A051448 -id:A051448 - cp's OEIS frontend

A164722 Numbers whose sum of distinct prime factors is a square.

1, 14, 28, 39, 46, 55, 56, 66, 92, 94, 98, 112, 117, 132, 155, 158, 183, 184, 186, 188, 196, 198, 203, 224, 255, 264, 275, 290, 291, 295, 299, 316, 323, 334, 351, 354, 368, 372, 376, 392, 396, 446, 448, 455, 506, 507, 528, 546, 549, 558, 579, 580, 583, 594

Offset: 1

Jonathan Vos Post, Aug 23 2009

This is to A008472 as A051448 is to A001414. It does seem that for any given k there should be a maximum n such that the sum of the prime factors of n = k^2, and a (perhaps different) maximum n such that the sum of distinct prime factors on n = k^2.

If k >= 3 and p = k^2 - 2 is prime (see A028870) then 2 * p is the term. - Marius A. Burtea, Jun 12 2019

			a(7) = 66 because 66 = 2 * 3 * 11 has sum of distinct prime factors 2 + 3 + 11 = 16 = 4^2. 8748 = 2^2 * 3^7 is the largest number whose prime factors (with multiplicity) add to 25 = 5^2, but it is not in this sequence because the sum of distinct prime factors of 8748 is 2 + 3 = 5, which is not a square.

Marius A. Burtea, Table of n, a(n) for n = 1..14587 (terms up to 10^6)

Cf. A000290, A001414, A008472, A028870, A051448.

[n:n in [1..600]| IsPower(&+PrimeDivisors(n), 2)]; // Marius A. Burtea, Jun 12 2019

Select[Range[600],IntegerQ[Sqrt[Total[Transpose[FactorInteger[#]] [[1]]]]]&] (* Harvey P. Dale, Mar 05 2014 *)

isOK(n) = local(fac, i); fac = factor(n); issquare(sum(i=1, matsize(fac)[1], fac[i, 1])); \\ Michel Marcus, Mar 19 2013

{n such that A008472(n) = k^2 for k an integer}.

{n such that A008472(n) is in A000290}.

More terms (including missing terms 56, 183, and 196) from Jon E. Schoenfield, May 27 2010

A359443 Primes p such that if q is the next prime, the sum (with multiplicity) of prime factors of p^2 + q^2 is a square.

Original entry on oeis.org

11, 17, 23, 79, 131, 229, 1019, 1123, 1583, 3299, 4019, 4091, 15307, 28813, 29147, 35083, 35933, 43427, 43597, 47809, 68683, 69029, 72047, 80173, 80513, 82483, 83257, 84263, 92567, 94583, 100693, 118603, 129517, 155317, 163243, 165553, 190181, 191021, 198901, 199211, 223439, 225721, 257273, 265117

Offset: 1

Robert Israel, Jan 01 2023

nonn

Suggested in an email by J. M. Bergot.

Primes prime(k) such that prime(k)^2 + prime(k+1)^2 is in A051448.

			a(3) = 23 is a term because 23 and 29 are consecutive primes with 23^2 + 29^2 = 1370 = 2*5*137, and 2+5+137 = 144 = 12^2.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A001414, A051448.

q:= 2: R:= NULL: count:=0:
while count < 100 do
  p:= q; q:= nextprime(p);
  s:= p^2 + q^2;
  if issqr(add(t[1]*t[2], t = ifactors(s)[2])) then
    R:= R, p; count:= count+1;
  fi;
od:
R;

A359445 Numbers k such that the sums (with multiplicity) of prime factors of k and k+1 are both squares.

Original entry on oeis.org

255, 290, 323, 578, 1484, 2219, 2418, 2491, 4370, 4706, 5243, 6075, 7139, 7930, 9378, 10082, 10554, 10603, 12716, 15872, 16739, 18146, 18938, 22424, 22842, 25227, 25283, 25959, 26910, 28364, 28448, 30255, 33669, 33698, 34316, 34317, 38895, 40179, 41261, 43343, 43999, 47384, 60400, 62695, 64970

Offset: 1

Robert Israel, Jan 01 2023

nonn

Numbers k such that k and k+1 are both in A051448.

Numbers k such that k, k+1 and k+2 are all in A051448 include 34316, 594044, and 869123. Are there numbers k for which k, k+1, k+2 and k+3 are all in A051448?

			a(3) = 323 is a term because 323 = 17*19 with 17+19 = 36 = 6^2 and 324 = 2^2*3^4 with 2*2 + 4*3 = 16 = 4^2.

Robert Israel, Table of n, a(n) for n = 1..750

Cf. A001414, A051448.

A:= select(proc(n) local t; issqr(add(t[1]*t[2], t=ifactors(n)[2])) end proc, {$1..10^5}):
B:= A intersect map(`-`,A,1):
sort(convert(B,list));

Module[{nn=65000,sq},sq=Table[If[IntegerQ[Sqrt[Total[Times@@@FactorInteger[n]]]],1,0],{n,nn}];SequencePosition[sq,{1,1}]][[;;,1]] (* Harvey P. Dale, Apr 12 2024 *)

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A164722 Numbers whose sum of distinct prime factors is a square.

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A359443 Primes p such that if q is the next prime, the sum (with multiplicity) of prime factors of p^2 + q^2 is a square.

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Maple

A359445 Numbers k such that the sums (with multiplicity) of prime factors of k and k+1 are both squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica