A276574 -id:A276574 - cp's OEIS frontend

A276573 The infinite trunk of least squares beanstalk: The only infinite sequence such that a(0) = 0 and a(n-1) = a(n) - least number of squares (A002828) that sum to a(n).

0, 3, 6, 8, 11, 15, 16, 18, 21, 24, 27, 30, 32, 35, 38, 40, 43, 45, 48, 51, 53, 56, 59, 63, 64, 67, 70, 72, 75, 78, 80, 83, 85, 88, 90, 93, 96, 99, 102, 105, 108, 112, 115, 117, 120, 123, 126, 128, 131, 134, 136, 139, 143, 144, 147, 149, 152, 155, 158, 160, 162, 165, 168, 171, 173, 176, 179, 183, 186, 189, 192, 195

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10028

Cf. A002828, A005563, A255131, A260731, A260733, A262689, A276572, A276574, A276575 (first differences), A277016 (squares present), A277015 (their square roots), A277888 (primes), A278486 (numbers one more than a prime), A278265, A278487, A278488, A278491 (another subsequence), A278497, A278498, A278499, A278513, A278516, A278517, A278518, A278519, A278521, A278522.
Cf. A277890 & A277891 (number of even and odd terms in each range. The latter seem to be slightly more numerous), A277889.
Positions of nonzero terms in A278515.
Subsequence of A278489, no common terms with A278490.
Cf. also A179016, A259934, A276583, A276613, A276623 for similar constructions.

(define (A276573 n) (A276574 (A276572 n)))

a(n) = A276574(A276572(n)).
Other identities and observations. For all n >= 0:
A260731(a(n)) = n.
a(A260733(n+1)) = A005563(n).
A278517(n) <= a(n) <= A278519(n).
A010873(a(n)) = A278499(n). [Terms reduced modulo 4.]
A010877(a(n)) = A278488(n). [modulo 8.]
A046523(a(n)) = A278497(n). [Least number with the same prime signature.]
A008683(a(n)) = A278513(n).
A065338(a(n)) = A278498(n).
A278509(a(n)) = A278265(n).
A278216(a(n)) = A278516(n). [Number of children the n-th node of the trunk has.]

Definition clarified and more identities added to the Formula section by Antti Karttunen, Nov 28 2016

A277487 a(n) = number of primes encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 2, 1, 2, 0, 3, 2, 0, 3, 0, 2, 0, 1, 4, 2, 3, 2, 4, 2, 0, 3, 3, 2, 5, 3, 4, 3, 3, 3, 2, 4, 2, 2, 4, 3, 3, 3, 6, 3, 1, 3, 4, 2, 6, 3, 3, 2, 5, 5, 5, 5, 4, 3, 7, 4, 4, 6, 4, 2, 4, 6, 5, 5, 5, 4, 7, 4, 4, 7, 4, 0, 5, 6, 7, 4, 4, 9, 4, 5, 2, 6, 6, 7, 11, 3, 6, 4, 9, 5, 7, 7, 7, 6, 8, 8, 7, 6, 4, 6, 5, 7, 8, 5, 9, 8, 8, 5, 12, 7, 5, 6

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

Number of primes on row n of A276574, after the initial zero-row.
Note how for the most n in range 1..10000, a(n) < A277486(n), even though for the most n in the same range A277890(n) < A277891(n). In range n=1..10000, there are only 209 cases where a(n) >= A277486(n).
On the other hand, when a(n) is compared to A277488(n), there is no such marked bias.

			For n=3, starting from k = ((3+1)^2)-1, and iterating k -> A255131(k), yields 15 -> 11 -> 8, where the iteration stops as the next lower number one less than a square has been reached. Of these numbers only 11 is a prime, thus a(3) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000290, A002828, A010052, A276574, A277486, A277488, A277890, A277891, A278167 (partial sums).

Cf. A277888.

istwo(n:int)=my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1
isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7
A002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))) \\ From Charles R Greathouse IV, Jul 19 2011
A277487(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + if(isprime(k),1,0); k = k - A002828(k)); s; };
for(n=1, 10000, write("b277487.txt", n, " ", A277487(n)));

(define (A277487 n) (let ((org_k (- (A000290 (+ 1 n)) 1))) (let loop ((k org_k) (s 0)) (if (and (< k org_k) (= 1 (A010052 (+ 1 k)))) s (loop (- k (A002828 k)) (+ s (A010051 k)))))))

a(n) <= A277891(n).

A277890 Number of even numbers encountered before (n^2)-1 is reached when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Original entry on oeis.org

0, 2, 0, 3, 2, 3, 1, 5, 3, 4, 4, 6, 3, 5, 3, 7, 8, 8, 6, 8, 9, 10, 6, 8, 10, 10, 7, 11, 10, 13, 11, 12, 12, 14, 10, 13, 12, 13, 14, 15, 13, 15, 15, 18, 18, 16, 15, 17, 21, 18, 18, 18, 19, 20, 16, 21, 20, 20, 22, 20, 23, 20, 22, 23, 21, 23, 23, 27, 25, 24, 22, 28, 22, 27, 24, 26, 25, 25, 29, 29, 28, 26, 30, 31, 28, 28, 31, 30, 32, 33, 27, 32, 34, 34, 30, 33, 33

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

The starting point ((n+1)^2)-1 of the iteration is included if it is even, but the ending point (n^2)-1 is never included in the count.
a(n) = number of even numbers on row n of A276574, after the initial zero-row.
See also comments in A277891.

			For n=6, we start iterating from k = ((6+1)^2)-1 = 48, and then 48 - A002828(48) = 45, 45 - A002828(45) = 43, 43 - A002828(43) = 40, 40 - A002828(40) = 38, and 38 - A002828(38) = 35 (which is 6^2 - 1), and three of these numbers are even, thus a(6) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000035, A000290, A002828, A010052, A260734, A276573, A276574, A277486, A277488, A277889, A277891.

istwo(n:int)=my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1
isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7
A002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))) \\ From Charles R Greathouse IV, Jul 19 2011
A277890(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + (1-(k%2)); k = k - A002828(k)); s; };
for(n=1, 10000, write("b277890.txt", n, " ", A277890(n)));

(define (A277890 n) (let ((org_k (- (A000290 (+ 1 n)) 1))) (let loop ((k org_k) (s 0)) (if (and (< k org_k) (= 1 (A010052 (+ 1 k)))) s (loop (- k (A002828 k)) (+ s (- 1 (A000035 k))))))))

a(n) + A277891(n) = A260734(n).
For n >= 2, a(n) >= A277486(n).
a(n) >= A277488(n).

A277488 a(n) = number of integers one less than a prime encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 0, 3, 2, 3, 2, 1, 1, 2, 1, 3, 2, 3, 1, 2, 4, 4, 1, 2, 3, 2, 2, 3, 4, 3, 3, 4, 3, 4, 3, 2, 4, 5, 5, 4, 4, 3, 4, 5, 4, 4, 3, 4, 6, 4, 6, 8, 4, 4, 3, 5, 7, 3, 7, 2, 6, 6, 4, 7, 7, 4, 5, 7, 4, 8, 6, 4, 4, 3, 6, 11, 4, 5, 5, 9, 6, 3, 6, 7, 6, 9, 9, 8, 11, 6, 5, 5, 7, 8, 7, 7, 5, 8, 9, 5, 7, 6, 5, 6, 7, 6, 8, 9, 6, 9, 6, 15, 8, 10, 9, 7, 10, 6, 6, 10

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

Only 325 cases in range n=1..10000 where a(n) >= A277486(n). See also comments in A277487.

			For n=6, we start iterating from k = ((6+1)^2)-1 = 48, and then 48 - A002828(48) = 45, 45 - A002828(45) = 43, 43 - A002828(43) = 40, 40 - A002828(40) = 38, and 38 - A002828(38) = 35 (which is 6^2 - 1), and when we add one to each, only 41 is prime, thus a(6) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000290, A010051, A010052, A276574, A277486, A277487, A277890, A278168 (partial sums).

istwo(n:int)=my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1
isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7
A002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))) \\ From Charles R Greathouse IV, Jul 19 2011
A277488(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + if(isprime(1+k),1,0); k = k - A002828(k)); s; };
for(n=1, 10000, write("b277488.txt", n, " ", A277488(n)));

(define (A277488 n) (let ((org_k (- (A000290 (+ 1 n)) 1))) (let loop ((k org_k) (s 0)) (if (and (< k org_k) (= 1 (A010052 (+ 1 k)))) s (loop (- k (A002828 k)) (+ s (A010051 (+ 1 k))))))))

For all n >= 1, a(n) <= A277890(n).

A277891 a(n) = number of odd numbers encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Original entry on oeis.org

1, 0, 2, 1, 2, 2, 4, 2, 4, 3, 4, 4, 6, 5, 7, 6, 5, 6, 7, 7, 6, 6, 11, 9, 9, 9, 12, 9, 10, 9, 11, 11, 12, 11, 14, 13, 15, 12, 14, 14, 16, 14, 15, 13, 15, 17, 18, 17, 14, 17, 19, 18, 20, 17, 22, 19, 22, 20, 20, 22, 20, 22, 23, 22, 24, 25, 22, 22, 25, 26, 26, 25, 28, 24, 30, 26, 28, 29, 27, 27, 28, 32, 29, 28, 32, 32, 29, 31, 30, 29, 35, 33, 32, 32, 35, 34, 35, 36

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

The starting point ((n+1)^2)-1 of the iteration is included if it is odd, but the ending point (n^2)-1 is never included in the count.
a(n) = number of odd numbers on row n of A276574, after the initial zero-row.
On the average, the odd terms in A276573 (A276574) seem to occur more frequently than the even terms. (The last point in range 1..10000 where a(n) <= A277890(n) is n=862). See also comments in A277487 and the plot of ratio a(n)/A277890(n), also the plot of A277889.

			For n=6, we start iterating from k = ((6+1)^2)-1 = 48, with k -> k - A002828(k), to obtain 48 -> 45 -> 43 -> 40 -> 38 before reaching 35 (which is 6^2 - 1, an ending point and thus not included in the count), and the only odd numbers before that were 45 and 43, thus a(6) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
A. Karttunen, Ratio a(n)/A277890(n) drawn with OEIS Plot 2 tool

Cf. A000035, A000290, A002828, A010052, A077773, A260733, A260734, A276574, A277191, A277192, A277487, A277889, A277890.

istwo(n:int)=my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1
isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7
A002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))) \\ From Charles R Greathouse IV, Jul 19 2011
A277891(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + (k%2); k = k - A002828(k)); s; };
for(n=1, 10000, write("b277891.txt", n, " ", A277891(n)));

(define (A277891 n) (let ((org_k (- (A000290 (+ 1 n)) 1))) (let loop ((k org_k) (s 0)) (if (and (< k org_k) (= 1 (A010052 (+ 1 k)))) s (loop (- k (A002828 k)) (+ s (A000035 k)))))))

a(n) + A277890(n) = A260734(n).

a(n) >= A277487(n).

A276572 Simple self-inverse permutation of natural numbers: after a(0)=0, list each block of A260734(n) numbers in reverse order, from A260732(n) to A260733(1+n).

Original entry on oeis.org

0, 1, 3, 2, 5, 4, 9, 8, 7, 6, 13, 12, 11, 10, 18, 17, 16, 15, 14, 23, 22, 21, 20, 19, 30, 29, 28, 27, 26, 25, 24, 37, 36, 35, 34, 33, 32, 31, 44, 43, 42, 41, 40, 39, 38, 52, 51, 50, 49, 48, 47, 46, 45, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 71, 70, 69, 68, 67, 66, 65, 64, 63, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Maps between A276573 and A276574.

Antti Karttunen, Table of n, a(n) for n = 0..10028
Index entries for sequences that are permutations of the natural numbers

Cf. A260732, A260733, A260734, A276573, A276574.

f[n_] := NestWhileList[# - (If[First@ # > 0, 1, Length[First@ Split@ #] + 1] &@ SquaresR[Range@ 4, #]) &, n^2, # != 0 &]; t = Table[Table[n, {Length[#] - 1 &@ NestWhileList[# - (If[First@ # > 0, 1, Length[First@ Split@ #] + 1] &@ SquaresR[Range@ 4, #]) &, ((n + 1)^2) - 1, # != (n^2) - 1 &]}], {n, 20}] // Flatten ; {0}~Join~Table[Length@ f@ t[[n]] - 1 + Length@ f[t[[n]] + 1] - n - 2, {n, 81}] (* Michael De Vlieger, Sep 08 2016 *)

(define (A276572 n) (if (zero? n) n (+ (- (A260733 (+ 1 (A276571 n))) n) (A260732 (A276571 n)))))

a(0) = 0; for n >= 1, a(n) = (A260733(1+A276571(n))-n)+A260732(A276571(n)).

cp's OEIS Frontend

A276573 The infinite trunk of least squares beanstalk: The only infinite sequence such that a(0) = 0 and a(n-1) = a(n) - least number of squares (A002828) that sum to a(n).

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A277487 a(n) = number of primes encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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A277890 Number of even numbers encountered before (n^2)-1 is reached when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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A277488 a(n) = number of integers one less than a prime encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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A277891 a(n) = number of odd numbers encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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A276572 Simple self-inverse permutation of natural numbers: after a(0)=0, list each block of A260734(n) numbers in reverse order, from A260732(n) to A260733(1+n).

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