A278494 -id:A278494 - cp's OEIS frontend

A278496 a(n) = A000196(A278494(n)).

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

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Each n occurs A278495(n) times.

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

Cf. A000196, A002828, A255131, A278494, A278495.

(define (A278496 n) (A000196 (A278494 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.

A278490 Leaves in the tree defined by edge relation A255131(child) = parent, the least squares beanstalk.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 12, 13, 14, 17, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 42, 44, 46, 47, 49, 52, 55, 57, 58, 60, 61, 62, 65, 68, 69, 76, 77, 79, 82, 84, 86, 89, 92, 94, 97, 98, 100, 101, 103, 106, 109, 110, 113, 116, 118, 119, 121, 122, 124, 125, 127, 132, 133, 140, 141, 142, 145, 148, 150, 153, 154, 156, 157

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Numbers n for which there are no solutions to k - A002828(k) = n for any k, in other words, numbers n such that (A002828(1+n) <> 1) and (A002828(2+n) <> 2) and (A002828(3+n) <> 3) and (A002828(4+n) <> 4), as the maximum value that A002828 may obtain is 4.

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

Complement: A278489.
Positions of zeros in A278216.
Cf. A278494 (primes in this sequence).

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

A278496 a(n) = A000196(A278494(n)).

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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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A278490 Leaves in the tree defined by edge relation A255131(child) = parent, the least squares beanstalk.

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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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Scheme

Formula