p-adic valuations of Yang's coefficients b_n: verification ok b_n from exp(-sum u**k/(k(k+1))) equals the classical recurrence, n <= 250 [ 1.9 s] ok 1/T(u) = (1-u)**(-1/u)/e has the OEIS coefficients A055505/A055535, n <= 17 [ 1.9 s] ok b_n = a_{n-1} - a_n with a_n = A055505(n)/A055535(n), n <= 17 [ 1.9 s] ok S(u) = 1/(1 - B(u)) = T(u)/(1-u), r <= 250 [ 2.9 s] ok b_1..b_6 = 1/2, 1/24, 1/48, 73/5760, 11/1280, 3625/580608 [ 2.9 s] ok p=3, b: brute-force M(n) = formula of Proposition 6, n <= 100 [ 2.9 s] ok p=3, b: unique optimum <=> criterion of Theorem 7, n <= 100 [ 2.9 s] ok p=3, b: v_p >= -M(n) (Lemma 4), n <= 100 [ 2.9 s] ok p=3, T: brute-force M(n) = formula of Proposition 6, n <= 100 [ 2.9 s] ok p=3, T: unique optimum <=> criterion of Theorem 7, n <= 100 [ 2.9 s] ok p=3, T: v_p >= -M(n) (Lemma 4), n <= 100 [ 2.9 s] ok p=5, b: brute-force M(n) = formula of Proposition 6, n <= 100 [ 2.9 s] ok p=5, b: unique optimum <=> criterion of Theorem 7, n <= 100 [ 2.9 s] ok p=5, b: v_p >= -M(n) (Lemma 4), n <= 100 [ 2.9 s] ok p=5, T: brute-force M(n) = formula of Proposition 6, n <= 100 [ 3.0 s] ok p=5, T: unique optimum <=> criterion of Theorem 7, n <= 100 [ 3.0 s] ok p=5, T: v_p >= -M(n) (Lemma 4), n <= 100 [ 3.0 s] ok p=7, b: brute-force M(n) = formula of Proposition 6, n <= 100 [ 3.0 s] ok p=7, b: unique optimum <=> criterion of Theorem 7, n <= 100 [ 3.0 s] ok p=7, b: v_p >= -M(n) (Lemma 4), n <= 100 [ 3.0 s] ok p=7, T: brute-force M(n) = formula of Proposition 6, n <= 100 [ 3.0 s] ok p=7, T: unique optimum <=> criterion of Theorem 7, n <= 100 [ 3.0 s] ok p=7, T: v_p >= -M(n) (Lemma 4), n <= 100 [ 3.0 s] ok p=11, b: brute-force M(n) = formula of Proposition 6, n <= 100 [ 3.0 s] ok p=11, b: unique optimum <=> criterion of Theorem 7, n <= 100 [ 3.0 s] ok p=11, b: v_p >= -M(n) (Lemma 4), n <= 100 [ 3.0 s] ok p=11, T: brute-force M(n) = formula of Proposition 6, n <= 100 [ 3.0 s] ok p=11, T: unique optimum <=> criterion of Theorem 7, n <= 100 [ 3.0 s] ok p=11, T: v_p >= -M(n) (Lemma 4), n <= 100 [ 3.0 s] ok p=13, b: brute-force M(n) = formula of Proposition 6, n <= 100 [ 3.0 s] ok p=13, b: unique optimum <=> criterion of Theorem 7, n <= 100 [ 3.0 s] ok p=13, b: v_p >= -M(n) (Lemma 4), n <= 100 [ 3.0 s] ok p=13, T: brute-force M(n) = formula of Proposition 6, n <= 100 [ 3.1 s] ok p=13, T: unique optimum <=> criterion of Theorem 7, n <= 100 [ 3.1 s] ok p=13, T: v_p >= -M(n) (Lemma 4), n <= 100 [ 3.1 s] ok p=3: v_3(b_n) > -M(n) at exactly 23 indices n <= 80, the smallest being 5 [ 3.1 s] ok p=3: the conditions for i < s_p(m) and for 1 <= i <= m agree, m < 3000 [ 5.7 s] ok p=5: the conditions for i < s_p(m) and for 1 <= i <= m agree, m < 3000 [ 6.6 s] ok p=7: the conditions for i < s_p(m) and for 1 <= i <= m agree, m < 3000 [ 7.3 s] ok p=11: the conditions for i < s_p(m) and for 1 <= i <= m agree, m < 3000 [ 7.7 s] ok p=13: the conditions for i < s_p(m) and for 1 <= i <= m agree, m < 3000 [ 8.0 s] ok p=3: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (64 indices <= 250) [ 8.0 s] ok p=5: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (19 indices <= 250) [ 8.0 s] ok p=7: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (8 indices <= 250) [ 8.0 s] ok p=11: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (5 indices <= 250) [ 8.0 s] ok p=13: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (3 indices <= 250) [ 8.0 s] ok p=17: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (1 indices <= 250) [ 8.0 s] ok p=19: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (1 indices <= 250) [ 8.0 s] ok p=23: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (1 indices <= 250) [ 8.0 s] ok p=29: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (1 indices <= 250) [ 8.0 s] ok p=31: v_p(b_n) = v_p(T_n) = -(m + v_p(m!)) on N_p (1 indices <= 250) [ 8.0 s] ok p=3 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=3 K=2: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=3 K=3: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=3 K=4: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=3 K=5: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=5 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=5 K=2: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=5 K=3: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=7 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=7 K=2: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=11 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=11 K=2: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=13 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=13 K=2: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=17 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=19 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=23 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=29 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok p=31 K=1: R_K in U_p, M(p**K-2) = D_K - 1, b, T, S minimal at p**K-1 only [ 8.0 s] ok M(p**K-2) <= D_K - 1 and R_K in U_p for p=3 (K<=7), p=5,7,11 (K<=4) [ 8.1 s] ok p=3: Delta = p-1 on U_p, and 30004 = #U_p in [1,60000] = X/(p-1) +4.00 within bounds [ 9.2 s] ok p=5: Delta = p-1 on U_p, and 15006 = #U_p in [1,60000] = X/(p-1) +6.00 within bounds [ 9.8 s] ok p=7: Delta = p-1 on U_p, and 10009 = #U_p in [1,60000] = X/(p-1) +9.00 within bounds [ 10.2 s] ok p=11: Delta = p-1 on U_p, and 6006 = #U_p in [1,60000] = X/(p-1) +6.00 within bounds [ 10.5 s] ok p=13: Delta = p-1 on U_p, and 5007 = #U_p in [1,60000] = X/(p-1) +7.00 within bounds [ 10.8 s] ok p=17: Delta = p-1 on U_p, and 3771 = #U_p in [1,60000] = X/(p-1) +21.00 within bounds [ 11.0 s] ok p=19: Delta = p-1 on U_p, and 3357 = #U_p in [1,60000] = X/(p-1) +23.67 within bounds [ 11.3 s] ok p=23: Delta = p-1 on U_p, and 2760 = #U_p in [1,60000] = X/(p-1) +32.73 within bounds [ 11.6 s] ok p=29: Delta = p-1 on U_p, and 2166 = #U_p in [1,60000] = X/(p-1) +23.14 within bounds [ 11.8 s] ok p=31: Delta = p-1 on U_p, and 2015 = #U_p in [1,60000] = X/(p-1) +15.00 within bounds [ 12.0 s] ALL CHECKS PASSED