Are there four numbers in AP such that their prime factors are also in AP?

Solution 1:

Not a 'real' answer, but it was too big for a comment. I think that you're looking for a solution without using a calculator or PC but maybe this gives some insight. I did a quick search where I look for in the range $0\le\text{n}\le10^6$.

I wrote and ran some Mathematica-code:

In[1]:=Clear["Global`*"];
ParallelTable[
  If[Length[
      DeleteCases[
       Table[If[PrimeQ[Part[Divisors[n], k]], Part[Divisors[n], k], 
         a], {k, 1, Length[Divisors[n]]}], a]] >= 3 && 
    Length[DeleteDuplicates[
       Differences[
        DeleteCases[
         Table[If[PrimeQ[Part[Divisors[n], k]], Part[Divisors[n], k], 
           a], {k, 1, Length[Divisors[n]]}], a]]]] == 1, n, 
   Nothing], {n, 0, 10^6}] //. {} -> Nothing

Running the code gives:

Out[1]={105, 231, 315, 525, 627, 693, 735, 897, 935, 945, 1575, 1581, 1617, 
1729, 1881, 2079, 2205, 2465, 2541, 2625, 2691, 2835, 2967, 3675, 
4123, 4301, 4675, 4715, 4725, 4743, 4851, 5145, 5487, 5643, 6237, 
6615, 6897, 7623, 7685, 7875, 7881, 8073, 8505, 8901, 9717, 10285, 
10707, 11025, 11319, 11339, 11661, 11913, 12103, 12325, 13125, 14175, 
14229, 14553, 14993, 15435, 15895, 16377, 16461, 16929, 17353, 17787, 
18375, 18711, 19845, 20213, 20631, 20691, 20915, 21505, 22477, 22869, 
23375, 23575, 23625, 23643, 23779, 24219, 25327, 25515, 25725, 26331, 
26703, 26765, 26877, 27951, 28861, 29151, 29341, 29607, 32021, 32121, 
32851, 33075, 33335, 33957, 34983, 35739, 36015, 38425, 39375, 40587, 
40807, 41905, 42525, 42687, 42911, 43659, 46305, 47311, 48635, 49011, 
49131, 49321, 49383, 50787, 51425, 53361, 54739, 55125, 55581, 55637, 
56133, 59535, 59563, 60297, 61625, 61893, 62073, 63017, 65625, 67731, 
68241, 68607, 69443, 70875, 70929, 71029, 71485, 72657, 73117, 75597, 
75867, 76545, 76751, 76985, 77175, 78337, 78993, 79233, 79475, 80109, 
80189, 80631, 83503, 83853, 84721, 86437, 87453, 88821, 91875, 95631, 
96363, 98923, 99225, 99485, 101065, 101177, 101303, 101871, 102131, 
102311, 104575, 104949, 107217, 107525, 108045, 108445, 111381, 
113135, 116875, 117875, 118125, 119377, 121471, 121761, 124509, 
124729, 127575, 127581, 127813, 128061, 128625, 130977, 131043, 
133825, 138915, 143479, 146481, 146969, 147033, 147393, 148149, 
151593, 152279, 152361, 157339, 160083, 160993, 163493, 164923, 
165375, 165831, 166453, 166675, 166743, 168335, 168399, 170097, 
174845, 176149, 177289, 178605, 180075, 180891, 184265, 185679, 
186219, 192125, 192763, 193315, 194937, 195657, 196875, 196883, 
202027, 203193, 204723, 205821, 207217, 208639, 209525, 210239, 
212201, 212625, 212787, 213931, 217167, 217971, 218285, 221757, 
222865, 226347, 226791, 227601, 228241, 229635, 229957, 231525, 
232667, 236555, 236979, 237699, 240327, 240463, 241893, 243175, 
251559, 252105, 257125, 258427, 260797, 262359, 263683, 265227, 
266463, 268203, 270215, 275625, 278185, 286893, 289089, 291597, 
292201, 294011, 294409, 296367, 297675, 298351, 305613, 307461, 
308125, 311023, 314847, 315935, 321651, 323317, 323637, 323733, 
324135, 328125, 328831, 329759, 334143, 334907, 337393, 343621, 
346317, 347687, 352231, 354375, 357425, 358343, 361691, 365283, 
365585, 373527, 375747, 381433, 382725, 382743, 383165, 384183, 
384569, 384925, 385875, 386389, 392931, 393129, 396341, 397375, 
397891, 398397, 407305, 412129, 412647, 415817, 416745, 416941, 
427063, 434797, 439443, 441099, 442179, 444447, 454779, 456909, 
457083, 459375, 460401, 467443, 470051, 472021, 474513, 477987, 
480249, 481213, 490141, 494615, 496125, 497087, 497203, 497425, 
497493, 499913, 500229, 505197, 505325, 506717, 509615, 510291, 
512029, 520421, 522875, 523979, 530491, 535815, 537065, 537625, 
540225, 542225, 542673, 547973, 548359, 554631, 557037, 558657, 
559551, 563473, 565675, 577527, 584375, 584521, 584811, 586177, 
586971, 589375, 590625, 593047, 598553, 600081, 609579, 614169, 
617463, 623645, 624169, 629821, 637875, 638361, 642061, 643125, 
645569, 651501, 653913, 654065, 654387, 663247, 665271, 669125, 
676133, 679041, 680373, 682803, 683243, 685115, 685279, 688905, 
694575, 704671, 710937, 711773, 712385, 713097, 719331, 720107, 
720981, 721927, 725679, 729973, 754677, 756315, 758951, 763873, 
767643, 782585, 785137, 787077, 790901, 795681, 797597, 799389, 
804287, 804609, 811927, 821197, 825373, 826875, 828733, 829961, 
831417, 833187, 833375, 834537, 839843, 841675, 850297, 860679, 
867267, 867981, 871563, 874225, 874791, 877951, 885167, 888681, 
889101, 893025, 894691, 899963, 900375, 916839, 921325, 922383, 
935857, 939401, 944541, 950669, 950779, 959239, 959435, 960625, 
964953, 966575, 970911, 971199, 972405, 983005, 984375}

So, I will pick one value to show that is true. When $\text{n}=959435$ we have the following distinct prime factors: $\left(5,311,617\right)$ and they are in arithmetic progression.