Fred Patrick Doty - cp's OEIS frontend

User: Fred Patrick Doty

Fred Patrick Doty has authored 2 sequences.

A347009 a(n) is the largest integer m such that e^-m > e - Sum_{k=0..n} 1/k!.

-1, 0, 1, 2, 4, 6, 8, 10, 12, 15, 17, 19, 22, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 61, 64, 67, 71, 74, 78, 81, 85, 88, 92, 95, 99, 102, 106, 110, 114, 117, 121, 125, 129, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192

Offset: 0

Fred Patrick Doty, Aug 10 2021

sign

From Jon E. Schoenfield, Aug 11 2021: (Start)
It follows from the definition that a(n) = floor(-log(e - Sum_{k=0..n} 1/k!)) = floor(-log(Sum_{k>=n+1} 1/k!)) = floor(f(n)) where, for large n, f(n) = (n + 3/2)*log(n) - n - zeta'(0) + (1/12)/n + 1/n^2 - (361/360)/n^3 - (3/2)/n^4 + (10081/1260)/n^5 - (61/6)/n^6 - 40/n^7 - ...
Conjecture: a(n) = floor((n + 3/2)*log(n) - n - zeta'(0) + (1/12)/n + 1/n^2) for n >= 4.
(End)

Cf. A001113.

Floor[Log[N[(1/(E - Sum[1/n!, {n, 0, #}] & /@ Range[50])), 2]]]

More terms from Jon E. Schoenfield, Aug 11 2021

A258801 Carmichael numbers divisible by 3.

Original entry on oeis.org

561, 62745, 656601, 11921001, 26719701, 45318561, 174352641, 230996949, 662086041, 684106401, 689880801, 1534274841, 1848112761, 2176838049, 3022354401, 5860426881, 6025532241, 6097778961, 7281824001, 7397902401, 10031651841, 10054063041, 10585115841

Offset: 1

Fred Patrick Doty, Jun 10 2015

nonn

Most Carmichael numbers are congruent to 1 modulo 6. Those that are not are observed to include numbers that are 5 modulo 6 as well as multiples of 3.
Subsequence of A008585 and of A205947.
No member of this sequence is divisible by any prime of the form 6k+1, hence all prime factors for this sequence are members of A045410.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..426, below 10^16, based on Richard Pinch data, from Giovanni Resta)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard G. E. Pinch, Tables relating to Carmichael numbers.
Index entries for sequences related to Carmichael numbers.

Cf. A002997 (Carmichael numbers), A205947 (Carmichael numbers not congruent to 1 modulo 6).
Cf. A008585 (3*n).
Cf. A045410 (primes not congruent to 1 modulo 6).

select(t -> t mod numtheory:-lambda(t) = 1, [seq(6*k+3,k=1..10^6)]); # Robert Israel, Jul 12 2015

Cases[Range[555,10^6,6],n_/;Mod[n,CarmichaelLambda[n]]==1]

Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
is(n)=n%6==3 && Korselt(n) && n>9 \\ Charles R Greathouse IV, Jul 20 2015

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User: Fred Patrick Doty

A347009 a(n) is the largest integer m such that e^-m > e - Sum_{k=0..n} 1/k!.

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