A095366 -id:A095366 - cp's OEIS frontend

A058255 Distinct values of lcm_{i=1..n} (p(i)-1), where p() are the primes.

1, 2, 4, 12, 60, 240, 720, 7920, 55440, 1275120, 16576560, 480720240, 19709529840, 39419059680, 197095298400, 3350620072800, 177582863858400, 532748591575200, 19711697888282400, 59135093664847200

Offset: 1

Labos Elemer, Dec 06 2000

nonn

The prime A095365(n) is the least prime yielding an LCM of a(n). This sequence and A095365 are related to A095366. - T. D. Noe, Jun 04 2004

			For p = 29, 31, 37, 41, 43 these LCMs are equal to 55440 = 11*7! = lcm[1, 2, 4, 6, 10, 12, 16, 22, 28, 30, 36, 40, 42] = lcm[1, 2, 4, 6, 10, 12, 16, 22, 28]. The values was put on the stage only once. Repetitions skipped.

Michael De Vlieger, Table of n, a(n) for n = 1..360

A058254 with duplicates removed.
Cf. A002110, A005867, A003418, A054451, A000142, A000010.
Cf. A095366 (least k > 1 such that k divides 1^n + 2^n + ... + (k-1)^n).

Union@ Table[LCM @@ (Prime@ Range[1, n] - 1), {n, 38}] (* Michael De Vlieger, Dec 06 2018 *)

A094756 a(n) = least k>1 such that (1+2+3+...+k) divides (1^n + 2^n + 3^n + ... + k^n).

Original entry on oeis.org

2, 4, 2, 7, 2, 4, 2, 7, 2, 4, 2, 16, 2, 4, 2, 7, 2, 4, 2, 7, 2, 4, 2, 16, 2, 4, 2, 7, 2, 4, 2, 7, 2, 4, 2, 16, 2, 4, 2, 7, 2, 4, 2, 7, 2, 4, 2, 22, 2, 4, 2, 7, 2, 4, 2, 7, 2, 4, 2, 16, 2, 4, 2, 7, 2, 4, 2, 7, 2, 4, 2, 16, 2, 4, 2, 7, 2, 4, 2, 7, 2, 4, 2, 16, 2, 4, 2, 7, 2, 4, 2, 7, 2, 4, 2, 22, 2, 4, 2, 7, 2, 4

Offset: 1

Amarnath Murthy, May 29 2004

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A094755, A095366.

f[n_] := Block[{k = 2}, While[ !IntegerQ[ 2Sum[i^n, {i, k}]/(k(k + 1))], k++ ]; k]; Table[ f[n], {n, 50}] (* Robert G. Wilson v, Jun 02 2004 *)

A094756(n) = { my(k=1,s1=1,s2=1); while(1, k++; s1 += k; s2 += (k^n); if(!(s2%s1), return(k))); }; \\ Antti Karttunen, Dec 19 2018

Formulae from Don Reble: If N is not divisible by 2, a(N) = 2.
Otherwise, if N is not divisible by 4, a(N) = 4.
Otherwise, if N is not divisible by 12, a(N) = 7.
Otherwise, if N is not divisible by 48, a(N) = 16.
Otherwise, if N is not divisible by 240, a(N) = 22 or 31. (31 if N is divisible by 528=11*48; otherwise 22).
Otherwise, if N is not divisible by 720, a(N) = 37.
Otherwise, if N is not divisible by 11 nor 23, a(N) = 46.
Otherwise, if N is not divisible by 77, a(N) = 58.
Otherwise, if N is not divisible by 13 nor 53, a(N) = 106.
Otherwise, if N is not divisible by 13, a(N) = 157.
Otherwise, if N is not divisible by 41 nor 83, a(N) = 166. ...
That works for N < 29549520 or so. But it is unlikely that any finite description of that kind is complete.

Edited and extended by Don Reble and Robert G. Wilson v, Jun 02 2004

A095365 Primes p such that f(q) < f(p) for all primes q < p, where f(p) = lcm(2-1, 3-1, 5-1, 7-1, 11-1, ..., p-1).

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 47, 53, 59, 83, 97, 101, 103, 107, 109, 149, 163, 167, 173, 179, 191, 193, 197, 227, 251, 257, 263, 269, 283, 293, 311, 317, 347, 359, 367, 383, 389, 467, 479, 487, 503, 509, 557, 563, 569, 587, 607, 619, 643, 653, 677, 709, 719

Offset: 1

T. D. Noe, Jun 03 2004

nonn

A058255 gives the values of f(p) for the primes p in this sequence. The odd primes in this sequence seem to be the only values appearing in A095366.

			13 is not a member of this sequence because f(11) = f(13) = 60.

nn=200; lst={}; lcm=1; Do[p=Prime[i]; newLCM=LCM[lcm, p-1]; If[newLCM>lcm, lcm=newLCM; AppendTo[lst, p]], {i, 2, nn}]; lst

A378640 Smallest m such that phi(m) does not divide n, where phi is the Euler totient function (A000010).

Original entry on oeis.org

3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 15, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5

Offset: 1

Paolo Xausa, Dec 05 2024

Up to n = 10^7 the distinct terms of the sequence (which are also the record values) are {3, 5, 7, 11, 15, 17, 19, 23, 29, 47, 51, 53}. Is this A076245 (for n >= 2)?
First differs from A095366 at n = 60.
It appears that a(n) = A095366(n) except when n = 60*(2*k + 1), with k >= 0, where a(n) = 15 while A095366(n) = 17.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000010, A076245, A095366, A378638, A378641, A378642.

A378640[n_] := If[OddQ[n], 3, Module[{m = 4}, While[Divisible[n, EulerPhi[++m]]]; m]];
Array[A378640, 100]

a(n) = 3 if n is odd.

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A058255 Distinct values of lcm_{i=1..n} (p(i)-1), where p() are the primes.

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A094756 a(n) = least k>1 such that (1+2+3+...+k) divides (1^n + 2^n + 3^n + ... + k^n).

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A095365 Primes p such that f(q) < f(p) for all primes q < p, where f(p) = lcm(2-1, 3-1, 5-1, 7-1, 11-1, ..., p-1).

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A378640 Smallest m such that phi(m) does not divide n, where phi is the Euler totient function (A000010).

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