A254004
Numbers that divide the reverse of the sum of their divisors.
Original entry on oeis.org
1, 14, 69, 102, 123, 134, 164, 276, 639, 2556, 9568, 1259196, 1333334, 1473381, 1741983, 133333334, 821554911, 929247534, 1333333334, 22214600673, 133333333334
Offset: 1
sigma(14) = 24, Rev(24) = 42 and 42 / 14 = 3.
sigma(69) = 96, Rev(96) = 69 and 69 / 69 = 1.
sigma(9568) = 21168, Rev(21168) = 86112 and 86112 / 9568 = 9.
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[n: n in [1..10^7] | Seqint(Reverse(Intseq(SumOfDivisors(n)))) mod n eq 0]; // Bruno Berselli, Jan 22 2015
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with(numtheory): T:=proc(w) local x,y,z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10);
od; y; end:
P:=proc(q) local n; for n from 1 to q do
if type(T(sigma(n))/n,integer) then print(n);
fi; od; end: P(10^9);
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Select[Range@ 2000000, Mod[FromDigits@ Reverse@ IntegerDigits@ DivisorSigma[1, #], #] == 0 &] (* Michael De Vlieger, May 09 2015 *)
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isok(n) = !(eval(concat(Vecrev(Str(sigma(n))))) % n); \\ Michel Marcus, Feb 27 2015
A380984
Primes p such that p*(p-1) consists of exactly two different decimal digits.
Original entry on oeis.org
5, 7, 11, 17, 67, 101, 167, 1667, 166667, 666667, 66666667, 666666667, 1666666667, 66666666667, 166666666667, 166666666666667, 66666666666666666667
Offset: 1
a(5) = 67 is a term because it is prime and 67 * 66 = 4422 consists of digits 2 and 4.
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p:= 1: R:= NULL: count:= 0:
while count < 11 do
p:= nextprime(p);
if nops(convert(convert(p*(p-1),base,10),set)) = 2 then
R:= R,p; count:= count+1
fi;
od:
R;
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Select[Prime[Range[10^6]],Length[Union[IntegerDigits[#(#-1)]]]==2&] (* James C. McMahon, Feb 13 2025 *)
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isok(k) = isprime(k) && #Set(digits(k*(k-1))) == 2; \\ Michel Marcus, Feb 11 2025
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from math import isqrt
from itertools import count, combinations, product, islice
from sympy import isprime
def A380984_gen(): # generator of terms
for n in count(1):
c = []
for a in combinations('0123456789',2):
if '0' in a or '2' in a or '6' in a:
for b in product(a,repeat=n):
if b[0] != '0' and b[-1] in {'0','2','6'} and b != (a[0],)*n and b != (a[1],)*n:
m = int(''.join(b))
q = isqrt(m)
if q*(q+1)==m and isprime(q+1):
c.append(q+1)
yield from sorted(c)
A380984_list = list(islice(A380984_gen(),10)) # Chai Wah Wu, Feb 19 2025
Original entry on oeis.org
0, 48, 4488, 444888, 44448888, 4444488888, 444444888888, 44444448888888, 4444444488888888, 444444444888888888, 44444444448888888888, 4444444444488888888888, 444444444444888888888888
Offset: 0
A294396
Numbers k such that 12*10^k + 1 is prime.
Original entry on oeis.org
0, 2, 38, 80, 9230, 25598, 39500
Offset: 1
13 and 1201 are prime, so 0 and 2 are the initial values.
Cf.
A001562,
A096507,
A056797,
A056807,
A056806,
A102940,
A056805,
A056804,
A096508,
A102975,
A289051,
A099017,
A295325,
A273002,
A102945,
A282456,
A267420.
Cf.
A062339 (primes with digit sum 4).
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ParallelMap[ If[ PrimeQ[12*10^# +1], #, Nothing] &, 2 + 6Range@ 4500] (* Robert G. Wilson v, Feb 13 2018 *)
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isok(k) = isprime(12*10^k + 1); \\ Altug Alkan, Mar 04 2018
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