A164788 -id:A164788 - cp's OEIS frontend

A165346 Numbers such that the sum of the distinct prime factors is a fourth power.

1, 39, 55, 66, 117, 132, 158, 198, 264, 275, 316, 351, 396, 507, 528, 594, 605, 632, 726, 792, 1053, 1056, 1095, 1188, 1255, 1264, 1375, 1452, 1491, 1506, 1521, 1584, 1782, 2112, 2130, 2178, 2211, 2376, 2528, 2904, 3012, 3025, 3111, 3159, 3168, 3285, 3363

Offset: 1

Jonathan Vos Post, Sep 15 2009

nonn

This is the 4th row of the infinite array A(k,n) = n-th positive integer such that the sum of the distinct prime factors is of the form j^k for integers j, k. The 2nd row is A164722 (hence the current sequence is a proper subset of A164722). The 3rd row is A164788. The smallest integers whose sum of distinct prime factors is 4^4 are {1255, 1506, 3012, ...}. The smallest integers whose sum of distinct prime factors is 5^4 are {9255, 21455, ...}. The smallest integers whose sum of distinct prime factors is 6^4 are {6455, 7746, ...}. The smallest integers whose sum of distinct prime factors is 7^4 are {4798, 9596, ...}.

			a(2) = 39, because 39 = 3*13, and 3+13 = 16 = 2^4.
a(7) = 158, because 158 = 2*79, and 2+79 = 81 = 3^4.

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Cf. A000583, A008472, A164722, A164788.

A008472 := proc(n) add( p, p = numtheory[factorset](n)) ; end: isA000583 := proc(n) iroot(n,4,'exct') ; exct ; end: A165346 := proc(n) if n = 1 then 1; else for a from procname(n-1)+1 do if isA000583(A008472(a)) then RETURN(a); fi; od: fi; end: seq(A165346(n),n=1..80) ; # R. J. Mathar, Sep 20 2009

a165346[n_] := Select[Range@n, IntegerQ[Power[Plus @@ Transpose[FactorInteger[#]][[1]], 1/4]] &]; a165346[3400] (* Michael De Vlieger, Jan 06 2015 *)

isok(n) = my(f=factor(n)); ispower(vecsum(f[,1]),4); \\ Michel Marcus, Jan 06 2015

{n such that A008472(n) = k^4 for k an integer}.

{n such that A008472(n) is in A000583}.

More terms from R. J. Mathar, Sep 20 2009

A367950 Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors (sopf) of a(n) + a(n + 1) is a perfect cube.

Original entry on oeis.org

1, 14, 31, 44, 91, 92, 43, 2, 13, 32, 103, 80, 55, 20, 25, 50, 85, 98, 37, 8, 7, 38, 97, 86, 49, 26, 19, 56, 79, 104, 121, 62, 73, 110, 115, 68, 67, 116, 109, 74, 61, 122, 163, 132, 3, 12, 33, 42, 93, 90, 45, 30, 15, 60, 75, 108, 27, 18, 57, 78, 105, 120, 63, 72, 111, 24, 21, 54, 81, 102

Offset: 1

Giorgos Kalogeropoulos and Eric Angelini, Dec 05 2023

nonn

			a(1) + a(2) =  1 + 14 =  15 whose sopf is  8, a perfect cube.
a(2) + a(3) = 14 + 31 =  45 whose sopf is  8, a perfect cube.
a(5) + a(6) = 91 + 92 = 183 whose sopf is 64, a perfect cube.

Éric Angelini, Sums of distinct prime factors, Personal blog, December 2023.

Cf. A008472, A164788.

a[1]=1;a[n_]:=a[n]=(k=1;While[MemberQ[ar=Array[a,n-1],k]|| !IntegerQ[Total[First/@FactorInteger[k+a[n-1]]]^(1/3)],k++];k);Array[a,70]

cp's OEIS Frontend

A165346 Numbers such that the sum of the distinct prime factors is a fourth power.

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