Comments on A139414, A155814, and 155925 From: Harry J. Smith (hjsmithh(AT)sbcglobal.net) The four polynomials of A139414, A155814, and 155925 look remarkable to me. Here are the first 100 values. The nonprime values have been replaced by zeros. pE = n^2 + n + 41. n p1 p2 p3 p4 pE(Euler's polynomial) 0 1373 1459 1301 1877 41 1 1231 1319 1163 1741 43 2 1097 1187 1033 1613 47 3 971 1063 911 1493 53 4 853 947 797 1381 61 5 743 839 691 1277 71 6 641 739 593 1181 83 7 547 647 503 1093 97 8 461 563 421 1013 113 9 383 487 347 941 131 10 313 419 281 877 151 11 251 359 223 821 173 12 197 307 173 773 197 13 151 263 131 733 223 14 113 227 97 701 251 15 83 199 71 677 281 16 61 179 53 661 313 17 47 167 43 653 347 18 41 163 41 653 383 19 43 167 47 661 421 20 53 179 61 677 461 21 71 199 83 701 503 22 97 227 113 733 547 23 131 263 151 773 593 24 173 307 197 821 641 25 223 359 251 877 691 26 281 419 313 941 743 27 347 487 383 1013 797 28 421 563 461 1093 853 29 503 647 547 1181 911 30 593 739 641 1277 971 31 691 839 743 1381 1033 32 797 947 853 1493 1097 33 911 1063 971 1613 1163 34 1033 1187 1097 1741 1231 35 1163 1319 1231 1877 1301 36 1301 1459 1373 0 1373 37 1447 1607 1523 0 1447 38 1601 0 0 2333 1523 39 0 0 1847 0 1601 40 1933 2099 0 2677 0 41 2111 0 2203 2861 0 42 2297 2467 2393 0 1847 43 0 2663 2591 3253 1933 44 2693 0 2797 3461 0 45 2903 3079 3011 3677 2111 46 3121 3299 0 0 2203 47 3347 3527 3463 4133 2297 48 3581 0 3701 4373 2393 49 3823 4007 3947 4621 0 50 4073 4259 4201 4877 2591 51 0 4519 4463 0 2693 52 4597 4787 4733 5413 2797 53 4871 0 5011 5693 2903 54 5153 5347 5297 5981 3011 55 5443 5639 5591 6277 3121 56 5741 5939 0 6581 0 57 6047 6247 6203 0 3347 58 6361 6563 6521 7213 3463 59 0 0 0 7541 3581 60 7013 7219 0 7877 3701 61 7351 7559 7523 8221 3823 62 0 7907 7873 8573 3947 63 0 8263 8231 8933 4073 64 0 8627 8597 0 4201 65 8783 8999 8971 9677 0 66 9161 0 0 10061 4463 67 9547 9767 9743 10453 4597 68 9941 10163 10141 10853 4733 69 10343 10567 0 11261 4871 70 10753 10979 0 11677 5011 71 11171 11399 11383 12101 5153 72 11597 11827 11813 0 5297 73 0 12263 12251 12973 5443 74 12473 0 12697 13421 5591 75 12923 13159 13151 13877 5741 76 13381 13619 13613 14341 0 77 0 14087 14083 14813 6047 78 14321 14563 14561 0 6203 79 0 0 0 0 6361 80 0 0 15541 0 6521 81 15791 0 0 0 0 82 0 16547 16553 17293 0 83 16811 0 0 0 7013 84 17333 0 17597 18341 0 85 17863 18119 18131 0 7351 86 18401 0 0 19421 7523 87 18947 19207 0 19973 0 88 19501 19763 0 20533 7873 89 20063 20327 20347 21101 0 90 0 20899 20921 0 8231 91 21211 0 21503 0 0 92 0 22067 22093 22853 8597 93 22391 0 22691 0 8783 94 22993 0 23297 24061 8971 95 23603 23879 23911 24677 9161 96 0 24499 24533 25301 0 97 24847 25127 25163 25933 9547 98 0 25763 25801 26573 9743 99 0 26407 0 0 9941 100 0 27059 0 0 10141