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.

Showing 1-10 of 20 results. Next

A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

Original entry on oeis.org

8, 32, 512, 8192, 33554432, 8589934592, 137438953472, 2305843009213693952, 2658455991569831745807614120560689152, 191561942608236107294793378393788647952342390272950272
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Comments

Sum of n-th even perfect number and n-th even superperfect number.
Also, sum of n-th perfect number and n-th superperfect number, if there are no odd perfect and odd superperfect numbers, then the n-th perfect number is the difference between a(n) and the n-th superperfect number (see A135652, A135653, A135654 and A135655).

Examples

			a(5) = 33554432 because A000043(5) = 13 and 2^(2*13 - 1) = 2^25 = 33554432.
Also, if there are no odd perfect and odd superperfect numbers then we can write a(5) = A000396(5) + A019279(5) = A000396(5) + A061652(5) = 33554432.

Links

Crossrefs

Cf. A000079, A000396, A019279, A061645, A061652, A133033, A135652, A135653, A135654, A135655, A139286, A139294, A139307.
Cf. A000043, A000668, A072868.

Programs

  • Mathematica
    2^(2 * MersennePrimeExponent[Range[10]] - 1) (* Amiram Eldar, Oct 17 2024 *)

Formula

a(n) = 2^(2*A000043(n) - 1). Also, a(n) = 2^A133033(n), if there are no odd perfect numbers. Also, a(n) = A000396(n) + A019279(n), if there are no odd perfect and odd superperfect numbers. Also, a(n) = A000396(n) + A061652(n), if there are no odd perfect numbers, then we can write: perfect number A000396(n) = a(n) - A061652(n).
a(n) = A061652(n)*(A000668(n)+1) = A061652(n)*A072868(n). - Omar E. Pol, Apr 13 2008

A139286 a(n) = 2^(2*prime(n) - 1).

Original entry on oeis.org

8, 32, 512, 8192, 2097152, 33554432, 8589934592, 137438953472, 35184372088832, 144115188075855872, 2305843009213693952, 9444732965739290427392, 2417851639229258349412352, 38685626227668133590597632, 9903520314283042199192993792
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A000040, A000079, A076274, A139287, A139288, A139289, A139290, A139291, A139292, A139293, A139294, A139306.

Programs

Extensions

Corrected and extended by Harvey P. Dale, Dec 01 2017

A139295 a(n) = 2^(2p - 1)-1, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

31, 8191, 2305843009213693951, 14474011154664524427946373126085988481658748083205070504932198000989141204991
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A000668, A139294, A139296, A139297, A139298, A139299, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

Programs

  • Mathematica
    a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 3) - 1; Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

Formula

a(n) = 2^(2*A000668(n)-1)-1 = A139294(n)-1.

A139296 a(n) = 2^(2p - 1)/2, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

16, 4096, 1152921504606846976, 7237005577332262213973186563042994240829374041602535252466099000494570602496
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Comments

a(5) has 4931 digits and is too large to include. - R. J. Mathar, May 30 2008

Crossrefs

Cf. A000668, A139286, A139294, A139295, A139297, A139298, A139299, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

Programs

  • Mathematica
    a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 4); Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

Formula

a(n) = 2^(2*A000668(n)-1)/2.

Extensions

One more term from R. J. Mathar, May 30 2008

A139297 a(n) = 2^(2p - 1)/2-1, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

15, 4095, 1152921504606846975, 7237005577332262213973186563042994240829374041602535252466099000494570602495
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A000668, A139294, A139295, A139296, A139298, A139299, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

Programs

  • Mathematica
    a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 4) - 1; Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

Formula

a(n) = 2^(2*A000668(n)-1)/2-1 = A139296(n)-1.

Extensions

a(4) from Amiram Eldar, Jul 10 2025

A139298 a(n) = 2^(2p - 1)/4, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

8, 2048, 576460752303423488, 3618502788666131106986593281521497120414687020801267626233049500247285301248
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A000668, A139294, A139295, A139296, A139297, A139299, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

Programs

  • Mathematica
    a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 5); Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

Formula

a(n) = 2^(2*A000668(n) - 1)/4 = A139294(n)/4.

A139299 a(n) = 2^(2p - 1)/4-1, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

7, 2047, 576460752303423487, 3618502788666131106986593281521497120414687020801267626233049500247285301247
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A000668, A139294, A139295, A139296, A139297, A139298, A139300, A139301, A139302, A139303, A139304, A139305, A139306.

Programs

  • Mathematica
    a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 5) - 1; Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

Formula

a(n) = 2^(2*A000668(n)-1)/4-1 = A139294(n)/4-1 = A139298(n)-1.

Extensions

a(4) from Amiram Eldar, Jul 10 2025

A139300 a(n) = 2^(2p - 1)/8, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

4, 1024, 288230376151711744, 1809251394333065553493296640760748560207343510400633813116524750123642650624
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A000668, A139294, A139295, A139296, A139297, A139298, A139299, A139301, A139302, A139303, A139304, A139305, A139306.

Programs

  • Mathematica
    a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 6); Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

Formula

a(n) = 2^(2*A000668(n)-1)/8 = A139294(n)/8.

Extensions

a(4) from Amiram Eldar, Jul 10 2025

A139301 a(n) = (2^(2p - 1)/8)-1, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 1023, 288230376151711743, 1809251394333065553493296640760748560207343510400633813116524750123642650623
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A000668, A139294, A139295, A139296, A139297, A139298, A139299, A139300, A139302, A139303, A139304, A139305, A139306.

Programs

  • Mathematica
    a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 6) - 1; Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

Formula

a(n) = (2^(2*A000668(n)-1)/8)-1 = (A139294(n)/8)-1.

Extensions

a(4) from Amiram Eldar, Jul 10 2025

A139302 a(n) = 2^(2p - 1)/16, where p is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

2, 512, 144115188075855872, 904625697166532776746648320380374280103671755200316906558262375061821325312
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A000668, A139294, A139295, A139296, A139297, A139298, A139299, A139300, A139301, A139303, A139304, A139305, A139306.

Programs

  • Mathematica
    a[n_] := 2^(2^(MersennePrimeExponent[n] + 1) - 7); Array[a, 4] (* Amiram Eldar, Jul 10 2025 *)

Formula

a(n) = 2^(2*A000668(n)-1)/16 = A139294(n)/16.

Extensions

a(4) from Amiram Eldar, Jul 10 2025
Showing 1-10 of 20 results. Next

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