How to apply a function to one part while keeping the rest

Question

I have a list and want to apply a function just on the second Part and keep the rest.

t1 = Table[{k, 2^k + 1}, {k, 2, 7}]

yielding

{{2, 5}, {3, 9}, {4, 17}, {5, 33}, {6, 65}, {7, 129}}

I want to FactorInteger the second parts to get

 {{2, {5}}, {3, {3}}, {4, {17}}, {5, {3, 11}}, {6, {5, 13}}, {7, {3, 43}}}

By not using Cases like

 Cases[t1, {a_, b_} :> {a, FactorInteger[b][[All, 1]]}]

but by applying a function (maybe Hold or HoldPattern) which leaves k untouched when applying FactorInteger on the whole of t1. (I hope it makes sense what I am looking for.)

Answer 1

MapAt[Map[First] @* FactorInteger, {All, 2}] @ t1

{{2, {5}}, {3, {3}}, {4, {17}}, {5, {3, 11}}, {6, {5, 13}}, {7, {3,  43}}}

SubsetMap[#[[All,All,1]]& @* FactorInteger, {All, 2}] @ t1

{{2, {5}}, {3, {3}}, {4, {17}}, {5, {3, 11}}, {6, {5, 13}}, {7, {3,  43}}}

{#, FactorInteger[#2][[All, 1]]} & @@@ t1

{{2, {5}}, {3, {3}}, {4, {17}}, {5, {3, 11}}, {6, {5, 13}}, {7, {3,  43}}}

Module[{t = #}, 
   t[[All, 2]] = FactorInteger[t[[All, 2]]][[All, All, 1]]; t] & @ t1

{{2, {5}}, {3, {3}}, {4, {17}}, {5, {3, 11}}, {6, {5, 13}}, {7, {3, 43}}}

Answer 2

MapAt[FactorInteger[#][[;; , 1]] &, t1, {All, 2}]

Answer 3

Rule approach starts with

rule1 = {x_,y_} :> {x, First /@ FactorInteger[y]};

and then

Replace[z1,rule1,{1}]

gives

    {{2,{5}},{3,{3}},{4,{17}},{5,{3,11}},{6,{5,13}},{7,{3,43}}}

Answer 4

t1//Transpose[{#[[All,1]],FactorInteger[#[[All,2]]][[All,All,1]]}]&

{{2, {5}}, {3, {3}}, {4, {17}}, {5, {3, 11}}, {6, {5, 13}}, {7, {3, 43}}}

As Inner may be thought of as a generalized form of dot, a function may also be applied only to the y values as follows:

Inner[Times,t1,{1,1},{#1,FactorInteger[#2][[All,1]]}&]

{2, {5}}, {3, {3}}, {4, {17}}, {5, {3, 11}}, {6, {5, 13}}, {7, {3, 43}}}

---

Fun with Inner/Dot

ll={{a,b},{c,d}}

To multiply all y values by 10:

ll.{{1,0},{0,10}}
Inner[#1 #2&, ll, {1,1},{#1,10 #2}&]

(*
  {{a, 10 b}, {c, 10 d}}
  {{a, 10 b}, {c, 10 d}} 
*)

Or:

ll.{{1,0},{0,10}}==
Inner[#1 #2&, ll, {1,1},{#1,10 #2}&]==
Inner[Times, ll, {1,1},{#1,10 #2}&]==
Inner[Times, ll, {{1,0},{0,10}}]

True

To apply a function only to the y values of ll:

Inner[Times,ll,{1,1},{#1,f@#2}&]

(* {{a, f[b]}, {c, f[d]}} *)

Answer 5

Another option, /@ is usually my go-to for these sorts of things

f := FactorInteger[#][[All,1]]&

{#[[1]], f@#[[2]]} & /@ t1