A117388 -id:A117388 - cp's OEIS frontend

A117406 Integer k such that 2^n + k = A117405(n).

3, 2, 0, 1, -1, 1, 1, 1, -2, -1, 3, -1, 1, 1, -2, -3, -5, 1, -2, 1, 1, -3, 7, -1, 3, -3, 3, 3, 1, 6, -3, 1, 1, -3, -3, -3, -3, -1, 18, 3, 1, -1, 3, 1, -3, 3, 7, -9, 3, -1, 7, -5, 3, 11, -3, -5, 6, -9, -3, -1, -3, 1, -2, 9, 1, 5, 3, -1, -5, -13, 9, -3, -7, -3

Offset: 0

Jonathan Vos Post, Mar 13 2006

After n=2, never again is a(n) = 0. Semiprime analog of A117388 Integer k such that 2^n + k = A117387(n). A117387(n) is prime nearest to 2^n. (In case of a tie, choose the smaller).

			a(0) = 3 because 2^0 + 3 = 4 = A001358(1) and no semiprime is closer to 2^0.
a(1) = 2 because 2^1 + 2 = 4 = A001358(1) and no semiprime is closer to 2^1.
a(2) = 0 because 2^2 + 0 = 4 = A001358(1) and no semiprime is closer to 2^2.
a(3) = 1 because 2^3 + 1 = 9 = 3^2 = A001358(3), no semiprime is closer to 2^3.
a(4) = -1 because 2^4 - 1 = 15 = 3 * 5 and no semiprime is closer.
a(5) = 1 because 2^5 + 1 = 33 = 3 * 11 and no semiprime is closer to 2^5.
a(6) = 1 because 2^6 + 1 = 65 = 5 * 13 and no semiprime is closer to 2^6.
a(7) = 1 because 2^7 + 1 = 129 = 3 * 43 and no semiprime is closer to 2^7.
a(8) = -2 because 2^8 - 2 = 254 = 2 * 127 and no semiprime is closer to 2^8.

Cf. A000079, A001358, A117387, A117405.

a[n_] := Catch@Block[{p = 2^n, k = 0}, While[True, If[p > k && PrimeOmega[p - k] == 2, Throw[-k]]; If[PrimeOmega[p + k] == 2, Throw[k]]; k++]]; a /@ Range[0, 80] a /@ Range[0, 80] (* Giovanni Resta, Jun 15 2016 *)

a(n) = A117405(n) - 2^n. a(n) = Min{k such that A001358(i) + k = 2^j}.

Corrected and extended by Giovanni Resta, Jun 15 2016

A113762 Numbers n with nonzero digits in their decimal representation such that when all numbers formed by inserting the exponentiation symbol between any two digits are added up, the sum is prime.

Original entry on oeis.org

21, 31, 51, 71, 121, 142, 161, 162, 164, 181, 211, 237, 326, 343, 412, 416, 456, 491, 494, 612, 616, 726, 817, 929, 1226, 1228, 1427, 1513, 1622, 1776, 1824, 1828, 1911, 1915, 1975, 2127, 2188, 3716, 5265, 6276, 6321, 6491, 6852, 7739, 14423, 14487, 15297, 16159

Offset: 1

Ray G. Opao, Jan 18 2006

			a(6) = 142 because 1^42+14^2 = 197, which is prime.

Michael S. Branicky, Table of n, a(n) for n = 1..55

Cf. A117388.

lst = {}; Do[ If[ Min@ IntegerDigits@n > 0, a=0; p=10; While[(w = Floor[n/p]) > 0, a += w^ Mod[n, p]; p*=10]; If[PrimeQ[a], Print[{n, a}]; AppendTo[lst, n]]], {n, 11, 9999}]; lst

from sympy import isprime
from itertools import count, islice, product
def agen():
    for d in count(2):
        for p in product("123456789", repeat=d):
            s = "".join(p)
            if isprime(sum(int(s[:i])**int(s[i:]) for i in range(1, d))):
                yield int(s)
print(list(islice(agen(), 44))) # Michael S. Branicky, Jun 27 2022

More terms from Giovanni Resta, Jan 19 2006
More terms from Robert G. Wilson v, Apr 27 2006
a(47) and beyond from Michael S. Branicky, Jun 27 2022

cp's OEIS Frontend

A117406 Integer k such that 2^n + k = A117405(n).

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