A276573 -id:A276573 - cp's OEIS frontend

A277888 Primes in A276573, the infinite trunk of least squares beanstalk.

3, 11, 43, 53, 59, 67, 83, 131, 139, 149, 173, 179, 227, 233, 251, 277, 283, 331, 347, 349, 419, 431, 491, 547, 557, 563, 571, 587, 617, 643, 659, 661, 683, 701, 733, 739, 743, 757, 821, 827, 907, 941, 947, 971, 1013, 1019, 1051, 1061, 1091, 1109, 1117, 1123, 1129, 1163, 1187, 1213, 1229, 1259, 1283, 1291, 1301, 1307, 1327, 1373, 1427, 1429, 1451, 1453

Offset: 1

Antti Karttunen, Nov 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4248

Intersection of A000040 and A276573.

Cf. A277887, A277487, A278167.

(define (A277888 n) (A276573 (A277887 n)))

a(n) = A276573(A277887(n)).

A277015 Numbers whose squares are present in A276573 (the infinite trunk of least squares beanstalk).

Original entry on oeis.org

0, 4, 8, 12, 16, 19, 20, 24, 36, 40, 45, 48, 56, 60, 68, 72, 80, 83, 84, 92, 96, 104, 109, 112, 120, 124, 132, 136, 140, 144, 147, 148, 156, 160, 164, 168, 173, 176, 180, 192, 204, 208, 211, 216, 220, 224, 228, 232, 237, 240, 252, 264, 272, 275, 276, 280, 284

Offset: 0

Antti Karttunen, Oct 03 2016

nonn

Indexing starts from zero because a(0)=0 is a special case in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..130

Cf. A000196, A002828, A276573, A277014, A277016 (the corresponding squares), A277025 (multiples of four present / 4).

Positions of squares in A277023, zeros in A277024.

(define (A277015 n) (A000196 (A277016 n)))

a(n) = A000196(A277016(n)) = A000196(A276573(A277014(n))).

A276575 After a(0)=0, the first differences of A276573.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10028
Burkard Polster, Root 2 and the deadly Marching Squares, Mathologer video (2015)

Cf. A001541, A002828, A132592, A260731, A260733, A276573.

a(n) = A002828(A276573(n)).
a(0) = 0; for n >= 1, a(n) = A276573(n) - A276573(n-1).
Other identities.
For all n >= 1, a(A260731(A132592(n))) = a(A260733(A001541(n))) = 2. [This is implied by the fact observed in the Polster video. Of course 2's occur at other points too.]

A277016 Squares present in A276573 (the infinite trunk of least squares beanstalk).

Original entry on oeis.org

0, 16, 64, 144, 256, 361, 400, 576, 1296, 1600, 2025, 2304, 3136, 3600, 4624, 5184, 6400, 6889, 7056, 8464, 9216, 10816, 11881, 12544, 14400, 15376, 17424, 18496, 19600, 20736, 21609, 21904, 24336, 25600, 26896, 28224, 29929, 30976, 32400, 36864, 41616, 43264, 44521, 46656, 48400, 50176, 51984, 53824, 56169, 57600, 63504, 69696, 73984, 75625, 76176, 78400, 80656

Offset: 0

Antti Karttunen, Oct 03 2016

nonn

Indexing starts from zero because a(0)=0 is a special case in this sequence.

Cf. A276573, A277014, A277015 (square roots of these numbers), A277025.

(define (A277016 n) (A276573 (A277014 n)))

a(n) = A276573(A277014(n)).

A277023 a(n) = A276573(A260732(n)); For n >= 1, a(n) = the next larger term right after each (n^2)-1 in the infinite trunk of least squares beanstalk.

Original entry on oeis.org

0, 3, 6, 11, 16, 27, 38, 51, 64, 83, 102, 123, 144, 171, 198, 227, 256, 291, 326, 361, 400, 444, 486, 531, 576, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1296, 1371, 1446, 1523, 1600, 1683, 1767, 1851, 1938, 2025, 2118, 2211, 2304, 2403, 2502, 2603, 2706, 2811, 2918, 3027, 3136, 3251, 3366, 3483, 3600, 3723, 3846

Offset: 0

Antti Karttunen, Oct 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..750

Cf. A002828, A260732, A276573.

Cf. A277015 (the positions of squares in this sequence), A277024, A277025 A277026.

(define (A277023 n) (A276573 (A260732 n)))

a(n) = A276573(A260732(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.

A278497 a(n) = Least number with the prime signature of A276573(n).

Original entry on oeis.org

2, 6, 8, 2, 6, 16, 12, 6, 24, 8, 30, 32, 6, 6, 24, 2, 12, 48, 6, 2, 24, 2, 12, 64, 2, 30, 72, 12, 30, 48, 2, 6, 24, 60, 6, 96, 12, 30, 30, 72, 48, 6, 12, 120, 6, 60, 128, 2, 6, 24, 2, 6, 144, 12, 2, 24, 6, 6, 96, 48, 30, 120, 12, 2, 48, 2, 6, 30, 24, 192, 30, 60, 72, 6, 12, 210, 6, 216, 6, 6, 96, 2, 30, 2, 12, 240, 32, 12, 24, 2, 30, 256, 6, 12, 120, 6, 120

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10028

Cf. A002828, A046523, A260733, A276573, A278160, A278498, A278499.
Cf. A277014 (gives the indices of squares).
Cf. also A278232.

(define (A278497 n) (A046523 (A276573 n)))

a(n) = A046523(A276573(n)).

For all n >= 1, a(A260733(1+n)) = A278160(n).

A278498 a(n) = A065338(A276573(n)).

Original entry on oeis.org

3, 6, 8, 3, 3, 16, 18, 9, 24, 27, 6, 32, 3, 6, 8, 3, 9, 48, 3, 1, 24, 3, 27, 64, 3, 6, 72, 3, 6, 16, 3, 1, 24, 18, 9, 96, 27, 6, 9, 108, 48, 3, 9, 24, 3, 54, 128, 3, 6, 8, 3, 3, 144, 27, 1, 24, 3, 6, 32, 162, 9, 72, 27, 1, 48, 3, 3, 18, 81, 192, 3, 54, 8, 3, 27, 18, 9, 216, 3, 1, 96, 3, 27, 1, 12, 48, 243, 9, 24, 3, 3, 256, 3, 9, 72, 3, 54, 16, 3, 1, 24, 3, 6

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10028

Cf. A002828, A065338, A276573, A278497, A278499.

(define (A278498 n) (A065338 (A276573 n)))

a(n) = A065338(A276573(n)).

A278499 a(n) = A276573(n) modulo 4.

Original entry on oeis.org

0, 3, 2, 0, 3, 3, 0, 2, 1, 0, 3, 2, 0, 3, 2, 0, 3, 1, 0, 3, 1, 0, 3, 3, 0, 3, 2, 0, 3, 2, 0, 3, 1, 0, 2, 1, 0, 3, 2, 1, 0, 0, 3, 1, 0, 3, 2, 0, 3, 2, 0, 3, 3, 0, 3, 1, 0, 3, 2, 0, 2, 1, 0, 3, 1, 0, 3, 3, 2, 1, 0, 3, 2, 0, 3, 3, 2, 1, 0, 3, 1, 0, 3, 3, 1, 0, 0, 3, 1, 0, 3, 3, 0, 3, 1, 0, 3, 2, 0, 3, 1, 0, 3, 2, 0, 3, 2, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 3, 2, 0, 3

Offset: 0

Antti Karttunen, Nov 25 2016

nonn

No subsequence 2, 3 will ever occur.

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

Cf. A002828, A010873, A276573, A276575, A278497, A278498.

(define (A278499 n) (modulo (A276573 n) 4))

a(n) = A010873(A276573(n)) = A276573(n) modulo 4.

A278521 a(n) = A276573(n) - A278517(n).

Original entry on oeis.org

0, 2, 2, 0, 2, 3, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 1, 2, 1, 2, 3, 2, 1, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 1, 3, 2, 2, 2, 2, 2, 0, 1, 2, 2, 2, 2, 2, 0, 0, 2, 3, 1, 2, 3, 2, 1, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0

Offset: 0

Antti Karttunen, Nov 28 2016

nonn

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

Cf. A276573, A278517, A278518, A278522.

(define (A278521 n) (- (A276573 n) (A278517 n)))

a(n) = A276573(n) - A278517(n).

cp's OEIS Frontend

A277888 Primes in A276573, the infinite trunk of least squares beanstalk.

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A277015 Numbers whose squares are present in A276573 (the infinite trunk of least squares beanstalk).

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A276575 After a(0)=0, the first differences of A276573.

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A277016 Squares present in A276573 (the infinite trunk of least squares beanstalk).

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Scheme

Formula

A277023 a(n) = A276573(A260732(n)); For n >= 1, a(n) = the next larger term right after each (n^2)-1 in the infinite trunk of least squares beanstalk.

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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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Scheme

Formula

A278497 a(n) = Least number with the prime signature of A276573(n).

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A278498 a(n) = A065338(A276573(n)).

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A278499 a(n) = A276573(n) modulo 4.

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