A014945 -id:A014945 - cp's OEIS frontend

A115976 Numbers k that divide 2^(k-2) + 1.

1, 3, 49737, 717027, 9723611, 21335267, 32390921, 38999627, 43091897, 86071337, 101848553, 102361457, 228911411, 302948067, 370219467, 393664027, 455781089, 483464027, 1040406177, 1272206987, 2371678553, 2571052241, 2648052857, 3054713937, 3597613307, 3782971499, 3917903851, 4005163577, 5419912241

Offset: 1

Max Alekseyev, Mar 15 2006

nonn

Some larger terms: 4465786944074559659, 1440261542571735083956640176981881665928575750093930787551969

Max Alekseyev, Table of n, a(n) for n = 1..708 (contains all terms below 10^15)

The odd elements of A244673.

Cf. A006521, A006517, A069927, A067945, A067946, A067947, A068382, A068383, A014945, A014946, A014949, A092028.

lst = {}; Do[ If[ PowerMod[2, 2n - 3, 2n - 1] == 2n - 2, AppendTo[lst, 2n - 1]], {n, 10^9}]; lst (* Robert G. Wilson v, Apr 04 2006 *)

More terms from Robert G. Wilson v, Apr 04 2006
Terms a(24) onward from Max Alekseyev, Feb 03 2015
b-file corrected and extended by Max Alekseyev, Oct 27 2018

A342320 Integers k such that Euler(k, 1) is an integer multiple of Bernoulli(k + 1, 1).

Original entry on oeis.org

0, 1, 5, 17, 41, 53, 125, 161, 293, 341, 377, 485, 881, 1025, 1133, 1313, 1457, 1805, 2057, 2393, 2645, 3077, 3401, 3941, 4373, 5333, 5417, 6173, 6497, 7181, 7937, 9197, 9233, 10205, 11825, 12641, 13121, 14153, 14405, 16001, 16253, 16757, 18521, 19493, 21545

Offset: 0

Peter Luschny, Mar 24 2021

nonn

			Let E(n) = Euler(n, 1) and B(n) = Bernoulli(n, 1).
2*E(0)  = 4*B(1) = 2;
2*E(1)  = 6*B(2) = 1;
2*E(5)  = 42*B(6) = 1;
2*E(17) = 58254*B(18) = 3202291;
2*E(41) = 418861572486*B(42) = 352552873457246307069012458671.

a(n) = A015942(n-1)-1 for n>=2, (a(n)+1)/2 = A014945(n) for n>=1.
a(n) = A014741(n+1) - 1. - Vaclav Kotesovec, Mar 24 2021
Cf. A341759 (subsequence of primes), A198631/A006519 (Euler), A164555/A027642 (Bernoulli).

Join[{0}, Select[Range[1000], BernoulliB[#+1, 1] != 0 && IntegerQ[EulerE[#, 1]/BernoulliB[#+1, 1]] &]] (* Vaclav Kotesovec, Mar 24 2021 *)
Select[Range[100000], IntegerQ[(2*(-1 + 2^#))/#] & ] - 1 (* Vaclav Kotesovec, Mar 24 2021 *)
L342320 := Select[Range[0, 10000], Divisible[2^(# + 2) - 2, # + 1] &];
A342320[n_] := L342320[[n + 1]] (* Peter Luschny, Apr 10 2021 *)

Numbers k such that k + 1 divides 2^(k + 2) - 2. - Vaclav Kotesovec, Mar 24 2021

A125227 A014741(n)/6 for n>2.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 49, 57, 63, 81, 147, 171, 189, 219, 243, 301, 343, 399, 441, 513, 567, 657, 729, 889, 903, 1029, 1083, 1197, 1323, 1533, 1539, 1701, 1971, 2107, 2187, 2359, 2401, 2667, 2709, 2793, 3087, 3249, 3591, 3969, 4161, 4401, 4599, 4617, 5103, 5913

Offset: 3

Alexander Adamchuk, Jan 15 2007

nonn

A014741(n) is divisible by 6 for n>2.
All powers of 3 are terms. All powers of 7 are terms. The prime divisors of terms of this sequence (for n up to 10^6) in order of their first appearance are 3, 7, 19, 73, 43, 127, 337, 163, 379, 571, 5419, 487, 2593, 439, 1459, 431, 883.
The sequence is multiplicative in the sense that if two numbers k and m are terms, then k*m is too.

			a(3) = A014741(3)/6 = 6/6 = 1.
a(4) = A014741(4)/6 = 18/6 = 3.

Amiram Eldar, Table of n, a(n) for n = 3..10000

Cf. A014741, A000295, A086787.

Select[Range[3,6000], PowerMod[2,#+1,# ]==2&]/6

a(n) = A014741(n)/6 = A014945(n-1)/3. - Max Alekseyev, Nov 14 2012

A323203 "Primitive" numbers k such that k divides 4^k - 1.

Original entry on oeis.org

1, 3, 21, 147, 171, 657, 903, 1029, 1197, 2667, 3249, 4599, 6321, 7077, 7203, 8379, 12483, 13203, 18669, 22743, 32193, 38829, 44247, 47961, 49539, 50421, 51471, 58653, 61731, 71631, 87381, 92421, 97641, 113799, 114681, 118341, 130683, 152019, 159201, 197757

Offset: 1

Bernard Schott, Jan 07 2019

In the comments of A014945, Charles R. Greathouse writes "this sequence is closed under multiplication". So, here, the terms are only the "primitive" integers which satisfy the definition and are not the product of two or more previous numbers of the sequence. This sequence is a subsequence of A014945.
Also numbers k in A014945 such that no divisors d > 1 of k exist where d and k/d are in A014945. - David A. Corneth, Jan 11 2019
Following an observation of David A. Corneth, yes, a(n) is divisible by 3 for n > 1, there is a proof by Robert Israel in A014945. - Bernard Schott, Jan 25 2019

			3 is a term because 3 * 21 = 4^3 - 1.
63 divides 4^63 - 1, but 63 is not a term because 63 = 3 * 21 with 3 which divides 4^3 - 1, and 21 which divides 4^21 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..3200

Cf. A014945, A024036.

filter:= proc(n) local d;
  if 4 &^ n - 1 mod n <> 0 then return false fi;
  for d in select(t -> t > 1 and t^2 <= n, numtheory:-divisors(n)) do
    if 4 &^ d - 1 mod d = 0 and 4 &^ (n/d) - 1 mod (n/d) = 0 then return false fi;
  od;
true
end proc:
select(filter, [$1..200000]); # Robert Israel, Jan 24 2019

is(n) = my(d=divisors(n)); if(Mod(4,n)^n != 1, return(0)); for(i = 2, (#d - 1) >> 1 + 1, if(Mod(4,d[i]) ^ d[i] == 1 && Mod(4, n/d[i]) ^ (n/d[i])==1, return(0))); 1
first(n) = n = max(n, 2); my(res = vector(n), t=1); res[1] = 1;forstep(i = 3, oo, 3, if(is(i), t++; res[t] = i; if(t==n, return(res)))) \\ David A. Corneth, Jan 11 2019

More terms (using b-file for A014945) from Jon E. Schoenfield, Jan 11 2019

Terms verified by Jon E. Schoenfield and David A. Corneth, Jan 12 2019

A014962 Odd numbers k that divide 25^k - 1.

Original entry on oeis.org

1, 3, 9, 21, 27, 63, 81, 93, 147, 171, 189, 243, 279, 441, 513, 567, 609, 651, 729, 837, 903, 1029, 1197, 1323, 1539, 1701, 1827, 1953, 2187, 2511, 2667, 2709, 2883, 2943, 3087, 3249, 3591, 3969, 4263, 4401, 4557, 4617, 5103, 5301, 5481, 5859, 6321

Offset: 1

Olivier Gérard

nonn

Also, numbers k such that k divides s(k), where s(1)=1, s(j) = s(j-1) + j*25^(j-1).

Equivalently, numbers k that divide ((24*k - 1)*25^k + 1) / 24^2 (cf. A014943).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014943, A014945, A014946, A014949, A014950, A014951, A014956, A014957, A128358, A128360, A014959, A014960.

select(t -> 25 &^ t - 1 mod t = 0, [seq(i,i=1..10^4,2)]); # Robert Israel, Oct 04 2020

Edited by Max Alekseyev, Nov 16 2019

A015942 Positive integers n such that n | (2^n + n/2 - 1).

Original entry on oeis.org

6, 18, 42, 54, 126, 162, 294, 342, 378, 486, 882, 1026, 1134, 1314, 1458, 1806, 2058, 2394, 2646, 3078, 3402, 3942, 4374, 5334, 5418, 6174, 6498, 7182, 7938, 9198, 9234, 10206, 11826, 12642, 13122, 14154, 14406, 16002, 16254, 16758

Offset: 1

Robert G. Wilson v

Cf. A015945.

A015942_list = [n for n in range(2,10**6,2) if pow(2,n,n) == n//2+1] # Chai Wah Wu, Mar 25 2021

a(n) = 2 * A014945(n). [Max Alekseyev, Aug 04 2011]

cp's OEIS Frontend

A115976 Numbers k that divide 2^(k-2) + 1.

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A342320 Integers k such that Euler(k, 1) is an integer multiple of Bernoulli(k + 1, 1).

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A125227 A014741(n)/6 for n>2.

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A323203 "Primitive" numbers k such that k divides 4^k - 1.

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A014962 Odd numbers k that divide 25^k - 1.

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Programs

Maple

Extensions

A015942 Positive integers n such that n | (2^n + n/2 - 1).

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Python

Formula