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URL: https://oeis.org/A391258

⇱ A391258 - OEIS


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A391258
Odd semiprimes p*q such that Stern polynomial B(p*q,x) is neither a multiple of B(p,x) nor of B(q,x).
3
25, 55, 65, 77, 91, 95, 115, 121, 133, 143, 145, 161, 169, 185, 203, 205, 209, 215, 235, 247, 253, 265, 289, 295, 299, 301, 305, 319, 323, 329, 335, 361, 377, 391, 395, 407, 415, 427, 445, 451, 469, 473, 481, 485, 493, 505, 515, 517, 529, 533, 545, 551, 559, 581, 583, 623, 629, 649, 655, 667, 671, 679, 689, 695
OFFSET
1,1
LINKS
EXAMPLE
Stern polynomial B(25,x) = B(5*5,x) = x^3 + 3*x^2 + 2*x + 1 is irreducible, thus is not a multiple of Stern polynomial B(5, x) = 2*x + 1, and therefore 25 is included in this sequence.
Stern polynomial B(481,x) = B(13*37,x) = x^7 + 2*x^6 + 3*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1, which factorizes as (x^3 + x^2 + 1)(x^4 + x^3 + 2*x^2 + 2*x + 1), and because neither of the factors is B(13, x) = 2*x^2 + 2*x + 1 nor B(37,x) = x^3 + 5*x^2 + 4*x + 1, 481 is included in this sequence. Note that 481 is the first term that represents a reducible polynomial (i.e., is listed in A391336 rather than A391337). [Edited by Peter Munn, Dec 18 2025]
PROG
(PARI)
memo_for_ps = Map();
ps(n) = if(n<2, n, my(v); if(mapisdefined(memo_for_ps, n, &v), v, v = if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)); mapput(memo_for_ps, n, v); (v)));
p2r(p) = { my(v=Vecrev(Vec(p))); prod(i=1, #v, prime(i)^v[i]); };
is_A391258(n) = if(!(n%2) || 2!=bigomega(n), 0, my(f=factor(n), a = f[1, 1], b = f[#f~, 1], Pa = ps(a), Pb = if(b==a, Pa, ps(b)), Pn = ps(n)); (0!=lift(Pn % Pa) && 0!=lift(Pn % Pb)));
CROSSREFS
Setwise difference A046315 \ A391257.
Cf. A125184, A260443 for a description of Stern polynomials.
Differs from its subsequence A391337 for the first time at n=43, where a(43) = 481, while A391337(43) = 485. See also A391336 and A389918.
Cf. also A391254.
Sequence in context: A157269 A371129 A186892 * A391337 A206075 A391254
KEYWORD
nonn,changed
AUTHOR
Antti Karttunen, Dec 07 2025
STATUS
approved