Subject: Prime of factorial and primorial Thema: Primzahlen aus Fakultaeten Main table of content: http://www.fermatquotient.com/ Definitions: n! ==> factorial n!2 ==> factorial with only odd or only even numbers p(n) ==> prime with number n n# ; p# ==> primorial p(0)# = 1# = 1 = 1! = 0! p#2 ==> primorial with only odd or only even numbers of primes References: http://oeis.org/A...... for example: http://oeis.org/A154525 https://t5k.org/top20/page.php?id=30 https://t5k.org/top20/page.php?id=5 n!+1 is prime for n=0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429 (2193027 digits, A002981). n!-1 is prime for n=3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003, 632760 (3395992 digits, A002982). n!^2+1 is prime for n=0, 1, 2, 3, 4, 5, 9, 10, 11, 13, 24, 65, 76 (223 digits, A046029). n!^4+1 is prime for n=0, 1, 2, 3, 4, 13, 112, 328, 11123 (160712 digits, A051855). p#+1 is prime for the primes p=1, 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, 5256037, 6369619, 7351117 (3191401 digits, A005234). p#-1 is prime for primes p=3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113, 4778027, 6354977, 6533299 (2835864 digits, A006794). p#^2+1 is prime for primes p=1, 2, 3, 7, 11, 19, 53, 1571 (1328 digits, A092061). p#^4+1 is prime for primes p=1, 2, 3, 7, 83, 383, 2281, 10477 (18042 digits, A133834). Sequence product: 1*2*3*4*5*7*9*11*13*17*19*23*25*29*31*37*41*43*47*49*53*...±1 n#*[Int(Sqrt(n))#]^2+1 is prime for n=1, 2, 3, 9, 29, 31, 37, 49, 59, 113, 151, 557, 907, 977, 1657, 2281, 3881, 5167, 15887, 22691, 57503, 76991 (33505 digits). n#*[Int(Sqrt(n))#]^2-1 is prime for n=3, 4, 7, 9, 13, 23, 37, 109, 313, 367, 2699, 2939, 7817, 10037, 17489, 22003, 24821, 34213, 40427 (17597 digits). Equivalent primes only with minimum n Sequence product for example n=49: 2^2*3^2*5^2*7^2*11*13*17*19*23*29*31*37*41*43*47±1 Sequence product: 1*2*3*2*5*7*3*11*13*17*19*23*5*29*31*37*41*43*47*7*53*59*...±1 n#*Int(Sqrt(n))#+1 is prime for n=1, 2, 3, 4, 5, 7, 13, 43, 53, 97, 109, 113, 269, 397, 439, 547, 733, 4007, 7309, 7793, 8377, 11369, 75557 (32789 digits). n#*Int(Sqrt(n))#-1 is prime for n=3, 4, 5, 7, 9, 11, 13, 17, 49, 97, 107, 127, 149, 193, 379, 421, 547, 563, 619, 859, 2447, 3923, 9973, 12967, 15077 (6523 digits). Equivalent primes only with minimum n Least common multiple (or lcm) of {1, 2, ..., n} lcm(1,2,...,n)+1 is prime for n=1, 2, 3, 4, 5, 7, 9, 19, 25, 31, 47, 89, 127, 139, 1369, 2251, 3271, 4253, 4373, 4549, 5449, 13331, 37123, 55291 (23998 digits, A051453 & A154525). lcm(1,2,..,n)-1 is prime for n=3, 4, 5, 7, 8, 19, 23, 29, 32, 47, 61, 97, 181, 233, 307, 401, 887, 1021, 1087, 1361, 1481, 2053, 2293, 5407, 5857, 11059, 14281, 27277, 27803, 36497, 44987, 53017 (23021 digits, A154524). lcm(1,2,..,n)^2+1 is prime for n=1, 2, 3, 7, 13, 89 (78 digits). Equivalent primes only with minimum n n!/n#+1 is prime for the primes n=1, 2, 3, 4, 8, 14, 20, 26, 34, 56, 104, 153, 182, 194, 217, 230, 280, 281, 462, 529, 1445, 2515, 3692, 6187, 6851, 13917, 17258, 48934 (187118 digits, A140294). n!/n#-1 is prime for the primes n=4, 5, 6, 8, 16, 17, 21, 34, 39, 45, 50, 72, 73, 76, 133, 164, 202, 216, 221, 280, 281, 496, 605, 2532, 2967, 3337, 8711, 10977, 13724, 15250, 18160, 20943, 33684, 41400 (155301 digits, A140293). (n!/n#)^2+1 is prime for the primes n=1, 2, 3, 4, 5, 6, 7, 12, 13, 22, 23, 39, 50, 54, 60, 61, 69, 182, 620, 767, 1308, 5129 (29223 digits, A108948). 2*n!+1 is prime for n=0, 1, 2, 3, 5, 12, 18, 35, 51, 53, 78, 209, 396, 4166, 9091, 9587, 13357, 15917, 17652, 46127 (195105 digits, A051915). 2*n!-1 is prime for n=2, 3, 4, 5, 6, 7, 14, 15, 17, 22, 28, 91, 253, 257, 298, 659, 832, 866, 1849, 2495, 2716, 2773, 2831, 3364, 5264, 7429, 28539, 32123, 37868 (156928 digits, A076133). n!/2+1 is prime for n=2, 4, 5, 7, 8, 13, 16, 30, 43, 49, 91, 119, 213, 1380, 1637, 2258, 4647, 9701, 12258 (44795 digits, A082672). n!/2-1 is prime for n=3, 4, 5, 6, 9, 31, 41, 373, 589, 812, 989, 1115, 1488, 1864, 1918, 4412, 4686, 5821, 13830 (51264 digits, A082671). 2*p#+1 is prime for n=1, 2, 3, 5, 7, 11, 31, 61, 83, 101, 113, 409, 659, 857, 1373, 2711, 5897, 6869, 7699, 32983, 37021, 44491 (19215 digits, A119834). 2*p#-1 is prime for n=2, 3, 5, 7, 17, 19, 37, 71, 79, 113, 857, 863, 16361, 62989 (27238 digits, A119833). p#/2+2 is prime for n=2, 3, 5, 7, 13, 29, 31, 37, 47, 59, 109, 223, 307, 389, 457, 1117, 1151, 2273, 9137, 10753, 15727, 25219, 26459, 29251, 30259, 52901, 194471 (84246 digits, A096177). p#/2-2 is prime for n=5, 7, 11, 13, 17, 19, 23, 31, 41, 53, 71, 103, 167, 431, 563, 673, 727, 829, 1801, 2699, 4481, 6121, 7283, 9413, 10321, 12491, 17807, 30307, 31891, 71917 (31096 digits, A096547). p#/2+4 is prime for n=2, 3, 5, 7, 17, 61, 73, 349, 383, 2141, 2287, 4021, 4157, 32003, 53551, 58217 (25215 digits). p#/2-4 is prime for n=5, 7, 11, 17, 37, 47, 101, 127, 229, 353, 1109, 1753, 2473, 12953, 16673, 34511, 42169, 168761, 169249 (73218 digits). Sequence product: 2*(1*3*5*7*11*...)±1 or 2*(2*4*6*8*10*12*...)±1 2*(n!2)+1 is prime for n=1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 18, 24, 28, 30, 39, 51, 149, 197, 355, 444, 623, 908, 1084, 1125, 1577, 1748, 2584, 2628, 2702, 3973, 5046, 6048, 6058, 7123, 9706, 14223 (26449 digits, A215775). 2*(n!2)-1 is prime for n=2, 3, 5, 9, 11, 13, 25, 35, 46, 56, 68, 69, 71, 84, 92, 103, 110, 121, 193, 259, 308, 368, 406, 426, 620, 627, 1206, 1631, 1845, 2108, 2210, 2786, 7931, 8008, 8367, 12181, 12291 (22467 digits, A215779). n!2+1 is prime for n=1, 2, 518, 33416, 37310, 52608, 123998, 220502 (541239 digits, A080778). n!2-1 is prime for n=3, 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, 728, 842, 888, 2328, 3326, 6404, 8670, 9682, 27056, 44318, 76190, 100654, 145706 (344538 digits, A007749). n!2+2 is prime for n=1, 3, 5, 7, 9, 21, 23, 27, 57, 75, 103, 169, 219, 245, 461, 695, 1169, 3597, 3637, 7495, 27743, 28799, 32501 (66266 digits, A076185). n!2-2 is prime for n=5, 7, 15, 17, 19, 51, 73, 89, 131, 153, 245, 333, 441, 463, 825, 1771, 2027, 9157, 10875, 20515 (39779 digits, A094144). n!2+4 is prime for n=1, 3, 5, 7, 11, 17, 29, 39, 43, 73, 93, 315, 549, 2059, 5543, 6937, 22819, 34523 (70841 digits, A076186). n!2-4 is prime for n=5, 7, 9, 11, 13, 15, 19, 25, 35, 79, 81, 105, 171, 243, 271, 295, 355, 523, 591, 1211, 3073, 11157, 12887, 19825 (38294 digits, A123910). n!2+(n-1)!2+2 is prime for n=2, 3, 4, 8, 10, 12, 14, 18, 30, 31, 44, 51, 99, 246, 494, 544, 565, 616, 730, 747, 854, 1077, 1297, 1839, 2673, 3349, 3758, 7959, 16761, 20012, 24430 (48297 digits). n!2+(n-1)!2-2 is prime for n=3, 6, 7, 8, 9, 10, 12, 13, 14, 15, 44, 46, 55, 56, 59, 116, 146, 154, 300, 575, 652, 939, 1642, 2077, 2410, 2662, 3307, 9273, 11411, 17413, 18924 (36363 digits). n!2-(n-1)!2+2 is prime for n=2, 3, 4, 7, 8, 9, 10, 13, 15, 20, 21, 24, 57, 59, 60, 62, 76, 98, 112, 121, 178, 290, 553, 883, 988, 2834, 3795, 4822, 5901, 10398, 12974, 20099 (38883 digits). n!2-(n-1)!2-2 is prime for n=4, 5, 6, 8, 11, 16, 38, 39, 40, 44, 69, 86, 172, 210, 320, 362, 445, 464, 681, 725, 751, 908, 947, 1016, 1033, 1725, 4182, 5666, 6403, 7386, 8381, 10262, 15366, 22571 (44234 digits). Primo-factorials: n!+n#+1 is prime for n=1, 2, 3, 4, 5, 6, 8, 17, 18, 24, 95, 96, 142, 1022, 1120, 1580, 6942, 19255, 19401 (74765 digits, A081710). n!+n#-1 is prime for n=2, 3, 4, 5, 8, 17, 23, 26, 35, 47, 82, 100, 147, 183, 271, 492, 708, 1116, 1538, 2491, 4207, 4468, 16878 (64022 digits, A081711). n!-n#+1 is prime for n=4, 6, 7, 8, 10, 20, 21, 26, 101, 119, 172, 409, 621, 1043, 1204, 1283, 1673, 2003, 4336, 5773, 12913, 13517 (49970 digits, A081712). n!-n#-1 is prime for n=4, 5, 20, 92, 106, 266, 308, 343, 583, 597, 903, 1021, 1239, 1314, 2458, 6160, 9627, 10649 (38265 digits, A081713). Sequence product for example n=7; p(7)=17: 2*5*11*17±3*7*13 = 1870±273 [p(n)#2]+[p(n-1)#2] is prime for p(n)=2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 59, 79, 101, 109, 137, 157, 271, 379, 701, 709, 863, 1259, 2393, 2939, 3527, 4943, 5023, 5237, 6373, 14401, 18047, 31849, 32569, 39799, 47717, 49939, 56843, 115807, 172199 (37265 digits). [p(n)#2]+[p(n-1)#2] is prime for n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 22, 26, 29, 33, 37, 58, 75, 126, 127, 150, 205, 356, 424, 492, 661, 674, 697, 831, 1687, 2068, 3422, 3495, 4183, 4920, 5128, 5764, 10943, 15683 (37265 digits, A095135). [p(n)#2]-[p(n-1)#2] is prime for p(n)=5, 7, 11, 13, 17, 59, 79, 97, 109, 139, 173, 271, 397, 523, 839, 1579, 4013, 5843, 7583, 9749, 9887, 11483, 12049, 12517, 14207, 15569, 17383, 23173, 26261, 46337, 47917, 49157, 60457, 69931, 149351, 167119 (36187 digits). [p(n)#2]-[p(n-1)#2] is prime for n=3, 4, 5, 6, 7, 17, 22, 25, 29, 34, 40, 58, 78, 99, 146, 249, 554, 767, 963, 1203, 1220, 1384, 1444, 1495, 1671, 1816, 1998, 2586, 2886, 4792, 4938, 5052, 6098, 6931, 13792, 15271 (36187 digits, A095138). [p(n)#2]^2+[p(n-1)#2]^2 is prime for p(n)=2, 3, 5, 7, 11, 13, 17, 29, 43, 61, 89, 113, 263, 409, 967, 22453, 22709, 49253 (21268 digits). [p(n)#2]^2+[p(n-1)#2]^2 is prime for n=1, 2, 3, 4, 5, 6, 7, 10, 14, 18, 24, 30, 56, 80, 163, 2512, 2537, 5062 (21268 digits). Left Factorial (L!n)/2 = [0!+1!+2!+...+(n-1)!]/2 (L!n)/2 is prime for n=3, 4, 5, 8, 9, 10, 11, 30, 76, 163, 271, 273, 354, 721, 1796, 3733, 4769, 9316, 12221, 41532 (173772 digits, A100614 & A124375). Left alternating Factorial La!n = Abs(1!-2!+3!-4!+-...)-(n-1)!+n! La!n is prime for n=3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961 (260448 digits, A001272). Left Primorial [L#p(n)]/2 = [p(1)#+p(2)#+p(3)#+...+p(n)#]/2 = [2#+3#+5#+...+p(n)#]/2 [L#p(n)]/2 is prime for p(n)=5, 11, 17, 31, 47, 211, 227, no more (89 digits). [L#p(n)]/2 is prime for n=3, 5, 7, 11, 15, 47, 49, no more (89 digits, A060389). [L#p(n)]-1 is prime for p(n)=3, 5, 11, 13, 17, 19, 31, 43, 73, 179, 181, 191, 373, 397, no more (162 digits) [L#p(n)]-1 is prime for n=2, 3, 5, 6, 7, 8, 11, 14, 21, 41, 42, 43, 74, 78, no more (162 digits, A285528) Left alternating Primorial [La#p(n)]/2 = [p(1)#-p(2)#+p(3)#-+...+-p(n-1)#+p(n)#]/2 = [2#-3#+5#-7#+11#-+...+-p(n-1)#+p(n)#]/2 [La#p(n)]/2 is prime for p(n)=5, 11, 17, 67, 137, 691, 2293, 4153, no more (1775 digits). [La#p(n)]/2 is prime for n=3, 5, 7, 19, 33, 125, 341, 571, no more (1775 digits). Calculate minimum until 33344 digits. New 04.12.2018 Update 05.03.2025 Richard Fischer