OEIS A369825. Michael Thomas De Vlieger, St. Louis, Missouri, 2024 0220 2200. Sequence definition: a(1) = 1, a(2) = 2. Q = Product a(1...n–1), R = rad(Q), r = rad(a(n–2)), s = R/r, rad(n) = A7947(n). a(n) = minimal m(s) * s != a(N), N < n. (Set counter m(s) = 1, incrementing upon the appearance of m(s) × s in the sequence as n increases.) 1. Definition implies incrementation of m until m*s != a(N), N < n. 2. Definition implies s | R, hence the source of primes q coprime to R must be m. 3. R is a primorial P(k) = A002110(k) as a consequence of incrementation of m. The index k is nondecreasing as n increases. Record setting m = prime(k) at n = N imply R = R*prime(k) at n = N+2. 4. Prime terms a(n) = p have the following sources (per David James Sycamore): i). m = p and s = 1, which implies both R = rad(a(n-2)) = P(k) and a(n) = prime(pi(gpf(R))+1). ii). m = 1 and s = p = prime(j), where s = R/r = P(k)/prime(j), 1 <= j <= k. 5. Let k = omega(R) = A1221(R). As k increases, there is a k/2^k = A000265(k)/A075101(k) chance of prime s, meaning that prime a(n) becomes increasingly less likely as n increases. 6. For n < 667722, with R = P(k+1) following prime(k+1) | a(A) at some A < n, we admit a(n-2) = P(k) and a(n) = prime(k+1) where n < B such that R = P(k+2) following prime(k+2) | a(B). For sufficiently small n, primes a(n) = prime(k+1) follow primorials a(n-2) = P(k) in the sequence. 7. Primorials and primes decouple such that a(676472) = P(18), but a(676474) = 4757 = prime(19)*prime(20). This is the result of R increasing twice (at n = 253724 and 667722), offering a greater degree of freedom for a(676474) than for terms that follow previous instances of primorials in the sequence. We also have a(9061722) = 73, following a(9061720) = P(23)/73. Conjecture: primorials and primes are not necessarily associated such that a(n-2) = P(k), a(n) = prime(k+1) for n >= 676472. 8. Primorials a(n) = P(k) do not appear in order. a(681764) = P(19) immediately precedes a(2191323) = P(21), followed by a(2348393) = P(20). This is a consequence of the increase of R as described above. 9. Primes do not appear in order for n > 667722. 10. Powerful numbers in the sequence are sparse, since they require w to appear m times, such that m*w is powerful: a(3) = 2^2, a(6) = 2^3, a(12) = 3^2, a(15) = 5^2, a(32) = 7^2, a(157) = 2^2*5^2, a(1101) = 17^2, a(1325) = 2^5*5, a(1329) = 5^3 are the only powerful numbers in the sequence for n <= 2^24. 11. Cellular automaton-like behavior. We can examine this sequence much like a linear cellular automaton where n represents iteration and the automata action spreads along the range of primes p(k). Generally, if p(k) | a(n) but p(k) does not divide a(n-1), then p(k) | a(n+1). As we proceed, p(k) does not divide either a(n+1) or a(n+2) unless p(k) | m for either n+1 or n+2. Therefore, through m, we perturb the general repetitive tendency of the prime factorization of terms a(n) as n increases. Examples of cellular automaton-like behavior: . indicates p divides neither s nor m(s), hence p does not divide a(n). o indicates p | s x indicates p | m(s). * indicates p divides both s and m(s). primes 1111 n a(n) 23571379 s m(s) ----------------------------------- 180 5720 *.o.oo.. 1430 4 181 102 oo....o. 102 1 182 1785 .oxo..o. 357 5 183 10010 x.oooo.. 5005 2 184 286 o...oo.. 286 1 185 51 .o....o. 51 1 186 3570 xooo..o. 1785 2 187 20020 *.oooo.. 10010 2 188 143 ....oo.. 143 1 189 153 .*....o. 51 3 190 7140 *ooo..o. 3570 2 191 30030 oxoooo.. 10010 3 P(6) 192 572 x...oo.. 143 4 193 17 ......o. 17 1 prime(7) 194 5355 .*oo..o. 1785 3 195 60060 *ooooo.. 30030 2 196 1144 *...oo.. 286 4 197 34 x.....o. 17 2 198 8925 .o*o..o. 1785 5 199 15015 .ooooo.. 15015 1 200 2288 *...oo.. 286 8 201 68 *.....o. 34 2 202 10710 x*oo..o. 1785 6 203 45045 .*oooo.. 15015 3 primes 111122 n a(n) 2357137939 s m(s) ----------------------------------------------- 1156 174 oo.......o 174 1 1157 44370 o*o...o..o 14790 3 1158 74364290 x.ooooooo. 37182145 2 1159 4811807 ...o*o.oo. 437437 11 1160 87 .o.......o 87 1 1161 59160 *oo...o..o 14790 4 1162 148728580 *.ooooooo. 74364290 2 1163 5686681 ...oo*.oo. 437437 13 1164 261 .*.......o 87 3 1165 73950 oo*...o..o 14790 5 1166 223092870 oxooooooo. 74364290 3 P(9) 1167 6998992 x..ooo.oo. 437437 16 1168 29 .........o 29 1 prime(10) 1169 36975 .o*...o..o 7395 5 1170 446185740 *oooooooo. 223092870 2 1171 7873866 ox.ooo.oo. 874874 9 1172 58 x........o 29 2 1173 2465 ..o...o..o 2465 1 1174 111546435 .oooooooo. 111546435 1 1175 10498488 *o.ooo.oo. 2624622 4 1176 116 *........o 58 2 1177 4930 x.o...o..o 2465 2 1178 334639305 .*ooooooo. 111546435 3 1179 9186177 .o.*oo.oo. 1312311 7 Some landmarks in the sequence given a dataset of 3355432 = 2^25 terms: Primes: n a(n-2) a(n-1) a(n) type ------------------------------------------------------------------------------- 2 - 1 2 given 4 2 4 3 i 7 6 8 5 i 16 30 25 7 i 47 210 165 11 ii 96 2310 462 13 ii 193 30030 572 17 ii 476 510510 37791 19 ii 697 9699690 298452 23 ii 1168 223092870 6998992 29 ii 1349 6469693230 35728 31 ii 4613 200560490130 33418770 37 ii 8898 7420738134810 10894807 41 ii 19728 304250263527210 8367028670 43 ii 40553 13082761331670030 997922298373 47 ii 49054 614889782588491410 383652161569677 53 ii 63802 32589158477190044730 63955203 59 ii 240925 1922760350154212639070 5857159 61 ii 681766<-A 7858321551080267055879090 39539899226751 71 ii 2191325 40729680599249024150621323470 799046620225530 79 ii 9061722<-B 3658418023140765087063342712230 132296516895946767 73 ii<-B 13178788 267064515689275851355624017992790 17306414990986836 89 ii 26120340 23768741896345550770650537601358310 8234657406320993934 97 ii Note A: a(681766) = 71 represents the first prime in the sequence not in order as n increases. Note B: R = P(23), rad(a(9061722)) = P(23)/prime(21). In all other cases (ii), we have R = P(k), a(n) = prime(k). Primorials: n a(n-2) a(n-1) a(n) ---------------------------------------------- 2 - 1 P(1) 5 4 3 P(2) 14 9 18 P(3) 45 22 308 P(4) 94 117 130 P(5) 191 153 7140 P(6) 474 171 6160 P(7) 695 46 1495 P(8) 1166 261 73950 P(9) 1347 62 44908305 P(10) 4611 333 222053 P(11) 8896 369 586449630 P(12) 19726 387 1563609 P(13) 40551 423 616170 P(14) 49052 477 9343900 P(15) 63800 118 30064173983690 P(16) 240923 549 20024790407671530 P(17) 676472 42813 847785601274157 P(18) 681764 639 14110830857890 P(19) 2191323 711 8053709723301242 P(21) 2348393 51903 920204380270594 P(20) 13178786 801 2746812890910600295 P(23) 26120338 194 559966942201079210 P(24) Sequence Product Kernels: k n a(n) ---------------------------------------- 1 2 2 2 4 3 3 7 5 4 16 7 5 35 55 6 52 91 7 154 119 8 268 8151 9 493 1771 10 1105 493 11 1333 155 12 3860 2312167 13 8029 2105719 14 13028 96422555 15 33417 759943 16 44018 7066543 17 58178 357599 18 221400 52130576535557 19 253724 259062133 20 667722<-C 1321500303643 21 944756 7441255 22 2012604 154889025528423719 23 4910792 11036458183015773022537 24 9957971 8889402465047819 25 18950928 12985293660308779 Note C: n = 667722 is the point where R advances to P(k+2) before P(k) enters the sequence. Full list of powerful terms for n <= 16777216: i n a(n-2) a(n-1) a(n) --------------------------------------------------- 1 3 1 2 4 2^2 2 6 3 6 8 2^3 3 12 10 20 9 3^2 4 15 6 30 25 5^2 5 32 30 50 49 7^2 6 145 2002 1287 225 3^2 * 5^2 7 157 51051 8580 100 2^2 * 5^2 8 1101 13123110 168 289 17^2 9 1120 462120945 8748740 196 2^2 * 7^2 10 1325 646969323 17864 160 2^5 * 5^1 11 1329 1293938646 31262 125 5^3