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 A260443 Prime factorization representation of Stern polynomials: a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = a(n)*a(n+1). 93
 1, 2, 3, 6, 5, 18, 15, 30, 7, 90, 75, 270, 35, 450, 105, 210, 11, 630, 525, 6750, 245, 20250, 2625, 9450, 77, 15750, 3675, 47250, 385, 22050, 1155, 2310, 13, 6930, 5775, 330750, 2695, 3543750, 128625, 1653750, 847, 4961250, 643125, 53156250, 18865, 24806250, 202125, 727650, 143, 1212750, 282975, 57881250, 29645, 173643750, 1414875, 18191250, 1001 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The exponents in the prime factorization of term a(n) give the coefficients of the n-th Stern polynomial. See A125184 and the examples. None of the terms have prime gaps in their factorization, i.e., all can be found in A073491. Contains neither perfect squares nor prime powers with exponent > 1. A277701 gives the positions of the terms that are 2*square. - Antti Karttunen, Oct 27 2016 Many of the derived sequences (like A002487) have similar "Fir forest" or "Gaudian cathedrals" style scatter plot. - Antti Karttunen, Mar 21 2017 LINKS Antti Karttunen, Table of n, a(n) for n = 0..1024 FORMULA a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = a(n)*a(n+1). Other identities. For all n >= 0: A001221(a(n)) = A277314(n). [#nonzero coefficients in each polynomial.] A001222(a(n)) = A002487(n). [When each polynomial is evaluated at x=1.] A048675(a(n)) = n. [at x=2.] A090880(a(n)) = A178590(n). [at x=3.] A248663(a(n)) = A264977(n). [at x=2 over the field GF(2).] A276075(a(n)) = A276081(n). ["at factorials".] A156552(a(n)) = A277020(n). [Converted to "unary-binary" encoding.] A051903(a(n)) = A277315(n). [Maximal coefficient.] A277322(a(n)) = A277013(n). [Number of irreducible polynomial factors.] A005361(a(n)) = A277325(n). [Product of nonzero coefficients.] A072411(a(n)) = A277326(n). [And their LCM.] A007913(a(n)) = A277330(n). [The squarefree part.] A000005(a(n)) = A277705(n). [Number of divisors.] A046523(a(n)) = A278243(n). [Filter-sequence.] A284010(a(n)) = A284011(n). [True for n > 1. Another filter-sequence.] A003415(a(n)) = A278544(n). [Arithmetic derivative.] A056239(a(n)) = A278530(n). [Weighted sum of coefficients.] A097249(a(n)) = A277899(n). a(A000079(n)) = A000040(n+1). a(A000225(n)) = A002110(n). a(A000051(n)) = 3*A002110(n). For n >= 1, a(A000918(n)) = A070826(n). A007949(a(n)) is the interleaving of A000035 and A005811, probably A101979. A061395(a(n)) = A277329(n). Also, for all n >= 1: A055396(a(n)) = A001511(n). A252735(a(n)) = A061395(a(n)) - 1 = A057526(n). a(A000040(n)) = A277316(n). a(A186891(1+n)) = A277318(n). [Subsequence for irreducible polynomials]. EXAMPLE n a(n) prime factorization Stern polynomial ------------------------------------------------------------ 0 1 (empty) B_0(x) = 0 1 2 p_1 B_1(x) = 1 2 3 p_2 B_2(x) = x 3 6 p_2 * p_1 B_3(x) = x + 1 4 5 p_3 B_4(x) = x^2 5 18 p_2^2 * p_1 B_5(x) = 2x + 1 6 15 p_3 * p_2 B_6(x) = x^2 + x 7 30 p_3 * p_2 * p_1 B_7(x) = x^2 + x + 1 8 7 p_4 B_8(x) = x^3 9 90 p_3 * p_2^2 * p_1 B_9(x) = x^2 + 2x + 1 MATHEMATICA a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[a@ n, {n, 0, 56}] (* Michael De Vlieger, Apr 05 2017 *) PROG (PARI) A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ After Charles R Greathouse IV's code for "ps" in A186891. \\ Antti Karttunen, Oct 11 2016 (Scheme) ;; Uses memoization-macro definec: (definec (A260443 n) (cond ((<= n 1) (+ 1 n)) ((even? n) (A003961 (A260443 (/ n 2)))) (else (* (A260443 (/ (- n 1) 2)) (A260443 (/ (+ n 1) 2)))))) ;; A more standalone version added Oct 10 2016, requiring only an implementation of A000040 and the memoization-macro definec: (define (A260443 n) (product_primes_to_kth_powers (A260443as_coeff_list n))) (define (product_primes_to_kth_powers nums) (let loop ((p 1) (nums nums) (i 1)) (cond ((null? nums) p) (else (loop (* p (expt (A000040 i) (car nums))) (cdr nums) (+ 1 i)))))) (definec (A260443as_coeff_list n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_coeff_list (/ n 2)))) (else (add_two_lists (A260443as_coeff_list (/ (- n 1) 2)) (A260443as_coeff_list (/ (+ n 1) 2)))))) (define (add_two_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (add_two_lists nums2 nums1)) (else (map + nums1 (append nums2 (make-list (- len1 len2) 0))))))) (Python) from sympy import factorint, prime, primepi from functools import reduce from operator import mul def a003961(n): F = factorint(n) return 1 if n==1 else reduce(mul, (prime(primepi(i) + 1)**F[i] for i in F)) def a(n): return n + 1 if n<2 else a003961(a(n//2)) if n%2==0 else a((n - 1)//2)*a((n + 1)//2) print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017 CROSSREFS Same sequence sorted into ascending order: A260442. Cf. A000040, A000079, A000225, A001222, A002487, A003415, A003961, A005811, A007949, A046523, A056239, A073491, A090880, A097249, A101979, A125184, A178590, A186891, A206284, A277314, A277315, A277325, A277326, A277329, A277330, A277701, A277705, A277899, A278243, A278530, A278544, A284010, A284011. Cf. also A048675, A277333 (left inverses). Cf. A277323, A277324 (bisections), A277200 (even terms sorted), A277197 (first differences), A277198. Cf. A277316 (values at primes), A277318. Cf. A023758 (positions of squarefree terms), A101082 (of terms not squarefree), A277702 (positions of records), A277703 (their values). Cf. A283992, A283993 (number of irreducible, reducible polynomials in range 1 .. n). Cf. also A206296 (Fibonacci polynomials similarly represented). Sequence in context: A329354 A332461 A319344 * A328316 A373003 A206242 Adjacent sequences: A260440 A260441 A260442 * A260444 A260445 A260446 KEYWORD nonn,look AUTHOR Antti Karttunen, Jul 28 2015 EXTENSIONS More linking formulas added by Antti Karttunen, Mar 21 2017 STATUS approved

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Last modified May 20 05:07 EDT 2024. Contains 372703 sequences. (Running on oeis4.)