A061652 -id:A061652 - cp's OEIS frontend

A139237 Second differences of even superperfect numbers A061652, divided by 2.

5, 18, 1992, 28704, 67584, 536641536, 576460751229812736, 154742503757751030292414464, 40564509722294095963578081214464, 42535214735633635831150802674328272896

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

Second differences of Mersenne primes A000668, divided by 4 (see A139232).

Also, second differences of superperfect numbers A019279, divided by 2, if there are no odd superperfect numbers.

Paolo Xausa, Table of n, a(n) for n = 1..16

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139232, A139233, A139234, A139235, A139236.

Differences[2^MersennePrimeExponent[Range[14]]-1,2]/4 (* Paolo Xausa, Oct 20 2023 *)

a(n) = A139236(n)/2.

More terms from Michel Marcus, Jul 09 2017

A139235 First differences of even superperfect numbers A061652, divided by 2.

Original entry on oeis.org

1, 6, 24, 2016, 30720, 98304, 536739840, 576460751766552576, 154742504334211782058967040, 40564664464798430175360140181504, 42535255300298100629580978034468454400

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

First differences of Mersenne primes A000668, divided by 4 (see A139231).

Also, first differences of superperfect numbers A019279, divided by 2, if there are no odd superperfect numbers.

Paolo Xausa, Table of n, a(n) for n = 1..17

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139232, A139233, A139234, A139236, A139237.

Differences[2^MersennePrimeExponent[Range[14]]-1]/4 (* Paolo Xausa, Oct 20 2023 *)

a(n) = A139234(n)/2.

a(8)-a(11) from Jinyuan Wang, Mar 04 2020

A139236 Second differences of even superperfect numbers A061652.

Original entry on oeis.org

10, 36, 3984, 57408, 135168, 1073283072, 1152921502459625472, 309485007515502060584828928, 81129019444588191927156162428928, 85070429471267271662301605348656545792

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

Second differences of Mersenne primes A000668, divided by 2 (see A139232).

Also, second differences of superperfect numbers A019279, if there are no odd superperfect numbers.

Jinyuan Wang, Table of n, a(n) for n = 1..16

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139232, A139233, A139234, A139235, A139237.

Differences[2^(Select[Range[512],PrimeQ[2^#-1]&]-1),2] (* Harvey P. Dale, Oct 15 2017 *)

a(n) = A139234(n+1) - A139234(n).

a(6) corrected and more terms from Joerg Arndt, Jul 09 2017

A139234 First differences of even superperfect numbers A061652.

Original entry on oeis.org

2, 12, 48, 4032, 61440, 196608, 1073479680, 1152921503533105152, 309485008668423564117934080, 81129328929596860350720280363008, 85070510600596201259161956068936908800

Offset: 1

Omar E. Pol, Apr 18 2008

nonn

First differences of Mersenne primes A000668, divided by 2 (see A139231).

Also, first differences of superperfect numbers A019279, if there are no odd superperfect numbers.

			a(2) = 12 because A061652(2) = 4 and A061652(3) = 16 then 16 - 4 = 12.

Cf. A000668, A019279, A061652, A139228, A139229, A139230, A139231, A139232, A139233, A139235, A139236, A139237.

Differences[Table[2^(MersennePrimeExponent[n]-1),{n,12}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 18 2020 *)

a(n) = A061652(n+1) - A061652(n) = A139231(n)/2. Also, a(n) = A019279(n+1) - A019279(n), if there are no odd superperfect numbers.

a(8)-a(11) from A139231(n)/2 by Jinyuan Wang, Mar 04 2020

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.

A138842 Concatenation of initial and final digits of n-th even superperfect number A061652(n).

Original entry on oeis.org

22, 44, 16, 64, 46, 66, 24, 14, 16, 36, 84, 84, 36, 24, 54, 74, 26, 16, 96, 14, 26, 16, 16, 26, 26, 26, 46, 24, 24, 26, 34, 84, 66, 24, 46, 36, 66, 26, 46, 64, 14, 64, 16, 66, 14, 86, 16

Offset: 1

Omar E. Pol, Apr 02 2008

Also, concatenation of initial and final digits of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Also, concatenation of A138124(n) and A138125(n).

Cf. A019279, A061652, A073729, A138124, A138125, A138838, A138840, A138841, A138843, A138844.

More terms from Jinyuan Wang, Mar 14 2020

A138125 Final digit of n-th even superperfect number A061652(n).

Original entry on oeis.org

2, 4, 6, 4, 6, 6, 4, 4, 6, 6, 4, 4, 6, 4, 4, 4, 6, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 4, 4, 6, 4, 6, 6, 6, 6, 6, 4, 4, 4, 6, 6, 4, 6, 6

Offset: 1

Omar E. Pol, Apr 01 2008, corrected Apr 03 2008

Also, final digit of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

			a(5)=6 because the 5th even superperfect number A061652(5) is 4096 and the final digit of 4096 is 6.
a(34)=4 because the final digit of 34th Mersenne prime is 7. a(39)=6 because the final digit of 39th Mersenne prime is 1.
.............................................................
............... SHORT TABLE OF FINAL DIGITS .................
.............................................................
Final digit of ..... Final digit of Even ..... Final digit of
Mersenne prime ..... Superperfect number ..... Perfect number
A000668 ............ A061652 ................. A000396........
(3) ................ (2) ..................... (6) ........... (For n=1, only)
(7) ................ (4) ..................... (8) ...........
(1) ................ (6) ..................... (6) ...........

Landon Curt Noll, Mersenne Prime Digits.

Cf. A000030, A000396, A000668, A077648, A019279, A061652, A135613, A135617, A138125.

Mod[#,10]&/@(2^(MersennePrimeExponent[Range[47]]-1)) (* Harvey P. Dale, Feb 23 2023 *)

a(1)=2. For n>1, if final digit of n-th Mersenne prime A000668(n) is equal to 1 then a(n)=6, otherwise a(n)=4.

a(40)-a(47) from Jinyuan Wang, Mar 14 2020

A138816 Concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th even superperfect number A061652(n) and initial digit of n-th perfect number A000396(n).

Original entry on oeis.org

326, 742, 314, 168, 843, 168, 521, 212, 212, 631, 181, 181, 632, 521, 155

Offset: 1

Omar E. Pol, Apr 01 2008

Also, concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th superperfect number A019279(n) and initial digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.

Also, concatenation of A135613(n), A138124(n) and A135617(n).

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

Cf. A000396, A000668, A019279, A061652, A135613, A135617, A138124, A138817.

a(13)-a(15) from Robert Price, Jun 16 2019

A138818 Concatenation of initial digit of n-th even superperfect number A061652(n), initial digit of n-th Mersenne prime A000668(n) and initial digit of n-th perfect number A000396(n).

Original entry on oeis.org

236, 472, 134, 618, 483, 618, 251, 122, 122, 361, 811, 811

Offset: 1

Omar E. Pol, Apr 05 2008

Also, concatenation of initial digit of n-th superperfect number A019279(n), initial digit of n-th Mersenne prime A000668(n) and initial digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.

Also, concatenation of A138124(n), A135613(n) and A135617(n).

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, A135613, A135617, A138124, A138816, A138817, A138819, A138841, A138842, A138843.

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.

cp's OEIS Frontend

A139237 Second differences of even superperfect numbers A061652, divided by 2.

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A139235 First differences of even superperfect numbers A061652, divided by 2.

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A139236 Second differences of even superperfect numbers A061652.

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A139234 First differences of even superperfect numbers A061652.

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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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A138842 Concatenation of initial and final digits of n-th even superperfect number A061652(n).

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A138125 Final digit of n-th even superperfect number A061652(n).

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A138816 Concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th even superperfect number A061652(n) and initial digit of n-th perfect number A000396(n).

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A138818 Concatenation of initial digit of n-th even superperfect number A061652(n), initial digit of n-th Mersenne prime A000668(n) and initial digit of n-th perfect number A000396(n).