Aaron T Cowan - cp's OEIS frontend

User: Aaron T Cowan

Aaron T Cowan has authored 2 sequences.

A361741 Starting positions of digit triples in the decimal expansion of Pi where the sum of the first 2 equals the third.

1, 3, 10, 29, 61, 73, 83, 106, 117, 132, 177, 192, 195, 198, 241, 248, 251, 281, 309, 311, 333, 362, 381, 393, 432, 477, 486, 494, 504, 508, 525, 532, 536, 555, 602, 611, 628, 647, 662, 674, 689, 699, 710, 747, 755, 760, 771, 806, 853, 856, 887, 899, 927, 934, 966, 969, 989

Offset: 1

Aaron T Cowan, Mar 22 2023

The first digit of Pi, "3", is reckoned as position 1.

This pattern happens from the first digit of Pi, so it seems to be pretty basic.

			1 is the first term, since the first two digits 3 and 1 add up to 4.
3 is the second term, since 4 + 1 = 5.
10 is next, since 3 + 5 = 8.

Index entries for sequences related to the number Pi

Cf. A000796, A110883.

p=char(vpa(pi,1000));p(2)='3';
for i=2:strlength(p)-2
  if str2num(p(i))+str2num(p(i+1))==str2num(p(i+2)) fprintf('%i,',i-1)  end
end

Integers k such that A000796(k) + A000796(k+1) = A000796(k+2).
Equivalently, A110883(k) = A000796(k+2).
In a random model of this sequence (call it A(n)), A(n) ~ kn with probability 1, where k = 200/11. - Charles R Greathouse IV, Mar 28 2023

A360790 Squared length of diagonal of right trapezoid with three consecutive prime length sides.

Original entry on oeis.org

8, 13, 41, 53, 137, 173, 305, 397, 533, 877, 977, 1373, 1697, 1885, 2245, 2813, 3517, 3737, 4493, 5077, 5345, 6277, 6953, 7937, 9413, 10217, 10613, 11465, 12077, 12785, 16165, 17165, 18869, 19325, 22237, 22837, 24665, 26605, 27925, 29933, 32141, 32765, 36497, 37253, 38953, 39745

Offset: 1

Aaron T Cowan, Feb 20 2023

nonn

The value d is the square of the length of the diagonal of a trapezoid with a height and bases that are consecutive primes, respectively. The diagonal length is calculated using the Pythagorean theorem, but this distance is squared so that the value is an integer.

			        p(2)=3
        _ _ _ _
a(1):  |        \  d^2=2^2+(5-3)^2=8
p(1)=2 |_ _ _ _ _\
        p(3)=5
        p(3)=5
        _ _ _ _ _ _
a(2):  |           \    d^2=3^2 + (7-5)^2 = 9+4 = 13
p(2)=3 |            \
       |_ _ _ _ _ _ _\
        p(4)=7
a(3)= 5^2+(11-7)^2 = 25+16 = 41
a(7)= 17^2+(23-19)^2=305 = 5*61

Aaron T Cowan, Table of n, a(n) for n = 1..500

Cf. A001248, A076821.

Cf. A131019, A106171.

%shorter 1 line version
arrayfun(@(p) p^2+(nextprime(nextprime(p+1)+1)-nextprime(p+1))^2,[primes(10^6)])

Map[(#[[1]]^2 + (#[[3]] - #[[2]])^2) &, Partition[Prime[Range[50]], 3, 1]] (* Amiram Eldar, Feb 24 2023 *)

a(n) = prime(n)^2 + (prime(n+2)-prime(n+1))^2; \\ Michel Marcus, Feb 23 2023

a(n) = prime(n)^2 + (prime(n+2)-prime(n+1))^2.

a(n) = A001248(n) + A076821(n+1). - Michel Marcus, Feb 23 2023

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User: Aaron T Cowan

A361741 Starting positions of digit triples in the decimal expansion of Pi where the sum of the first 2 equals the third.

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