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 1-8 of 8 results.

A349758 Nobly abundant numbers: numbers k such that both d(k) = A000005(k) and sigma(k) = A000203(k) are abundant numbers (A005101).

Original entry on oeis.org

60, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 180, 198, 204, 220, 224, 228, 234, 240, 252, 260, 276, 294, 300, 306, 308, 315, 336, 340, 342, 348, 350, 352, 360, 364, 372, 380, 396, 414, 416, 420, 432, 444, 460, 476, 480, 486, 490, 492, 495, 500, 504, 516
Offset: 1

Views

Author

Amiram Eldar, Nov 29 2021

Keywords

Comments

Analogous to sublime numbers (A081357), with abundant numbers instead of perfect numbers.
The least odd term is a(27) = 315 and the least term that is coprime to 6 is a(298) = 1925.

Examples

			60 is a term since both d(60) = 12 and sigma(60) = 168 are abundant numbers: sigma(12) = 28 > 2*12 = 24 and sigma(168) = 480 > 2*168 = 336.

References

  • József Sándor and E. Egri, Arithmetical functions in algebra, geometry and analysis, Advanced Studies in Contemporary Mathematics, Vol. 14, No. 2 (2007), pp. 163-213.

Links

Crossrefs

Cf. A000005, A000203, A005101, A081357, A349759.
A349760 is a subsequence.

Programs

  • Mathematica
    abQ[n_] := DivisorSigma[1, n] > 2*n; nobAbQ[n_] := And @@ abQ /@ DivisorSigma[{0, 1}, n]; Select[Range[500], nobAbQ]
  • PARI
    isab(k) = sigma(k) > 2*k; \\ A005101
isok(k) = my(f=factor(k)); isab(numdiv(f)) && isab(sigma(f)); \\ Michel Marcus, Dec 02 2021

A349759 Nobly deficient numbers: numbers k such that both d(k) = A000005(k) and sigma(k) = A000203(k) are deficient numbers (A005100).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 13, 16, 21, 25, 31, 36, 37, 43, 48, 49, 61, 64, 67, 73, 81, 93, 97, 100, 109, 111, 112, 121, 127, 128, 144, 151, 157, 162, 163, 169, 181, 183, 192, 193, 196, 208, 211, 217, 219, 225, 229, 241, 256, 277, 283, 289, 313, 324, 331, 337, 361, 373
Offset: 1

Views

Author

Amiram Eldar, Nov 29 2021

Keywords

Comments

Analogous to sublime numbers (A081357), with deficient numbers instead of perfect numbers.
If p != 5 is a prime such that (p+1)/2 is also a prime (i.e., p is in A005383 \ {5}), then p is a term of this sequence.

Examples

			2 is a term since both d(2) = 2 and sigma(2) = 3 are deficient numbers.

Links

Crossrefs

Cf. A000005, A000203, A005100, A005383, A081357, A349758.

Programs

  • Mathematica
    defQ[n_] := DivisorSigma[1, n] < 2*n; nobDefQ[n_] := And @@ defQ /@ DivisorSigma[{0, 1}, n]; Select[Range[400], nobDefQ]
  • PARI
    isdef(k) = sigma(k) < 2*k; \\ A005100
isok(k) = my(f=factor(k)); isdef(numdiv(f)) && isdef(sigma(f)); \\ Michel Marcus, Dec 03 2021

A146542 Numbers m such that sigma(m) is a perfect number.

Original entry on oeis.org

5, 12, 427, 10924032, 16125952, 22017387, 24376323, 32501857, 33288097, 3757637632, 6241076643, 8522760577, 45091651584, 66563866624, 86692869921, 137421905953, 137437511683, 727145809044307968, 1152771972099211264, 845044701535107443245558061611352064
Offset: 1

Views

Author

Howard Berman (howard_berman(AT)hotmail.com), Oct 31 2008

Keywords

Examples

			The divisors of 5 are 1 and 5, which add up to 6. 6 is a perfect number because its proper divisors are 1, 2 and 3, which also add up to 6.

Links

Crossrefs

Cf. A000396, A066961, A081357.

Programs

  • Maple
    with(numtheory); P:=proc(q) local n; for n from 1  to q do
if sigma(sigma(n))=2*sigma(n) then print(n);
fi; od; end: P(10^9); # Paolo P. Lava, Oct 22 2013
  • PARI
    isok(n) = sigma(sigma(n)) == 2*sigma(n); \\ Michel Marcus, Oct 22 2013

Extensions

Two missing terms added and a(10)-a(19) from Donovan Johnson, Jan 20 2012
a(20) from Daniel Suteu, May 23 2022

A382506 a(n) is the smallest k such that sigma(n) + k is a perfect number.

Original entry on oeis.org

5, 3, 2, 21, 0, 16, 20, 13, 15, 10, 16, 0, 14, 4, 4, 465, 10, 457, 8, 454, 464, 460, 4, 436, 465, 454, 456, 440, 466, 424, 464, 433, 448, 442, 448, 405, 458, 436, 440, 406, 454, 400, 452, 412, 418, 424, 448, 372, 439, 403, 424, 398, 442, 376, 424, 376, 416, 406, 436, 328, 434, 400
Offset: 1

Views

Author

Leo Hennig, Mar 29 2025

Keywords

Examples

			sigma(1) = 1, 1 + 5 = 6, k = 5.
sigma(6) = 12, 12 + 16 = 28, k = 16.
sigma(180) = 546, 546 + 7582 = 8128, k = 7582.
As sigma(3000) = 9360 and the smallest perfect number at least as large as 9360 is 2^12 * (2^13 - 1) = 33550336 we have a(3000) = 33550336 - sigma(3000) = 33540976. - _David A. Corneth_, Apr 10 2025

Links

Crossrefs

Cf. A000203, A000396, A081357, A146542, A382929.

Programs

  • Mathematica
    Do[k=0;s=DivisorSigma[1,n];While[DivisorSigma[1,s+k]!=2*(s+k),k++];a[n]=k,{n,62}];Array[a,62] (* James C. McMahon, Apr 10 2025 *)
  • PARI
    a(n) = my(s=sigma(n),k=0); while (sigma(s+k) != 2*(s+k), k++); k; \\ Michel Marcus, Mar 30 2025
  • PARI
    a(n) = {my(s = sigma(n));
    forprime(p = 2, oo,
        my(c = 2^p-1);
        if(isprime(c) && binomial(c+1, 2) >= s,
           return(binomial(c+1, 2) - s)))
} \\ David A. Corneth, Apr 10 2025

Formula

a(A081357(n)) = 0 and a(A146542(n)) = 0.

A233482 Numbers for which the number of divisors and the sum of the distinct prime divisors are both perfect.

Original entry on oeis.org

575, 2057, 2645, 3179, 4416, 8512, 12275, 33534, 94272, 138431, 203075, 218176, 392747, 715878, 918592, 982157, 991841, 1082176, 1205405, 1244387, 1559616, 1690432, 1966912, 2344079, 2464576, 2982976, 3386176, 3452992, 3625792, 3821632, 3867712, 3900497
Offset: 1

Views

Author

Michel Lagneau, Dec 11 2013

Keywords

Comments

Numbers n such that A000005(n) and A008472(n) are in the sequence A000396. See the sequence A081357 for the sublime numbers.

Examples

			575 is in the sequence because tau(575) = 6 and sopf(575) = 28,
4416 is in the sequence because tau(4416) = 28 and sopf(4416) = 28,
12275 is in the sequence because tau(12275) = 6 and sopf(12275) = 496,
203075 is in the sequence because tau(203075) = 6 and sopf(203075) = 8128.

Links

Crossrefs

Cf A000005, A000396, A081357, A008472.

Programs

  • Maple
    with(numtheory): lst:={6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216} :n1:=nops(lst): for n from 1 to 1000000 do :x:=factorset(n):n2:=nops(x): s:=sum('x[i]', 'i'=1..n2):
ii:=0:for m from 1 to n1 do:if s=lst[m] then ii:=1:else fi:od:jj:=0:for p from 1 to n1 do:if tau(n)=lst[p] then jj:=1:else fi:od:if ii=1 and jj=1 then printf(`%d, `,n):else fi:od:
  • Mathematica
    Select[Range[4*10^6],AllTrue[{DivisorSigma[0,#],Total[FactorInteger[#][[All,1]]]},PerfectNumberQ]&] (* Harvey P. Dale, Aug 11 2021 *)

A382483 a(n) = smallest number k such that at least one of sigma(n) - k and sigma(n) + k is a perfect number.

Original entry on oeis.org

5, 3, 2, 1, 0, 6, 2, 9, 7, 10, 6, 0, 8, 4, 4, 3, 10, 11, 8, 14, 4, 8, 4, 32, 3, 14, 12, 28, 2, 44, 4, 35, 20, 26, 20, 63, 10, 32, 28, 62, 14, 68, 16, 56, 50, 44, 20, 96, 29, 65, 44, 70, 26, 92, 44, 92, 52, 62, 32, 140, 34, 68, 76, 99, 56, 116, 40, 98, 68, 116, 44, 167, 46, 86, 96, 112, 68, 140
Offset: 1

Views

Author

Leo Hennig, Mar 27 2025

Keywords

Examples

			sigma(6) = 12, the nearest perfect number is 6, thus a(6) = 12 - 6 = 6.
sigma(26) = 42, the nearest perfect number is 28, thus a(26) = 42 - 28 = 14.

Crossrefs

Cf. A000396, A081357, A146542, A382506.

Programs

  • Maple
    isA000396 := proc(n::integer)
    if n < 6 then
        false ;
    elif numtheory[sigma](n) = 2*n then
        true;
    else
        false;
    end if;
end proc:
A382483 := proc(n)
    local k ;
    for k from 0 do
        if isA000396(numtheory[sigma](n)-k) or isA000396(numtheory[sigma](n)+k)  then
            return k;
        end if;
    end do:
end proc:
seq(A382483(n),n=1..50) ; # R. J. Mathar, Apr 01 2025
  • PARI
    isp(x) = if (x>0, sigma(x) == 2*x);
a(n) = my(k=0, s=sigma(n)); while (!(isp(s-k) || isp(s+k)), k++); k; \\ Michel Marcus, Apr 01 2025

Formula

a(A081357(k)) = 0.
a(A146542(k)) = 0.
a(A000396(k)) = A000396(k).

A233563 Numbers for which the number of prime divisors counted with multiplicity and the sum of the distinct prime divisors are both perfect.

Original entry on oeis.org

1104, 1656, 2128, 2484, 3726, 4620, 6930, 7448, 11550, 12285, 12696, 16170, 19044, 20216, 20475, 23568, 25410, 26068, 28566, 28665, 34125, 35352, 47775, 53028, 53235, 54544, 66885, 70756, 71875, 79542, 88725, 91238, 124215, 146004, 190904, 192052, 201180
Offset: 1

Views

Author

Michel Lagneau, Dec 13 2013

Keywords

Comments

Numbers n such that A001222(n) and A008472(n) are in the sequence A000396.

Examples

			1104 is in the sequence because bigomega(1104) = 6 and sopf(1104) = 28,
23568 is in the sequence because bigomega(23568) = 6 and sopf(23568) = 496,
389904 is in the sequence because bigomega(389904) = 6 and sopf(389904) = 8128.

Links

Crossrefs

Cf. A000396, A081357, A008472, A001222, A233482.

Programs

  • Maple
    with(numtheory): lst:={6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216} :n1:=nops(lst): for n from 1 to 1000000 do :x:=factorset(n):n2:=nops(x): s:=sum('x[i]', 'i'=1..n2):
ii:=0:for m from 1 to n1 do:if s=lst[m] then ii:=1:else fi:od:jj:=0:for p from 1 to n1 do:if bigomega(n)=lst[p] then jj:=1:else fi:od:if ii=1 and jj=1 then printf(`%d, `, n):else fi:od:

A290149 Totient sublime numbers: numbers k such that the number of terms in the iterations of phi(k) from k to 1, A032358(k)+2, and their sum, A092693(k) are both perfect totient numbers (A082897).

Original entry on oeis.org

6, 2916, 4374, 109100, 113708, 3188646
Offset: 1

Views

Author

Amiram Eldar, Jul 21 2017

Keywords

Comments

Analogous to A081357 (sublime numbers), as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).
No other terms below 10^8.

Examples

			There are 9 terms in the iterations of phi(k) for 2916: 2916, 972, 324, 108, 36, 12, 4, 2, 1. Their sum is 4375. Both 9 and 4375 are perfect totient numbers (A082897).

Crossrefs

Cf. A032358, A081357, A082897, A092693.

Programs

  • Mathematica
    iterList [n_] := FixedPointList[EulerPhi@# &, n]; sumIter [n_] := Plus @@ iterList[n] - 1; numIter[n_] := Length[iterList[n]] - 1; perfTotQ[n_] := sumIter[n] == 2 n; totSublimeQ[n_] := perfTotQ[numIter[n]] && perfTotQ[sumIter[n]]; Select[Range [10^8], totSublimeQ]
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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