Will Gosnell - cp's OEIS frontend

A385681 a(n) is the least k > 1 such that n^2 == k (mod sopfr(k)) and k^2 == n (mod sopfr(n)), or -1 if there is no such k, where sopfr = A001414.

2, 3, 2, 5, 4, 7, -1, 3, -1, 11, -1, 13, -1, -1, 4, 17, -1, 19, -1, 9, 120, 23, -1, 5, -1, 3, -1, 29, 30, 31, -1, -1, -1, -1, 4, 37, -1, -1, -1, 41, -1, 43, -1, 45, 36, 47, 2, 7, -1, -1, 16, 53, -1, -1, 2, -1, -1, 59, 30, 61, -1, -1, 2, -1, -1, 67, -1, 2745, 70, 71, 60, 73, -1, 150, -1, -1, -1

Offset: 2

Will Gosnell and Robert Israel, Aug 04 2025

sign

a(n) = -1 if n is in A385679.  Conjecture: a(n) > 0 if n is not in A385679.
0 < a(n) < n if and only if n is in A385664.
If p is prime, then a(p) = p.

			a(6) = 4 because sopfr(6) = 5, sopfr(4) = 4, 6^2 == 4 == 0 (mod 4) and 4^2 == 6 == 1 (mod 5), and neither 2 nor 3 works.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A001414, A385664, A385679.

sopfr:= proc(n) option remember; local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
f:= proc(x) local sx,R,y,X,r,k;
  sx:= sopfr(x);
  R:= sort(map(t -> rhs(op(t)), [msolve(X^2 = x,sx)]));
  if R = [] then return -1 fi;
  for k from 0 do
    for r in R do
      y:= r + k*sx;
      if y < 2 then next fi;
      if x^2 - y  mod sopfr(y) = 0 then return y fi
  od od;
end proc:
map(f, [$2 .. 100]);

A385664 Numbers y such that there is at least one x with 2 <= x < y, x^2 == y (mod sopfr(y)) and y^2 == x (mod sopfr(x)), where sopfr = A001414.

Original entry on oeis.org

4, 6, 9, 16, 21, 25, 27, 36, 46, 48, 49, 52, 56, 60, 64, 72, 81, 84, 90, 100, 102, 104, 108, 120, 121, 126, 128, 135, 141, 144, 150, 156, 160, 162, 166, 169, 174, 176, 180, 196, 204, 207, 216, 220, 225, 231, 237, 238, 240, 244, 245, 246, 248, 253, 256, 261, 264, 276, 280, 286, 288, 289, 294, 301

Offset: 1

Will Gosnell and Robert Israel, Aug 03 2025

nonn

Contains no primes, but all squares of primes (with y = p^2 for p an odd prime, x = p works). Does the sequence contain all squares > 1?

			a(5) = 21 is a term because with x = 9, we have sopfr(9) = 6, sopfr(21) = 10, 9^2 == 21 == 0 (mod 10) and 21^2 == 9 (mod 16).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001414.

spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
filter:= proc(y)
  ormap(x -> x^2 - y mod spf(y) = 0 and y^2 - x mod spf(x) = 0, [$2..y-1])
end proc:
select(filter, [$2 .. 500]);

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

Will Gosnell and Robert Israel, Jul 27 2025

nonn

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

			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.

Robert Israel, Table of n, a(n) for n = 1..3000

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

filter:= proc(n) local t;
   issqr(n + add(t[1]^3*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^6]);

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 *)

A386623 Numbers k such that k - A224787(k) is a square.

Original entry on oeis.org

1, 64, 630, 1225, 1296, 1750, 1925, 2079, 3125, 3402, 3888, 7150, 11495, 13000, 16445, 16464, 17160, 17500, 25578, 25935, 26082, 27508, 36975, 39083, 42688, 47125, 55955, 57188, 61740, 66671, 85085, 88451, 99372, 104544, 111375, 120736, 122452, 128898, 137547, 141427, 145509, 146927, 152592

Offset: 1

Will Gosnell and Robert Israel, Jul 27 2025

nonn

Numbers k such that k is the sum of a square and the cubes of the prime factors of k, counted with multiplicity.
Except for a(1) = 1, all terms are the product of at least 4 (not necessarily distinct) primes.
a(1) = 1 and a(7) = 1925 have k - A224787(k) = 1.  Are there any others? - Will Gosnell and Robert Israel, Aug 01 2025

			a(3) = 630 = 2 * 3^2 * 5 * 7 is a term because 630 - 2^3 - 2 * 3^3 - 5^3 - 7^3 = 100 is a square.

Robert Israel, Table of n, a(n) for n = 1..2511

Cf. A224787, A385238, A386245, A386304, A386640.

filter:= proc(n) local t;
   issqr(n - add(t[1]^3*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^6]);

A385238 Numbers k such that A224787(k) - k is a square.

Original entry on oeis.org

8, 16, 20, 25, 95, 169, 221, 234, 295, 312, 323, 410, 543, 1027, 1681, 3071, 3419, 3721, 4183, 4352, 6649, 7448, 7979, 8188, 9047, 9200, 10108, 11203, 12769, 15732, 16240, 20303, 22819, 25351, 26291, 28769, 32761, 33728, 42880, 51198, 51338, 52206, 53613, 55303, 56800, 63731, 65567, 71531, 77550

Offset: 1

Will Gosnell and Robert Israel, Jul 28 2025

nonn

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

Includes p^2 for p in A027862.

			a(3) = 20 = 2^2 * 5 is a term because 2*2^3 + 5^3 - 20 = 121 is a square.

Robert Israel, Table of n, a(n) for n = 1..2451

Cf. A027862, A224787, A386623, A386640.

filter:= proc(n) local t;
   issqr(add(t[1]^3*t[2], t=ifactors(n)[2]) - n)
end proc:
select(filter, [$1..10^5]);

A386257 Numbers k such that k + A067666(k) is a square.

Original entry on oeis.org

1, 15, 80, 192, 1472, 1482, 1512, 1539, 1938, 2090, 2197, 2370, 2805, 3045, 4095, 4356, 4557, 5796, 5978, 6018, 6156, 7130, 7920, 11445, 12125, 12852, 13578, 13800, 15435, 20405, 26562, 29375, 29592, 30996, 31141, 31682, 32205, 42975, 45733, 46060, 49218, 50652, 51645, 51834, 52767, 54272, 55272

Offset: 1

Will Gosnell and Robert Israel, Jul 16 2025

nonn

Numbers k such that the sum of k and the squares of its prime factors with multiplicity is a square.
The only term that is a semiprime is 15.
The generalized Bunyakovsky conjecture implies that there are infinitely many pairs of primes (p,q) with 4 * q = 21 * p^2 - 10 * p - 99.  For such p and q, 5*p*q is a term.

			a(4) = 192 is a term because 192 = 2^6 * 3 and 192 + 6 * 2^2 + 3^2 = 225 = 15^2 is a square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A067666, A386304.

filter:= proc(n) local t;
  issqr(n + add(t[1]^2*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^5]);

spf[{p_,e_}]:=e*p^2;Q[k_]:=IntegerQ[Sqrt[k+Total[spf/@FactorInteger[k]]]];Join[{1},Select[Range[56000],Q[#]&]] (* James C. McMahon, Jul 23 2025 *)

isok(k) = my(f=factor(k)); issquare(k + sum(i=1, #f~, f[i, 1]^2*f[i, 2])); \\ Michel Marcus, Jul 20 2025

A386304 Numbers k such that k - A067666(k) is a square.

Original entry on oeis.org

1, 16, 27, 75, 128, 343, 475, 600, 663, 715, 759, 1015, 1845, 2679, 3717, 3933, 4440, 5083, 5325, 5467, 6120, 6210, 6325, 6405, 6859, 7029, 8349, 8541, 8664, 9125, 9960, 12045, 12427, 12535, 13509, 15067, 16677, 18693, 18711, 21783, 22797, 23250, 23560, 24605, 25527, 26496, 26967, 27117, 28557

Offset: 1

Will Gosnell and Robert Israel, Jul 17 2025

nonn

Numbers k such that k minus the sum of the squares of its prime factors with multiplicity is a square.
Is there any number other than 1 in both this sequence and A386257?
Contains no semiprimes.

			a(4) = 75 is a term because 75 = 3 * 5^2 and 75 - 3^2 - 2 * 5^2 = 16 = 4^2 is a square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A067666, A386257.

filter:= proc(n) local t;
  issqr(n - add(t[1]^2*t[2], t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^5]);

spf[{p_,e_}]:=e*p^2;Q[k_]:=IntegerQ[Sqrt[k-Total[spf/@FactorInteger[k]]]];Select[Range[29000],Q[#]&] (* James C. McMahon, Jul 23 2025 *)

isok(k) = my(f=factor(k)); issquare(k - sum(i=1, #f~, f[i, 1]^2*f[i, 2])); \\ Michel Marcus, Jul 20 2025

A386228 Primes that are the sum of prime factors (with multiplicity) of a triprime which is the concatenation of three consecutive primes.

Original entry on oeis.org

8539, 11813, 19181, 27827, 45013, 52859, 64621, 64969, 81077, 103583, 105373, 127493, 228203, 264791, 297397, 318161, 324491, 439753, 466247, 480299, 491353, 496631, 561091, 613559, 638431, 678943, 779981, 822631, 827537, 906673, 908893, 1039477, 1046029, 1079927, 1090577, 1176871, 1220327

Offset: 1

Will Gosnell and Robert Israel, Jul 15 2025

Numbers that are the sum of prime factors (with multiplicity) of at least one member of A385968.

			a(3) = 19181 is a term because 487491499 = A385968(4) is the concatenation of consecutive primes 487, 491, 499 and 487491499 = 11 * 2689 * 16481 with 11 + 2689 + 16481 = 19181 prime.
The only term < 3 * 10^9 that arises in more than one way is
a(756) = 149573911 = 53281 + 121110841 + 28409789
                 = 143597911 + 524453 + 5451547
where 53281 * 121110841 * 28409789 = 183325718332591833269 = A385968(3382)
and 143597911 * 524453 * 5451547 = 410557941055894105601 = A385968(6601).

Robert Israel, Table of n, a(n) for n = 1..5098

Cf. A385968.

tcat:= proc(a, b, c);
  c + 10^(1+ilog10(c))*(b + 10^(1+ilog10(b))*a)
end proc:
xmax:= 10^15: Bmax:= 3*10^6:
B:= NULL: count:= 0:
q:= 2: r:= 3:
do
  p:= q; q:= r; r:= nextprime(r);
  x:= tcat(p, q, r);
  if x > xmax then break fi;
  F:= ifactors(x)[2];
  if add(t[2], t=F) = 3 then
     b:= add(t[1]*t[2], t=F);
     if b <= Bmax and isprime(b) then
       count:= count+1; B:= B, b;
  fi fi;
od:
sort(convert({B},list));

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

Will Gosnell and Robert Israel, Jul 16 2025

nonn

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?

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

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A056899, A075255, A386246.

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]);

A386246 Composite numbers k such that A075254(k) is a square.

Original entry on oeis.org

27, 108, 171, 240, 456, 603, 744, 936, 988, 1424, 1702, 1737, 1820, 1899, 1904, 1989, 2166, 2261, 2366, 2817, 2873, 3283, 3553, 3681, 3728, 3784, 3852, 3894, 4266, 4437, 4700, 4923, 4975, 5005, 5008, 5073, 5117, 5193, 5278, 5356, 5418, 5820, 6050, 6486, 6576, 6627, 6651, 6775, 7947, 8250, 9116

Offset: 1

Will Gosnell and Robert Israel, Jul 16 2025

nonn

Composite numbers k such that k + sopfr(k) is a square, where sopfr(k) is the sum of prime factors of k with multiplicity.
Contains no semiprimes.
Includes 9*p if p is a prime of the form (x^2-6)/10 where x == 4 or 6 (mod 10).
Is this sequence disjoint from A386245?

			a(3) = 171 is a term because 171 = 3^2 * 19 is composite and 171 + 3 + 3 + 19 = 196 = 14^2 is a square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A075254, A386245.

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..10000]);

cp's OEIS Frontend

User: Will Gosnell

A385681 a(n) is the least k > 1 such that n^2 == k (mod sopfr(k)) and k^2 == n (mod sopfr(n)), or -1 if there is no such k, where sopfr = A001414.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A385664 Numbers y such that there is at least one x with 2 <= x < y, x^2 == y (mod sopfr(y)) and y^2 == x (mod sopfr(x)), where sopfr = A001414.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A386623 Numbers k such that k - A224787(k) is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A385238 Numbers k such that A224787(k) - k is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A386257 Numbers k such that k + A067666(k) is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A386304 Numbers k such that k - A067666(k) is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A386228 Primes that are the sum of prime factors (with multiplicity) of a triprime which is the concatenation of three consecutive primes.

Views

Author