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.

User: Ophir Spector

Ophir Spector's wiki page.

Ophir Spector has authored 4 sequences.

A377766 Even numbers whose sum of proper (or aliquot) divisors is a prime.

Original entry on oeis.org

4, 8, 32, 50, 98, 128, 242, 324, 338, 392, 722, 784, 800, 1058, 1250, 1444, 2304, 2312, 2450, 2704, 2738, 3600, 3872, 5408, 5476, 5618, 6272, 6728, 7442, 7688, 8192, 9248, 11552, 12482, 12800, 14400, 14884, 15488, 15842, 16562, 16900, 16928, 17672, 18050, 19208, 21632, 21904, 22500, 23762, 25088
Offset: 1

Views

Author

Ophir Spector, Nov 06 2024

Keywords

Comments

Even terms of A037020.
Numbers from A088827 (2n^2 or 4n^2) are the only aliquot sum transition from even to odd.

Examples

			The aliquot divisors of 32 are 1, 2, 4, 8 and 16, whose sum is 31, a prime, so 32 is a term.

Links

Crossrefs

Intersection of A005843 and A037020.
Cf. A088827.

Programs

  • Mathematica
    Select[2Range[13000],PrimeQ[DivisorSigma[1,#]-#] &] (* Stefano Spezia, Nov 08 2024 *)
  • PARI
    is_a377766(n) = !(n%2) && isprime(sigma(n)-n) \\ Hugo Pfoertner, Nov 07 2024

A287615 Let r = prime(n). Then a(n) is the smallest prime p such that there is a prime q with p > q > r and p mod q = r.

Original entry on oeis.org

5, 13, 19, 29, 37, 47, 79, 101, 97, 103, 113, 131, 127, 137, 181, 199, 181, 227, 233, 229, 239, 257, 277, 283, 311, 307, 317, 409, 383, 367, 389, 409, 439, 521, 463, 509, 491, 509, 613, 571, 541, 563, 577, 587, 619, 653, 677, 677, 709, 743, 787, 853, 743, 877
Offset: 1

Views

Author

Ophir Spector, May 27 2017

Keywords

Comments

Prime p such that p = k * q + r, r < q < p primes; k even multiples such that p is minimal.

Examples

			a(1) = 5, as r = 2, q = 3, p = 5, is the smallest prime such that 5 = 2 mod 3.
a(9) = 97, as r = 23, q = 37, p = 97.  97 = 2 * 37 + 23 is smaller than 139 = 4 * 29 + 23 (A129919).

Links

Crossrefs

Cf. A129919.

Programs

  • Maple
    f:= proc(n) local p,q,r;
  r:= ithprime(n);
  p:= r+1;
  do
   p:= nextprime(p);
   q:= max(numtheory:-factorset(p-r));
   if q > r then return p fi
  od:
end proc:
map(f, [$1..100]); # Robert Israel, Jun 05 2017
  • Mathematica
    a[n_] := Module[{p, q, r}, r = Prime[n]; p = r+1; While[True, p = NextPrime[p]; q = Max[FactorInteger[p-r][[All, 1]]]; If[q>r, Return[p]]] ];
Array[a, 100] (* Jean-François Alcover, Oct 06 2020, after Robert Israel *)
  • PARI
    findfirstTerms(n)=my(t:small=0,a:vec=[]);forprime(r=2,,forprime(p=r+2,,forprime(q=r+2,p-2,if(p%q==r,a=concat(a,[p]);if(t++==n,a[1]-=2;return(a),break(2)))))) \\ R. J. Cano, Jun 06 2017
  • PARI
    first(n)=my(v=vector(n),best,k=1); v[1]=5; forprime(r=3,prime(n), best=oo; forprime(q=r+2,, if(q>=best, v[k++]=best; next(2)); forstep(p=r+2*q,best,2*q, if(isprime(p), best=p; break)))); v \\ Charles R Greathouse IV, Jun 07 2017

A135245 Aliquot predecessors with the largest degrees.

Original entry on oeis.org

0, 0, 4, 9, 0, 25, 8, 49, 15, 14, 21, 121, 35, 169, 33, 12, 55, 289, 65, 361, 91, 20, 85, 529, 143, 46, 133, 28, 187, 841, 161, 961, 247, 62, 253, 24, 323, 1369, 217, 81, 391, 1681, 341, 1849, 403, 86, 493, 2209, 551, 40, 481, 0, 667, 2809, 533, 106, 703, 68, 697, 3481
Offset: 1

Views

Author

Ophir Spector, ospectoro (AT) yahoo.com, Nov 25 2007

Keywords

Comments

Find each node's predecessors in aliquot sequences and choose the node with largest number of predecessors.
Climb the aliquot trees on thickest branches (see A135244 = Climb the aliquot trees on shortest paths).

Examples

			a(25) = 143 since 25 has 3 predecessors (95,119,143) with degrees (4,5,7), 143 having the largest degree. a(5) = 0 since it has no predecessors (see Untouchables - A005114).

Links

Crossrefs

Cf. A001065 A005114 A125601 A135244 A057709 A057710 A063769 A080907 A121507 A037020 A126016.

A135244 Largest m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

Original entry on oeis.org

0, 4, 9, 0, 25, 8, 49, 15, 14, 21, 121, 35, 169, 33, 26, 55, 289, 77, 361, 91, 38, 85, 529, 143, 46, 133, 28, 187, 841, 221, 961, 247, 62, 253, 24, 323, 1369, 217, 81, 391, 1681, 437, 1849, 403, 86, 493, 2209, 551, 94, 589, 0, 667, 2809, 713, 106, 703, 68, 697, 3481
Offset: 2

Views

Author

Ophir Spector (ospectoro(AT)yahoo.com), Nov 25 2007

Keywords

Comments

Previous name: Aliquot predecessors with the largest values.
Find each node's predecessors in aliquot sequences and choose the largest predecessor.
Climb the aliquot trees on shortest paths (see A135245 = Climb the aliquot trees on thickest branches).
The sequence starts at offset 2, since all primes satisfy sigma(n)-n = 1. - Michel Marcus, Nov 11 2014

Examples

			a(25) = 143 since 25 has 3 predecessors (95,119,143), 143 being the largest.
a(5) = 0 since it has no predecessors (see Untouchables - A005114).

Links

Crossrefs

Cf. A001065, A005114, A125601, A135245, A057709, A057710, A063769, A080907, A121507, A037020, A126016.

Programs

  • Mathematica
    seq[max_] := Module[{s = Table[0, {n, 1, max}], i}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]] = Max[s[[i]], n]], {n, 2, (max - 1)^2}]; Rest @ s]; seq[50]
  • PARI
    lista(nn) = {for (n=2, nn, k = (n-1)^2; while(k && (sigma(k)-k != n), k--); print1(k, ", "););} \\ Michel Marcus, Nov 11 2014

Extensions

a(1)=0 removed and offset set to 2 by Michel Marcus, Nov 11 2014
New name from Michel Marcus, Oct 31 2023

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