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A347038
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Primes p such that there are no solutions to d(k+p) = sigma(k).
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0
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29, 37, 41, 53, 67, 89, 101, 109, 113, 127, 137, 151, 157, 173, 181, 197, 227, 229, 233, 257, 269, 277, 281, 293, 313, 349, 373, 389, 401, 409, 421, 439, 461, 557, 587, 593, 601, 613, 617, 641, 643, 653, 661, 673, 677, 701, 709, 739, 761, 773, 787, 821, 829
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OFFSET
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1,1
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COMMENTS
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If p is not in the sequence and d(k+p) = sigma(k), then k <= 1+2*sqrt(p). Proof: We have d(m) <= 2*sqrt(m) (see formula in A000005), so 2*sqrt(k+p) >= d(k+p) = sigma(k) >= k+1 (if k > 1). After squaring and simplifying, we get k <= 1+2*sqrt(p). - Pontus von Brömssen, Aug 20 2021
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LINKS
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PROG
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(Python)
from sympy import divisor_count as d, divisor_sigma as sigma, primerange
from math import isqrt
a = []
for p in primerange(2, pmax + 1):
if not any(d(k + p) == sigma(k) for k in range(1, 2 + isqrt(4 * p))):
a.append(p)
return a # Pontus von Brömssen, Aug 20 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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