bibkey: cami2015a112041 authors: Pierre Cami; Harvey P. Dale year: 2015 title: “OEIS A112041, Numbers k such that 1k + 1, 3k + 1, 9k + 1, 27k + 1 are all primes” doi: null url: https://oeis.org/A112041 claim: “A112041 %N: Numbers k such that 1k + 1, 3k + 1, 9k + 1, 27k + 1 are all primes. A112041 %C: Conjecture: all terms except the first are multiples of 10. - Harvey P. Dale, Mar 26 2015” strata_touched:

• D5/S3/Arith/Congruence/DaleTripleScalingPrimeMultiplesOfTen license: citation-only triage: anchor

OEIS A112041

The NAME of A112041 states:

Numbers k such that 1k + 1, 3k + 1, 9k + 1, 27k + 1 are all primes.

Harvey P. Dale’s COMMENT of March 26, 2015 states:

Conjecture: all terms except the first are multiples of 10. - Harvey P. Dale, Mar 26 2015

The AUTHOR line states:

Pierre CAMI, Aug 26 2005

The sequence author is therefore listed first. The year 2015 records the conjecture year, while the AUTHOR line is dated August 26, 2005.

Verified locator

• URL: https://oeis.org/A112041
• NAME (verbatim): Numbers k such that 1k + 1, 3k + 1, 9k + 1, 27k + 1 are all primes.
• COMMENT, Mar 26 2015 (verbatim): Conjecture: all terms except the first are multiples of 10. - Harvey P. Dale, Mar 26 2015
• AUTHOR (verbatim): Pierre CAMI, Aug 26 2005

The formal theorem proves that every positive k for which k+1, 3k+1, 9k+1, and 27k+1 are prime satisfies k=4 or is divisible by 10. The excepted first term is k=4, for which 5, 13, 37, and 109 are prime.