A260443 -id:A260443 - cp's OEIS frontend

A277316 Prime-factorization representation of the prime-th Stern-polynomial: a(n) = A260443(A000040(n)).

3, 6, 18, 30, 270, 450, 630, 6750, 9450, 22050, 2310, 3543750, 4961250, 53156250, 727650, 173643750, 25467750, 2668050, 40020750, 891371250, 9550406250, 1400726250, 3190703906250, 467969906250, 173423250, 16378946718750, 1715889656250, 245684200781250, 25738344843750, 8497739250, 510510, 6763506750, 66919696593750

Offset: 1

Antti Karttunen, Oct 10 2016

nonn

If the conjecture by Ulas and Ulas is true, then all these terms can be found from A206284 and then this is also a subsequence of A277318.

Antti Karttunen, Table of n, a(n) for n = 1..1028
Maciej Ulas and Oliwia Ulas, On certain arithmetic properties of Stern polynomials, arXiv:1102.5109 [math.CO], 2011.

Cf. A000040, A206284, A260443.
Cf. A277317 (same sequence sorted into ascending order) is a subsequence of A277319.
Differs from A277318 for the first time at n=10, where A277318(10) = 15750, a term which is missing from this sequence.

(define (A277316 n) (A260443 (A000040 n)))

a(n) = A260443(A000040(n)).
Other identities.
For all n >= 1, a(A059305(n)) = A002110(A000043(n)).

A277318 Prime-factorization representation of irreducible (non-constant) Stern-polynomials B(n,x), listed in the order of increasing index n: a(n) = A260443(A186891(n+1)).

Original entry on oeis.org

3, 6, 18, 30, 270, 450, 630, 6750, 9450, 15750, 22050, 2310, 3543750, 4961250, 53156250, 727650, 173643750, 18191250, 25467750, 2668050, 90090, 40020750, 891371250, 9550406250, 212713593750, 1400726250, 3190703906250, 467969906250, 5013963281250, 104053950, 173423250, 16378946718750, 1715889656250, 245684200781250

Offset: 1

Antti Karttunen, Oct 11 2016

nonn

Cf. A186891, A260443.
All terms are included in A206284 and in A277200 (apart from initial 3).
Cf. A277316 (conjectured subsequence), from which this differs for the first time at n=10, where a(10) = 15750 , while A277316(10) = 22050.

A284576 a(n) = A059896(A260443(n), A260443(1+n)).

Original entry on oeis.org

2, 6, 6, 30, 90, 270, 30, 210, 630, 6750, 6750, 1890, 15750, 47250, 210, 2310, 6930, 47250, 47250, 330750, 992250, 425250, 47250, 103950, 173250, 2315250, 2315250, 519750, 8489250, 25467750, 2310, 30030, 90090, 519750, 25467750, 3638250, 1910081250, 13023281250, 1447031250, 1400726250, 4202178750, 104186250, 2604656250

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A059896, A260443, A277324, A284577, A284578, A285106.

(define (A284576 n) (A059896bi (A260443 n) (A260443 (+ 1 n)))) ;; For A059896bi, see A059896.

a(n) = A059896(A260443(n), A260443(1+n)).
a(n) = A284577(n) * A284578(n).
a(n) = A277324(n) / A284578(n).

A284577 a(n) = A059897(A260443(n), A260443(1+n)).

Original entry on oeis.org

2, 6, 2, 30, 90, 270, 2, 210, 630, 6750, 2250, 378, 15750, 47250, 2, 2310, 6930, 6750, 630, 66150, 198450, 3402, 90, 14850, 24750, 92610, 30870, 14850, 8489250, 25467750, 2, 30030, 90090, 6750, 339570, 14850, 382016250, 372093750, 9843750, 1400726250, 4202178750, 3402, 198450, 20465156250, 7796250, 83531250, 90, 859950, 1433250, 1890

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A059897, A260443, A277324, A284576, A284578, A285107.

(define (A284577 n) (A059897bi (A260443 n) (A260443 (+ 1 n)))) ;; For A059897bi, see A059897.

a(n) = A059897(A260443(n), A260443(1+n)).
a(n) = A277324(n) / A000290(A284578(n)).
A001222(a(n)) = A285107(n).

A284578 a(n) = A059895(A260443(n), A260443(1+n)).

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 5, 1, 1, 105, 1, 1, 7, 75, 5, 5, 125, 525, 7, 7, 25, 75, 35, 1, 1, 1155, 1, 1, 77, 75, 245, 5, 35, 147, 1, 1, 30625, 13125, 7, 245, 245, 40425, 11, 11, 13475, 1029, 245, 245, 1715, 1617, 11, 77, 1225, 3675, 385, 1, 1, 15015, 1, 1, 1001, 75, 245, 2695, 1715, 3, 1, 7, 48125, 7203, 1, 35, 300125, 363, 1, 1, 75625

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A000196, A059895, A260443, A277324, A284576, A284577, A285108.

(define (A284578 n) (A059895bi (A260443 n) (A260443 (+ 1 n)))) ;; For A059895bi, see A059895.

a(n) = A059895(A260443(n), A260443(1+n)).
a(n) = A277324(n) / A284576(n) = A000196(A277324(n)/A284577(n)).
a(n) = A284576(n) / A284577(n).
A001222(a(n)) = A285108(n).

A277200 Even terms in A260442 (in A260443).

Original entry on oeis.org

2, 6, 18, 30, 90, 210, 270, 450, 630, 2310, 6750, 6930, 9450, 15750, 20250, 22050, 30030, 47250, 90090, 330750, 510510, 727650, 1212750, 1531530, 1653750, 2668050, 3543750, 4961250, 8489250, 9699690, 18191250, 24806250, 25467750, 29099070, 40020750, 53156250, 57881250, 104053950, 173423250, 173643750

Offset: 1

Antti Karttunen, Oct 14 2016

nonn

All odd terms larger > 1 in A260442 can be obtained from these terms by shifting their prime factorization some number of steps towards larger primes with A003961.

Cf. A003961, A260442, A260443.
Sequence A277324 sorted into ascending order.
Subsequence of A055932.
Cf. A002110, A277317 (subsequences, apart from their initial terms).
Also all terms of A277318 apart from initial 3 are included in this sequence.

allocatemem(2^30);
A061395(n) =  if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
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))));
isA277200(n) = (!(n%2) && (omega(n) == A061395(n)) && (A260443(A048675(n)) == n));
i=1; n=0; while(i < 100, n++; if(isA277200(n), write("b277200.txt", i, " ", n); i++));

A277327 Number of distinct primes dividing gcd(A260443(n), A260443(n+1)): a(n) = A001221(A277198(n)).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 0, 2, 2, 1, 1, 2, 3, 0, 0, 3, 2, 1, 1, 2, 3, 1, 1, 3, 3, 2, 2, 3, 4, 0, 0, 4, 3, 2, 2, 3, 3, 1, 1, 3, 3, 2, 2, 3, 4, 1, 1, 4, 3, 2, 2, 3, 4, 2, 2, 4, 4, 3, 3, 4, 5, 0, 0, 5, 4, 3, 3, 4, 4, 2, 2, 4, 3, 2, 2, 3, 4, 1, 1, 4, 3, 2, 2, 3, 4, 2, 2, 4, 4, 3, 3, 4, 5, 1, 1, 5, 4, 3, 3, 4, 4, 2, 2, 4, 4, 3, 3, 4, 5, 2, 2, 5, 4, 3, 3, 4, 5, 3, 3

Offset: 0

Antti Karttunen, Oct 13 2016

nonn

a(n) = number of column positions where both row n and n+1 of A125184 have nonzero number present (when scanned from left), in other words, the number of k such that the term t^k has a nonzero coefficient in both Stern polynomials, B(n,t) and B(n+1,t).

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A001221, A125184, A260443, A277198, A277328.

(define (A277327 n) (A001221 (A277198 n)))
;; A standalone implementation:
(define (A277327 n) (length (filter positive? (gcd_of_exp_lists (A260443as_coeff_list n) (A260443as_coeff_list (+ 1 n))))))
(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)))))))
(define (gcd_of_exp_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (gcd_of_exp_lists nums2 nums1)) (else (map min nums1 (append nums2 (make-list (- len1 len2) 0)))))))

a(n) = A001221(A277198(n)).

a(n) <= A277328(n).

A277899 a(n) = A097249(A260443(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 2, 0, 1, 0, 0, 0, 3, 2, 3, 1, 3, 2, 2, 0, 2, 1, 2, 0, 1, 0, 0, 0, 4, 3, 4, 2, 4, 3, 3, 1, 3, 3, 3, 2, 3, 2, 2, 0, 3, 2, 3, 1, 3, 2, 2, 0, 2, 1, 2, 0, 1, 0, 0, 0, 5, 4, 5, 3, 5, 4, 4, 2, 4, 4, 4, 3, 4, 3, 4, 1, 4, 3, 4, 3, 4, 3, 4, 2, 4, 3, 4, 2, 3, 2, 2, 0, 4, 3, 4, 2, 4, 3, 3, 1, 3, 3, 3, 2, 3, 2, 3, 0, 3, 2, 3, 1, 3, 2, 2, 0

Offset: 0

Antti Karttunen, Nov 15 2016

nonn

a(n) = number of times we must iterate A097246, starting at A260443(n), before the result is squarefree.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A097246, A097249, A260443, A277905.

Cf. A023758 (positions of zeros).

(define (A277899 n) (A097249_for_coeff_list (A260443as_coeff_list n)))
(define (A097249_for_coeff_list nums) (let loop ((nums nums) (s 0)) (if (<= (reduce max 0 nums) 1) s (loop (A097246_for_coeff_list nums) (+ 1 s)))))
(define (A097246_for_coeff_list nums) (add_two_lists (map A000035 nums) (cons 0 (map A004526 nums))))
;; For the other required functions, see A260443.

a(n) = A097249(A260443(n)).

A283983 Square root of the largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A000188(A260443(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 5, 3, 1, 15, 1, 1, 1, 3, 5, 15, 7, 45, 5, 15, 1, 15, 35, 15, 1, 105, 1, 1, 1, 3, 5, 105, 7, 225, 35, 525, 11, 1575, 175, 1125, 7, 1575, 35, 105, 1, 105, 35, 525, 77, 1575, 35, 525, 1, 105, 385, 105, 1, 1155, 1, 1, 1, 3, 5, 1155, 7, 1575, 385, 3675, 11, 7875, 1225, 275625, 77, 55125, 2695, 5775, 13, 17325, 13475, 275625, 539

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A000188, A000196, A001222, A003961, A260443, A283989, A284264, A283484 (odd bisection).

Cf. A023758 (positions of ones).

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n]  (* after Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]];  A000188[n_]:= Sum[Boole[Mod[i^2, n] == 0], {i, n}]; Table[A000188[A260443[n]], {n, 0, 50}] (* Indranil Ghosh, Mar 28 2017 *)

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)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A000188(n) = core(n, 1)[2]; \\ This function from Michel Marcus, Feb 27 2013
A283983(n) = A000188(A260443(n));

(define (A283983 n) (A000188 (A260443 n)))
(define (A283983 n) (A000196 (A283989 n)))

a(n) = A000188(A260443(n)).
a(n) = A000196(A283989(n)).
Other identities. For all n >= 0:
a(2n) = A003961(a(n)).
A001222(a(n)) = A284264(n).

A277323 Even bisection of A260443 (the odd terms): a(n) = A260443(2*n).

Original entry on oeis.org

1, 3, 5, 15, 7, 75, 35, 105, 11, 525, 245, 2625, 77, 3675, 385, 1155, 13, 5775, 2695, 128625, 847, 643125, 18865, 202125, 143, 282975, 29645, 1414875, 1001, 444675, 5005, 15015, 17, 75075, 35035, 15563625, 11011, 346644375, 2282665, 108945375, 1859, 544726875, 15978655, 12132553125, 121121, 3813088125, 2697695

Offset: 0

Antti Karttunen, Oct 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A000035, A005811, A007949, A101979, A112765, A260443, A277324.

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[2 n], {n, 0, 46}] (* Michael De Vlieger, Apr 05 2017 *)

a(n) = A260443(2*n).
a(0) = 1; for n >= 1, a(n) = A003961(A260443(n)).
Other identities. For all n >= 0:
A007949(a(n)) = A000035(n).
A112765(a(n)) is the interleaving of A000035 and A005811, probably A101979.

cp's OEIS Frontend

A277316 Prime-factorization representation of the prime-th Stern-polynomial: a(n) = A260443(A000040(n)).

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A277318 Prime-factorization representation of irreducible (non-constant) Stern-polynomials B(n,x), listed in the order of increasing index n: a(n) = A260443(A186891(n+1)).

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A284576 a(n) = A059896(A260443(n), A260443(1+n)).

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A284577 a(n) = A059897(A260443(n), A260443(1+n)).

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A284578 a(n) = A059895(A260443(n), A260443(1+n)).

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A277200 Even terms in A260442 (in A260443).

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A277327 Number of distinct primes dividing gcd(A260443(n), A260443(n+1)): a(n) = A001221(A277198(n)).

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A277899 a(n) = A097249(A260443(n)).

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A283983 Square root of the largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A000188(A260443(n)).

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