A350590
Prime numbers p such that iterating the map m -> m^2 + 1 on p generates a number ending with p.
Original entry on oeis.org
2, 5, 7, 677, 948901, 55904677, 88948901, 36414201356422028396069993813455904677, 8964456980291877636414201356422028396069993813455904677, 711873588184178964456980291877636414201356422028396069993813455904677
Offset: 1
2 is a term because 2 is a prime and iterating the map on 2 gives: 2 -> 5 -> 26 -> 677 -> 458330 -> 210066388901 -> 44127887745906175987802, which ends with 2.
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from sympy import isprime; R = []
for i in range(1, 100):
m = 1; L = [m]; m = (m*m+1)%10**i
while m not in L: L.append(m); m = (m*m+1)%10**i
del L[:L.index(m)]; {R.append(j) for j in L if isprime(j) and j not in R}
R.sort(); print(*R, sep = ", ")
A356130
a(n) = Sum_{k=1..n} sigma_{n-1}(k).
Original entry on oeis.org
1, 4, 16, 111, 999, 12513, 185683, 3316418, 67810767, 1576561677, 40862702931, 1171104916405, 36722498575799, 1251419967587955, 46034784688102781, 1818440444592581068, 76763036794222996512, 3448830049286378614987, 164309958491233496689189
Offset: 1
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a[n_] := Sum[DivisorSigma[n-1, k], {k, 1, n}]; Array[a, 19] (* Amiram Eldar, Jul 28 2022 *)
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a(n) = sum(k=1, n, sigma(k, n-1));
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a(n) = sum(k=1, n, k^(n-1)*(n\k));
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from math import isqrt
from sympy import bernoulli
def A350130(n): return (((s:=isqrt(n))+1)*((b:=bernoulli(n))-bernoulli(n, s+1))+sum(k**(n-1)*n*((q:=n//k)+1)-b+bernoulli(n, q+1) for k in range(1,s+1)))//n if n>1 else 1 # Chai Wah Wu, Oct 21 2023
A354832
Integers m such that iterating the map f(x) = x^2 + 1 on m generates a number ending with m in binary format.
Original entry on oeis.org
0, 1, 2, 5, 10, 26, 37, 90, 165, 421, 933, 1957, 4005, 8101, 8282, 24666, 40869, 106405, 237477, 286810, 811098, 1286053, 3383205, 5005402, 11771813, 28549029, 38559834, 105668698, 239886426, 296984485, 833855397, 1313628250, 3461111898, 7756079194, 9423789989
Offset: 1
26 is a term because iterating the map on 26 gives, in binary format, 11010 -> 1010100101 -> 1101111111001011010, which ends with 11010.
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R = []
for i in range(0, 34):
t = 2**i; L = []
while t not in L: L.append(t); t = (t*t + 1) % 2**(i+1)
{R.append(j) for j in {L[-1], L[-2]} if j not in R}
R.sort(); print(*R, sep = ', ')
Showing 1-3 of 3 results.
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