 9, 43, 44, 18, 8, 43, 48, 23

Offset: 1

Omar E. Pol, Apr 02 2008

Also, concatenation of A135617(n) and A094540(n), divided by 2.

			a(5)=18 because the 5th perfect number A000396(5) is 33550336 and the concatenation of initial and final digits of 33550336 is 36 and 36/2 = 18.

Cf. A000396, A073729, A135617, A094540, A138843.

cp's OEIS Frontend

A138843 Concatenation of initial and final digits of n-th perfect number.

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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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A104511 Last 3 digits of the n-th even perfect number.

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A135617 a(n) is the initial digit of n-th even perfect number.

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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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A138839 Concatenation of initial and final digits of n-th perfect number, divided by 2.

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