A277486 -id:A277486 - cp's OEIS frontend

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).

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).

A278487 Primes p such that p+1 is in A276573, the infinite trunk of least squares beanstalk.

Original entry on oeis.org

2, 5, 7, 17, 23, 29, 31, 37, 47, 71, 79, 89, 101, 107, 127, 151, 157, 167, 191, 197, 199, 223, 239, 263, 269, 271, 293, 311, 317, 337, 359, 367, 383, 389, 421, 433, 439, 443, 449, 461, 463, 479, 487, 503, 509, 521, 541, 593, 599, 607, 619, 631, 647, 653, 677, 709, 719, 727, 751, 773, 797, 809, 823, 839, 857, 863, 881, 887, 911, 919

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

These seem to be substantially more common than A277888, even though odd terms are slightly more common in A276573 than the even terms. See also comments in A277487.

Antti Karttunen, Table of n, a(n) for n = 1..5050

One less than A278486.
No common terms with A277888, some common terms with A278494.
Cf. A277486 (gives the count of these primes in each range [n^2, (n+1)^2]).
Cf. A002828, A276573, A278485.

(define (A278487 n) (+ -1 (A276573 (A278485 n))))

a(n) = A278486(n) - 1 = A276573(A278485(n)) - 1.

A278166 a(n) = number of integers one more than a prime encountered before reaching 0 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Original entry on oeis.org

1, 3, 3, 5, 7, 9, 9, 11, 12, 14, 15, 18, 19, 22, 23, 26, 29, 31, 34, 37, 42, 46, 47, 51, 54, 58, 60, 64, 68, 70, 74, 78, 82, 85, 88, 92, 95, 99, 104, 109, 114, 118, 122, 128, 134, 137, 140, 149, 153, 158, 164, 173, 177, 183, 187, 191, 199, 205, 210, 217, 222, 231, 236, 241, 248, 256, 262, 273, 278, 287, 291, 298, 307, 316, 322, 332

Offset: 1

Antti Karttunen, Nov 13 2016

nonn

			For n=4, starting from k = ((4+1)^2)-1, and iterating k -> A255131(k), yields 24 -> 21 -> 18 -> 16 -> 15 -> 11 -> 8 -> 6 -> 3 before 0 is reached. Subtracting one from each gives [23, 20, 17, 15, 14, 10, 7, 5, 2], of which only 23, 17, 7, 5 and 2 are primes, thus a(4) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
A. Karttunen, Ratio a(n)/A278168(n) plotted with OEIS Plot 2 tool

Partial sums of A277486.

Cf. A002828, A278167, A278168.

(definec (A278166 n) (if (= 1 n) (A277486 n) (+ (A277486 n) (A278166 (- n 1)))))

a(1) = A277486(1); for n > 1, a(n) = A277486(n) + a(n-1).

A278495 a(n) = number of primes in range [n^2, (n+1)^2] that are leaves in "the least squares beanstalk" tree.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 1, 2, 4, 1, 2, 1, 3, 2, 4, 3, 3, 3, 5, 3, 2, 2, 4, 4, 4, 4, 3, 4, 4, 4, 4, 2, 3, 3, 2, 4, 2, 5, 4, 6, 3, 5, 4, 5, 5, 4, 6, 3, 3, 6, 8, 4, 5, 3, 5, 5, 5, 4, 6, 6, 7, 5, 5, 7, 6, 8, 8, 8, 8, 5, 5, 5, 8, 7, 7, 7, 3, 13, 5, 8, 6, 8, 7, 8, 5, 14, 7, 8, 8, 10, 7, 5, 8, 6, 7, 6, 9, 4, 10, 4, 9, 8, 6, 8, 8, 8, 6, 10, 11, 13, 9

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Number of terms of A278494 in range [n^2, (n+1)^2], where A278494 are primes p for which there does not exist any such integer k that k - A002828(k) = p.
In other words, number of primes p in range [n^2, (n+1)^2] for which (A002828(1+p) <> 1) and (A002828(2+p) <> 2) and (A002828(3+p) <> 3) and (A002828(4+p) <> 4).
Conjecture: a(n) > 0 for all n >= 1.
Similar guesses are easy to make but hard to prove. I also conjecture that A277487(n) > 0 for all n > 80, and that both A277486(n) > 0 and A277488(n) > 0 for all n > 7. If any of these claims were proved true, it would imply the proof of Legendre's conjecture as well. See also comments in A014085 and sequences A277888 & A278487.

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

Cf. A000290, A002828, A010051, A010052, A014085 (an upper bound), A278216, A278494 (primes that are counted), A278496.

Cf. also A277486, A277487, A277488.

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
A278495(n) = { my(s = 0); for(k=(n^2),(n+1)^2, if((isprime(k) && (A002828(1+k) <> 1) && (A002828(2+k) <> 2) && (A002828(3+k) <> 3) && (A002828(4+k) <> 4)),s = s+1) ); s; };
for(n=1, 10000, write("b278495.txt", n, " ", A278495(n)));

(define (A278495 n) (let loop ((k (+ -1 (A000290 (+ 1 n)))) (s 0)) (if (= 1 (A010052 k)) s (loop (- k 1) (+ s (* (A010051 k) (if (zero? (A278216 k)) 1 0)))))))

For all n >= 1, a(n) <= A014085(n).

cp's OEIS Frontend

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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A278487 Primes p such that p+1 is in A276573, the infinite trunk of least squares beanstalk.

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A278166 a(n) = number of integers one more than a prime encountered before reaching 0 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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A278495 a(n) = number of primes in range [n^2, (n+1)^2] that are leaves in "the least squares beanstalk" tree.

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