A000668 -id:A000668 - cp's OEIS frontend

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).

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.

A138863 Concatenation of first two digits and last two digits of n-th Mersenne prime A000668(n).

Original entry on oeis.org

33, 77, 3131, 1227, 8191, 1371, 5287, 2147, 2351, 6111, 1627, 1727, 6851, 5327, 1087, 1407, 4451, 2571, 1991, 2807, 4711, 3451, 2891, 4371, 4451, 4011, 8571, 5307, 5207, 5111, 7447, 1787, 1291, 4127, 8111, 6251, 1271, 4391, 9271, 1247, 2907, 1247, 3171, 1271, 2027, 1651, 3111

Offset: 1

Omar E. Pol, Apr 02 2008

L. C. Noll, Mersenne Prime Digits.

Cf. A000668, A080173, A104511, A138816, A138817, A138840, A138841, A138862, A138864, A138865, A138866.

More terms from Jinyuan Wang, Mar 14 2020

A139223 M*(M-1), where M is Mersenne prime A000668(n).

Original entry on oeis.org

6, 42, 930, 16002, 67084290, 17179475970, 274876334082, 4611686011984936962, 5316911983139663484697699213480296450, 383123885216472214589586754930667236976614368197214210

Offset: 1

Omar E. Pol, May 10 2008

nonn

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000668, A036689, A139115, A139116, A139224, A139225, A139226, A139256.

a(n) = A000668(n)*(A000668(n)-1).

More terms from R. J. Mathar, Jun 24 2009

A139225 M(M-1)/3, where M is Mersenne prime A000668(n).

Original entry on oeis.org

2, 14, 310, 5334, 22361430, 5726491990, 91625444694, 1537228670661645654, 1772303994379887828232566404493432150, 127707961738824071529862251643555745658871456065738070, 8776024305713098891493168973639040433964428736682367693182293334, 9649340769776349618630915417390658987602357538676244438223111363610210030934

Offset: 1

Omar E. Pol, May 10 2008

nonn

Terms from a(13) on have 314 or more digits and are not listed for that reason. - R. J. Mathar, May 11 2008

Cf. A000668, A036689, A139115, A139116, A139223, A139224, A139226.

a(n)=A000668(n)*(A000668(n)-1)/3.

More terms from R. J. Mathar, May 11 2008

A139226 M(M-1)/6, where M is Mersenne prime A000668(n).

Original entry on oeis.org

1, 7, 155, 2667, 11180715, 2863245995, 45812722347, 768614335330822827, 886151997189943914116283202246716075, 63853980869412035764931125821777872829435728032869035, 4388012152856549445746584486819520216982214368341183846591146667, 4824670384888174809315457708695329493801178769338122219111555681805105015467

Offset: 1

Omar E. Pol, May 10 2008

nonn

Perfect number A000396(n) minus Mersenne prime A000668(n), divided by 3.

Terms from a(13) on have 313 or more digits and are not listed for that reason. - R. J. Mathar, May 11 2008

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000217, A000396, A000668, A036689, A139096, A139115, A139116, A139223, A139224, A139225, A139256, A139306.

a(n) = A000668(n)*(A000668(n)-1)/6 = A139223(n)/6 = A139224(n)/3.

a(n) = (A000396(n)-A000668(n))/3.

More terms from R. J. Mathar, May 11 2008

A146768 a(n) is the number k such that 2^(2k+1)-1 = A000668(n+1).

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 15, 30, 44, 53, 63, 260, 303, 639, 1101, 1140, 1608, 2126, 2211, 4844, 4970, 5606, 9968, 10850, 11604, 22248, 43121, 55251, 66024, 108045, 378419, 429716, 628893, 699134, 1488110, 1510688, 3486296, 6733458, 10498005, 12018291, 12982475, 15201228

Offset: 1

Artur Jasinski, Nov 02 2008

nonn

The least common multiple of an even superperfect number greater than 2 and its arithmetic derivative divided by the number itself, i.e., lcm(A061652(i), A061652(i)')/A061652(i). - Giorgio Balzarotti, Apr 21 2011

Amiram Eldar, Table of n, a(n) for n = 1..46
C. K. Caldwell, Top 20 Mersenne primes
Bernhard Helmes, Prime generator f(n)=2n^2-1
George Woltman, Great Internet Mersenne Prime Search

Cf. A000043, A000668, A061652.

(MersennePrimeExponent[Range[2, 47]] - 1)/2 (* Amiram Eldar, Mar 29 2020 *)

a(n) = (A000043(n+1) - 1)/2.

2^(2*a(n) + 1) - 1 = A000668(n+1). - M. F. Hasler, Jan 27 2020

Term for the 39th Mersenne prime added by Roderick MacPhee, Oct 05 2009
Formula and edits from Charles R Greathouse IV, Aug 14 2010
Updated to include 40th Mersenne prime by Michael B. Porter, Nov 26 2010
a(40)-a(42) from Amiram Eldar, Mar 29 2020

A153474 Sum of first n Mersenne primes A000668.

Original entry on oeis.org

3, 10, 41, 168, 8359, 139430, 663717, 2148147364, 2305843011361841315, 618970021948533148811403426, 162259895799235311924726821691553, 170141345720365030966999228442705797280

Offset: 1

Omar E. Pol, Dec 27 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..18

Cf. A007504, A092336, A000668, A153475.

Accumulate[2^MersennePrimeExponent[Range[15]]-1] (* Harvey P. Dale, Feb 28 2023 *)

More terms from Max Alekseyev, Apr 13 2009

A171251 Sums of two (not necessarily distinct) Mersenne primes (A000668).

Original entry on oeis.org

6, 10, 14, 34, 38, 62, 130, 134, 158, 254, 8194, 8198, 8222, 8318, 16382, 131074, 131078, 131102, 131198, 139262, 262142, 524290, 524294, 524318, 524414, 532478, 655358, 1048574, 2147483650, 2147483654, 2147483678, 2147483774, 2147491838

Offset: 1

M. F. Hasler, Mar 06 2010

nonn

All terms are even, since all terms of A000668 are odd. This motivates to introduce A171252 = (1/2) * A171251, see there for further information.

Jonathan Vos Post, Sums of three Mersenne primes, and prime sums of three Mersenne primes, SeqFan list, Mar 5, 2010.

Cf. A000668, A171252.

With[{mps=Select[2^Range[50]-1,PrimeQ]},Union[Total/@Tuples[mps,2]]] (* Harvey P. Dale, Aug 11 2012 *)
With[{mps=2^MersennePrimeExponent[Range[10]]-1},Union[Total/@Tuples[mps,2]]] (* Harvey P. Dale, Mar 20 2025 *)

concat(vector(#A000668,i,vector(i,j,A000668[i]+A000668[j]))) /* having defined A000668 as vector with initial terms of A000668 */

A171251(n) = 2*A171252(n) = A000668(i) + A000668(j) where n = i*(i-1)/2+j.

A171253 Semi-sums (average) of any two distinct Mersenne primes (A000668).

Original entry on oeis.org

5, 17, 19, 65, 67, 79, 4097, 4099, 4111, 4159, 65537, 65539, 65551, 65599, 69631, 262145, 262147, 262159, 262207, 266239, 327679, 1073741825, 1073741827, 1073741839, 1073741887, 1073745919, 1073807359, 1074003967

Offset: 1

M. F. Hasler, Mar 06 2010

nonn

Subsequence of A171252, excluding the terms of A000668 itself.

			a(1) = (A000668(2)+A000668(1))/2, a(2) = (A000668(3)+A000668(1))/2, a(3) = (A000668(3)+A000668(2))/2, a(4) = (A000668(4)+A000668(1))/2, ,...

Jonathan Vos Post, Sums of three Mersenne primes, and prime sums of three Mersenne primes, SeqFan list, Mar 5, 2010.

Cf. A171252 (includes terms of A000668), A171255 (primes in this sequence).

concat(vector(#A000668,i,vector(i-1,j,A000668[i]+A000668[j])))/2 /* having defined A000668 as vector with initial terms of A000668 */

a(n) = A171252(A014132(n)).

A171253 = A171252 \ A000688.

cp's OEIS Frontend

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).

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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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A138863 Concatenation of first two digits and last two digits of n-th Mersenne prime A000668(n).

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A139223 M*(M-1), where M is Mersenne prime A000668(n).

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A139225 M(M-1)/3, where M is Mersenne prime A000668(n).

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A139226 M(M-1)/6, where M is Mersenne prime A000668(n).

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A146768 a(n) is the number k such that 2^(2k+1)-1 = A000668(n+1).

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Formula

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A153474 Sum of first n Mersenne primes A000668.

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Mathematica

Extensions

A171251 Sums of two (not necessarily distinct) Mersenne primes (A000668).

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Mathematica

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A171253 Semi-sums (average) of any two distinct Mersenne primes (A000668).