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.

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A228434 Primes expressible as sigma(n) + sigma(sigma(n)), in order of their occurrence.

Original entry on oeis.org

2, 7, 11, 23, 127, 167, 137, 269, 547, 547, 383, 547, 269, 431, 547, 547, 293, 383, 431, 1171, 1039, 1171, 641, 1039, 1103, 1171, 887, 1361, 2551, 1861, 3001, 2753, 1193, 2963, 1499, 2153, 2753, 2551, 2963, 4327, 5281, 1823, 2963, 4219, 4327, 3593, 3583, 6763
Offset: 1

Views

Author

K. D. Bajpai, Nov 10 2013

Keywords

Examples

			a(6)= 167: sigma(32)+sigma(sigma(32))= 63+104= 167, which is prime.
a(11)= 383: sigma(93)+sigma(sigma(93))= 128+255= 383, which is prime.

Links

Crossrefs

Cf. A000203 (sigma(n): sum of divisors of n).
Cf. A019279 (superperfect numbers: sigma(sigma(n))).
Cf. A033632 (numbers n: sigma(n)is prime).
Cf. A051027 (a(n)= sigma(sigma(n))).

Programs

  • Maple
    with(numtheory):KD := proc() local a; a:= sigma(n)+sigma(sigma(n));if isprime(a) then RETURN (a);fi;end:seq(KD(),n=1..5000);

A233865 Numbers n such that sigma(sigma(n))+1 is prime.

Original entry on oeis.org

1, 2, 5, 6, 11, 14, 15, 19, 20, 23, 26, 29, 36, 37, 41, 61, 63, 67, 68, 72, 74, 76, 82, 85, 86, 88, 90, 100, 102, 103, 104, 105, 107, 110, 113, 116, 117, 118, 120, 128, 129, 131, 133, 139, 141, 142, 144, 145, 146, 149, 153, 155, 157, 159, 161, 172, 174, 179, 181
Offset: 1

Views

Author

K. D. Bajpai, Dec 17 2013

Keywords

Examples

			6 is in the sequence because sigma(sigma(6))+1= 29, which is prime.
19 is in the sequence because sigma(sigma(19))+1= 43, which is prime.

Links

Crossrefs

Cf. A000203 (sigma(n): sum of divisors of n).
Cf. A019279 (superperfect numbers: sigma(sigma(n))).
Cf. A023194 (numbers n: sigma(n)is prime).
Cf. A228567 (primes: sigma(sigma(n))-sigma(n) is prime).

Programs

  • Maple
    with(numtheory): KD := proc() local a; a:= sigma(sigma(n))+1; if isprime(a) then RETURN (n); fi; end: seq(KD(),n=1..500);
  • Mathematica
    Select[Range[200],PrimeQ[DivisorSigma[1,DivisorSigma[1,#]]+1]&] (* Harvey P. Dale, Jan 29 2021 *)
  • PARI
    is(n)=isprime(sigma(sigma(n))+1) \\ Charles R Greathouse IV, Jun 13 2017

A290169 a(n) = least k such that both the sum of the smallest n divisors of k and the sum of its greatest n divisors are prime numbers.

Original entry on oeis.org

2, 4, 30, 16, 140, 64, 264, 144, 336, 525, 144, 800, 1200, 576, 1600, 2016, 1440, 1296, 2160, 2304, 7980, 6440, 3360, 8360, 4080, 3960, 2772, 16100, 9108, 10608, 7392, 12320, 14688, 37240, 21780, 18200, 45760, 20160, 9240, 24624, 14364, 8400, 22176, 23760
Offset: 2

Views

Author

Michel Lagneau, Jul 23 2017

Keywords

Comments

The corresponding pairs of primes are (3, 3), (7, 7), (11, 61), (31, 31), (29, 307), (127, 127), (47, 673), (61, 379), (73, 919), ...
The sequence contains a subsequence of numbers having the property that the sum of the first n divisors is equal to the sum of the last n divisors; for instance, for a(n) = 2, 4, 16 and 64 with n = 2, 3, 5 and 7. Is it possible to conjecture that this subsequence contains all the superperfect numbers (A019279)? The answer is no: for instance, A019279(5) = 4096 = 2^12 => the sum of the 13 terms 1 + 2 + 4 + 8 + ... + 4096 = 8191 is a Mersenne prime, but a(13) = 800 instead 4096 > 800, and we obtain the corresponding pair of primes (293, 1933) instead (8191, 8191).
The squares of the terms of the sequence are 4, 16, 64, 144, 576, 1296, 1600, 2304, ...

Examples

			a(4)=30 because both the sum of the first 4 divisors of 30 (1 + 2 + 3 + 5 = 11) and the sum of its last 4 divisors (30 + 15 + 10 + 6 = 61) are prime numbers.

Links

Crossrefs

Cf. A000043, A000668, A019279, A289776, A290126.

Programs

  • Mathematica
    Table[k=1;While[Nand[Length@#>=n,PrimeQ[Total@Take[PadRight[#,n],n]]]||Nand[Length@#>=n,PrimeQ[Total@Take[PadLeft[#,n],n]]]&@Divisors@k,k++];k,{n,2,10}]

A292369 Numbers n such that f(f(f(n))) = f(f(n)) + f(n) where f = A000203.

Original entry on oeis.org

2, 4, 16, 25, 64, 4096, 65536, 262144, 1073741824
Offset: 1

Views

Author

Altug Alkan, Sep 15 2017

Keywords

Comments

Numbers n such that A066971(n) = A051027(n) + A000203(n).
A061652 is a subsequence.
Are there any odd terms other than 25?

Examples

			25 = 5^2 is a term because sigma(sigma(sigma(5^2))) = sigma(2^5) = sigma(sigma(5^2)) + sigma(5^2).

Crossrefs

Cf. A000203, A019279, A051027, A061652, A066971.

Programs

  • Mathematica
    f[n_] := DivisorSigma[1, n]; fQ[n_] := f[f[f[n]]] == f[f[n]] + f[n]; Select[ Range@1000000, fQ] (* Robert G. Wilson v, Sep 23 2017 *)
  • PARI
    s(n) = sigma(n);
isok(n) = s(s(s(n)))==s(s(n))+s(n);

Extensions

a(9) from Giovanni Resta, Sep 15 2017

A324255 Lesser of super amicable pair m < n defined by sigma(sigma(m)) = sigma(sigma(n)) = m + n.

Original entry on oeis.org

23, 14999, 34673, 55373, 2056961, 2458187, 4586987, 5174363, 6204767, 15788453, 20699927, 22239647, 25748273, 43428947, 53636627, 74005637, 92626307, 132253139, 134131379, 158486747, 167677187, 183614987, 307412933, 385042487, 413859143, 418168343, 683772017
Offset: 1

Views

Author

Amiram Eldar, Feb 19 2019

Keywords

Comments

The larger numbers in each pair are in A324256.
Analogous to A002025 as A019279 is analogous to A000396.

Examples

			(23, 37) are the first pair since sigma(sigma(23)) = sigma(sigma(37)) = 60 = 23 + 37.

Crossrefs

Cf. A000203, A000396, A002025, A019279, A045613 (unitary analog), A051027, A324256.

Programs

  • Mathematica
    seq={}; s[n_]:=DivisorSigma[1,DivisorSigma[1,n]]-n; Do[m=s[n];If[m>n && s[m]==n, AppendTo[seq, n]], {n,1,60000}]; seq
  • PARI
    f(n) = sigma(sigma(n)) - n;
lista(nn) = {for (n=1, nn, my(fn = f(n)); if ((fn > n) && (f(fn) == n), print1(n, ", ")););} \\ Michel Marcus, Feb 20 2019

A329881 Nonunitary doubly superperfect numbers: numbers k such that nusigma(nusigma(k)) = 2*k, where nusigma(k) = sigma(k) - usigma(k) is the sum of nonunitary divisors of k (A048146).

Original entry on oeis.org

4032, 13104, 58032, 69648, 237744, 278592, 365652, 1114368, 15333552, 71319552, 245364912, 981465264, 1141112832, 4564451328, 873139150710, 4020089387184
Offset: 1

Views

Author

Amiram Eldar, Nov 23 2019

Keywords

Comments

Analogous to superperfect numbers (A019279) as nonunitary doubly perfect numbers (A064592) is analogous to perfect numbers (A000396).
If n = 2^k*3*1451 and nusigma(n) = 2^5*3*11^2*p, with p > 11 prime, then n is a term. This happens for k = 4, 6, 8, 14, 18, 20, 32, 62, 90, 108, 128, 522, 608, ... . Similarly, if p=2^k-1 is prime (A000043), then 2^4*3^2*13*p is a term for k > 2. - Giovanni Resta, Nov 23 2019
a(17) > 6*10^12. - Giovanni Resta, Nov 24 2019

Crossrefs

Cf. A000396, A019279, A034448, A038843, A048146, A064592, A328120, A329884.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; Select[Range[3*10^5], nusigma[nusigma[#]] == 2*# &]

Extensions

a(15)-a(16) from Giovanni Resta, Nov 24 2019

A135612 Even superperfect numbers divided by 2, written in base 2.

Original entry on oeis.org

1, 10, 1000, 100000, 100000000000, 1000000000000000, 100000000000000000, 100000000000000000000000000000, 100000000000000000000000000000000000000000000000000000000000
Offset: 1

Views

Author

Omar E. Pol, Mar 01 2008

Keywords

Comments

Also, concatenation of "1" and A000043(n)-2 digits "0".
The number of divisors of a(n) is equal to the number of its digits. This number is equal to A090748(n)=A000043(n)-1.

Examples

			a(3)=1000 because A134708(n)=8 and 8 written in base 2 is 1000.

Crossrefs

Even superperfect numbers divided by 2: A134708. Cf. A000043, A019279, A090748, A135651, A135656.

Formula

a(n)=A134708(n) written in base 2.

A138728 Array read by rows: row n lists n-th even superperfect number A061652(n), n-th Mersenne prime A000668(n) and n-th perfect number A000396(n).

Original entry on oeis.org

2, 3, 6, 4, 7, 28, 16, 31, 496, 64, 127, 8128, 4096, 8191, 33550336, 65536, 131071, 8589869056, 262144, 524287, 137438691328, 1073741824, 2147483647, 2305843008139952128, 1152921504606846976, 2305843009213693951
Offset: 1

Views

Author

Omar E. Pol, Apr 07 2008

Keywords

Examples

			Array begins:
2, 3, 6
4, 7, 28
16, 31, 496

Links

Crossrefs

Cf. A000396, A000668, A019279, A061652, A138818, A138819.

A138827 Sum of digits of n-th even superperfect number A061652(n).

Original entry on oeis.org

2, 4, 7, 10, 19, 25, 19, 37, 82, 115, 142, 172, 745, 856, 1603, 2980, 3079, 4240, 5812, 5842, 13165, 13480, 15136, 27052, 29455, 31339, 60112, 116386, 149239, 179821, 291511, 1025683, 1164760, 1703224, 1891861, 4038604, 4091767, 9440575, 18241120, 28450837, 32576617
Offset: 1

Views

Author

Omar E. Pol, Apr 01 2008

Keywords

Comments

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

Crossrefs

Cf. A007953, A061652.
Cf. A007605, A019279, A066538.

Formula

a(n) = A007953(A061652(n)). - Michel Marcus, Mar 15 2020

Extensions

a(10)-a(33) from R. J. Mathar, May 22 2008
More terms from Jinyuan Wang, Mar 15 2020

A138835 Bisection of even superperfect numbers A061652.

Original entry on oeis.org

4, 64, 65536, 1073741824, 309485009821345068724781056, 85070591730234615865843651857942052864
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Comments

Also, bisection of superperfect numbers A019279, if there are no odd superperfect numbers.
Next terms have 183, 663 and more digits and are therefore not listed. - R. J. Mathar, May 22 2008

Crossrefs

Cf. A019279, A061652, A099057, A099058, A134721, A134722, A138834.
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