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 1-3 of 3 results.

A386245 Composite numbers k such that A075255(k) is a square.

Original entry on oeis.org

4, 6, 22, 135, 166, 444, 454, 636, 650, 854, 886, 1086, 1122, 1196, 1431, 1928, 2182, 2244, 2316, 2702, 3046, 3464, 3510, 3770, 4004, 4054, 4125, 4476, 4671, 5052, 5106, 5394, 5450, 6435, 6502, 6750, 8076, 8264, 8500, 9170, 9471, 9726, 10035, 10386, 10648, 10659, 11228, 11495, 11515, 11935, 12732
Offset: 1

Views

Author

Will Gosnell and Robert Israel, Jul 16 2025

Keywords

Comments

Composite numbers k such that k - sopfr(k) is a square, where sopfr(k) is the sum of prime factors of k with multiplicity.
Includes 2*p for p in A056899, but no odd semiprimes.
Is this sequence disjoint from A386246?

Examples

			a(3) = 22 is a term because 22 = 2 * 11 is composite and 22 - (2 + 11) = 9 is a square.

Links

Crossrefs

Cf. A056899, A075255, A386246.

Programs

  • Maple
    filter:= proc(n) local t;
 if isprime(n) then return false fi;
 issqr(n - add(t[1]*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$4..20000]);

A386640 Numbers k such that k + A224787(k) is a square.

Original entry on oeis.org

1, 225, 270, 1900, 4988, 5656, 6120, 8704, 11180, 16588, 17710, 19228, 24475, 28449, 29458, 32330, 34606, 38088, 39292, 40221, 41181, 42476, 48545, 48640, 53795, 56832, 57288, 64975, 78793, 84925, 86242, 117116, 124135, 128478, 129673, 134044, 136224, 136896, 147149, 150528, 168055, 183141
Offset: 1

Views

Author

Will Gosnell and Robert Israel, Jul 27 2025

Keywords

Comments

Numbers k such that the sum of k and the cubes of the prime factors of k, counted with multiplicity, is a square.

Examples

			a(3) = 270 = 2 * 3^3 * 5 is a term because 270 + 2^3 + 3 * 3^3 + 5^3 =  484 = 22^2 is a square.

Links

Crossrefs

Cf. A224787, A385238, A386246, A386257, A386623.

Programs

  • Maple
    filter:= proc(n) local t;
   issqr(n + add(t[1]^3*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^6]);
  • Mathematica
    lim=184000;f[{p_,e_}]:=e*p^3;a224787[k_]:=If[k==1,0,Total[f/@FactorInteger[k]]];q[k_]:=IntegerQ[Sqrt[k+a224787[k]]];Select[Range[lim],q[#]&] (* James C. McMahon, Jul 30 2025 *)

A386991 Numbers k such that k^2 + sopfr(k)^2 is a square, where sopfr = A001414.

Original entry on oeis.org

1, 8, 15, 35, 112, 143, 323, 899, 1763, 3599, 5183, 10403, 11663, 19043, 22499, 32399, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 656099, 675683, 685583, 736163
Offset: 1

Views

Author

Robert Israel, Aug 12 2025

Keywords

Comments

Includes A037074 because if k = p*(p+2) where p and p+2 are primes, k^2 + sopfr(k)^2 = p^2*(p+2)^2 + (2*p+2)^2 = (p^2 + 2*p + 2)^2.
Are 1, 8 and 112 the only terms not in A037074?

Examples

			a(3) = 15 is a term because the sum of prime factors of 15 is 3+5 = 8 and 15^2 + 8^2 = 289 = 17^2.

Links

Crossrefs

Cf. A001414, A386246. Includes A037074.

Programs

  • Maple
    sopfr:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
filter:= t -> issqr(t^2 + sopfr(t)^2):
select(filter, [$1..10^5]);
  • Mathematica
    Sopfr[1]=0;Sopfr[n_]:= Plus @@ Times @@@ FactorInteger@ n;Select[Range[500000],IntegerQ[Sqrt[#^2+Sopfr[#]^2]]&] (* James C. McMahon, Aug 14 2025 *)
Showing 1-3 of 3 results.

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

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