A191474 -id:A191474 - cp's OEIS frontend

A215946 Smallest prime q such that q + prime(n) is power of 6 or 0 if no such q exists.

0, 3, 31, 29, 0, 23, 19, 17, 13, 7, 0, 179, 0, 173, 1249, 163, 157, 0, 149, 0, 1223, 137, 1213, 127, 46559, 0, 113, 109, 107, 103, 89, 0, 79, 78364163957, 67, 0, 59, 53, 1129, 43, 37, 0, 0, 23, 19, 17, 0, 60465953, 1069, 7547, 1063, 7537, 0, 0, 1039, 1033

Offset: 1

Zak Seidov, Aug 28 2012

nonn

Such q certainly do not exist if p == 1 (mod 10).

Corresponding exponents of 6 (0 if a(n)=0): 0, 0, 2, 2, 0, 2, 2, 2, 2, 2, 0, 3, 0, 3, 4, 3, 3, 0.

			n=2: 3+3=6^1, n=3: 5+31=6^2, n=4: 7+29=6^2, n=6: 13+23=6^2.

Zak Seidov, Table of n, a(n) for n = 1..162

Cf. A191474.

s={0,3}; Do[p=Prime[n]; If[Mod[p,10]<2, AppendTo[s,0]; Goto[ne]]; m=Ceiling[Log[6,p]]; While[!PrimeQ[q=6^m-p], m++]; AppendTo[s,q]; Label[ne], {n, 3, 162}]; s

A382035 a(n) is the smallest prime q such that q + prime(n) is of form 10^k or 2*10^k, or 0 if no such prime exists.

Original entry on oeis.org

0, 7, 5, 3, 89, 7, 3, 181, 977, 71, 1999969, 163, 59, 157, 53, 47, 41, 139, 1933, 29, 127, 199921, 17, 11, 3, 999999999899, 97, 999999893, 19891, 887, 73, 9999999999999999999869, 863, 61, 9851, 1999999999849, 43, 37, 9833, 827, 821, 19, 809, 7, 3, 1801, 1789

Offset: 1

Steven Lu, Mar 12 2025

nonn

a(1) is not the only term equal to 0.
For example, a(37145)=0, since prime(37145)=442609, and:
  10^k - 442609 is a multiple of 3, for k>=6,
  2*10^(2*k) - 442609 is a multiple of 11, for k>=3,
  2*10^(6*k+1) - 442609 is a multiple of 7, for k>=1,
  2*10^(6*k+3) - 442609 is a multiple of 13, for k>=1,
  2*10^(6*k+5) - 442609 is a multiple of 37, for k>=1.

			For n=11 (prime(n)=31):
For all positive integer k, 10^k-31 is multiple of 3.
200 - 31 = 169 = 13 * 13
2000 - 31 = 1969 = 11 * 179
20000 - 31 = 19969 = 19 * 1051
200000 - 31 = 199969 = 7 * 7 * 7 * 11 * 53
2000000 - 31 = 1999969 is a prime number.
thus a(11) = 1999969.

Cf. A191474 (base 2 version of this sequence).

Table[If[MissingQ[#], 0, # - Prime[i]] &@SelectFirst[Flatten[Table[{10^j, 2 10^j}, {j, 100}]], # > Prime[i] && PrimeQ[# - Prime[i]] &], {i, 1, 47}]

cp's OEIS Frontend

A215946 Smallest prime q such that q + prime(n) is power of 6 or 0 if no such q exists.

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