A001348 -id:A001348 - cp's OEIS frontend

A088863 Number of prime factors of n-th Mersenne number M(p_n).

1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 2, 2, 3, 3, 3, 2, 1, 2, 3, 3, 3, 2, 1, 2, 2, 2, 1, 2, 5, 1, 2, 2, 2, 2, 5, 4, 5, 2, 4, 3, 4, 5, 3, 2, 2, 3, 6, 2, 4, 4, 6, 2, 5, 3, 4, 2, 2, 3, 2, 3, 2, 5, 3, 4, 4, 3, 5, 2, 3, 3, 6, 5, 2, 2, 5, 3, 9, 4, 3, 5, 2, 8, 4, 4, 3, 5, 2, 4, 6, 3, 4, 2, 7, 3, 4, 4, 1, 2, 5, 4, 5, 3, 5, 4

Offset: 1

Jeppe Stig Nielsen, Nov 25 2003

nonn

			a(5)=2 because M(p_5)=M(11)=2047 has 2 (not necessarily distinct) prime factors.

Max Alekseyev, Table of n, a(n) for n = 1..197 (first 137 terms from Herman Jamke)
Herman Jamke, The first 137 terms in detail

Cf. A001348, A003260, A046051, A046932.

seq(nops(ifactor(2^ithprime(n)-1)),n=1..32); # Emeric Deutsch, Dec 23 2004

Do[m = 2^Prime[n] - 1; Print[Plus @@ Last /@ FactorInteger[m]], {n, 1, 50}] (* Ryan Propper, Jul 31 2005 *)

for(n=1,137,print1(bigomega(2^prime(n)-1)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

a(n) = A001222(A001348(n)) = A001222(A000225(A000040(n))).

14 more terms from Emeric Deutsch, Dec 23 2004
More terms from Ryan Propper, Jul 31 2005
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

A303941 Triangle read by rows: T(0,0) = 1; T(n,k) = 3T(n-1,k) - 2T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. Triangle of coefficients of Fermat polynomials.

Original entry on oeis.org

1, 3, 9, -2, 27, -12, 81, -54, 4, 243, -216, 36, 729, -810, 216, -8, 2187, -2916, 1080, -96, 6561, -10206, 4860, -720, 16, 19683, -34992, 20412, -4320, 240, 59049, -118098, 81648, -22680, 2160, -32, 177147, -393660, 314928, -108864, 15120, -576, 531441, -1299078, 1180980, -489888, 90720, -6048, 64

Offset: 0

Zagros Lalo, May 03 2018

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A303901 ((3-2x)^n).
Row n gives coefficients of Fermat polynomial.
The coefficients in the expansion of 1/(1-3x+2x^2) are given by the sequence generated by the row sums.

			Triangle begins:
n\k |       0         1        2         3        4        5      6     7
----+--------------------------------------------------------------------
   0|       1
   1|       3
   2|       9        -2
   3|      27       -12
   4|      81       -54        4
   5|     243      -216       36
   6|     729      -810      216        -8
   7|    2187     -2916     1080       -96
   8|    6561    -10206     4860      -720       16
   9|   19683    -34992    20412     -4320      240
  10|   59049   -118098    81648    -22680     2160      -32
  11|  177147   -393660   314928   -108864    15120     -576
  12|  531441  -1299078  1180980   -489888    90720    -6048     64
  13| 1594323  -4251528  4330260  -2099520   489888   -48384   1344
  14| 4782969 -13817466 15588936  -8660520  2449440  -326592  16128  -128
  15|14348907 -44641044 55269864 -34642080 11547360 -1959552 145152 -3072

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 104, 394, 395.

Zagros Lalo, Left-justified triangle
Eric Weisstein's World of Mathematics, Fermat Polynomial.

Row sums give A000225.
Some row sums give A001348.
Cf. A303901.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 3 t[n - 1, k] - 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 14}, {k, 0, Floor[n/2]}] // Flatten

T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 3*T(n-1,k) - 2*T(n-2,k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 10 2018

A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.

Original entry on oeis.org

3, 7, 12, 13, 18, 19, 24, 28, 36, 37, 40, 43, 66, 67, 79, 88, 97, 103, 109, 124, 127, 163, 184, 193, 223, 229, 232, 277, 307, 313, 328, 349, 379, 397, 424, 439, 457, 463, 487, 499, 508, 613, 643, 664, 673, 712, 739, 757, 769, 823, 853, 859, 877, 883, 904, 907

Offset: 1

Labos Elemer, Aug 17 2000

nonn

In contrast with A015913, composite solutions are not rare. Prime solutions are common.
From Kevin J. Gomez, Mar 02 2016: (Start)
Composite solutions have two known forms:
  n such that n = 4 * (2^p - 1) where 2^p - 1 is a Mersenne prime. (A001348)
  n such that n = 8q where q is a Sophie Germain prime. (A005394)
There are composite solutions (such as 36) that do not fit either of these forms. (End)

			For k = 1048: phi(1048) = 520, phi(1048+4) = 524.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Cf. A000010, A015913 (sigma(k+4) = sigma(k) + 4).

Cf. A001838 (k=2), this sequence (k=4), A262084 (k=6), A262085 (k=8), A262086 (k=10).

[n: n in [1..1000] | EulerPhi(n+4) eq EulerPhi(n)+4]; // Vincenzo Librandi, Sep 11 2015

Select[Range@1000, EulerPhi@(# + 4)== EulerPhi[#] + 4 &] (* Vincenzo Librandi, Sep 11 2015 *)
Position[Partition[EulerPhi[Range[1000]],5,1],?(#[[1]]+4==#[[5]]&),1, Heads-> False]//Flatten (* _Harvey P. Dale, Dec 18 2019 *)

isok(n) = eulerphi(n+4) == eulerphi(n) + 4; \\ Michel Marcus, Sep 11 2015

A108974 Sort the primes (except 2) according to the multiplicative order of 2 modulo that prime. If two primes have the same order of 2, they are arranged numerically.

Original entry on oeis.org

3, 7, 5, 31, 127, 17, 73, 11, 23, 89, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 47, 178481, 241, 601, 1801, 2731, 262657, 29, 113, 233, 1103, 2089, 331, 2147483647, 65537, 599479, 43691, 71, 122921, 37, 109, 223, 616318177, 174763, 79

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Jul 27 2005

nonn

Or, primitive prime divisors of the Mersenne numbers 2^n-1 (see A000225) in their order of occurrence.
Of course the Mersenne primes 2^p-1 (cf. A000043) appear in this sequence.
If all odd positive numbers, not just the odd primes, are sorted in this way, the result is A059912. - Jeppe Stig Nielsen, Feb 13 2020

			The order of 2 modulo 3 is 2 and the order of 2 modulo 7 is 3. So 3 comes before 7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..4275
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
Jeppe Stig Nielsen, A108974 arranged as an irregular array.
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265-284.

Cf. A000043, A000225, A001348, A014664, A059912, A086251.

a = 1; DeleteDuplicates[Flatten[#[[All, 1]] & /@ FactorInteger[Table[a = 2 a + 1, {i, 1, 30}]]]] (* Horst H. Manninger, Mar 20 2021 *)

do(n)=my(v=List(),P=1,g,t,f); for(k=2,n, t=2^k-1; g=P; while((g=gcd(g,t))>1, t/=g); f=factor(t)[,1]; for(i=1,#f, listput(v,f[i])); P*=t); Vec(v) \\ Charles R Greathouse IV, Sep 23 2016

More terms from Martin Fuller, Sep 25 2006

A131458 Residues of 3^(2^(p(n)-1)-1) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 6, 30, 126, 1565, 8190, 131070, 524286, 7511964, 89777599, 2147483646, 20166585982, 840455563322, 4787976306682, 5519162753736, 2617809209727498, 334169564069012755, 2305843009213693950, 47306781863857413639

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 13 2007, Jul 20 2007

nonn

Mp is prime iff 3^(2^(p(n)-1)-1) is congruent to (-1) Mod Mp. Thus M7 = 127 is prime because 3^63 Mod 127 = 126 (=127-1) while M11 = 2047 is composite because 3^1023 Mod 2047 <> 2046.

			a(5) = 3^(2^(11-1)-1) Mod 2^11-1 = 3^1023 Mod 2047 = 1565

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131459, A131460, A131461, A131462, A131463.

a(n) = 3^(2^(p(n)-1)-1) Mod 2^p(n)-1

A131459 Residues of 3^(2^(p(n)-1)) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 4, 28, 124, 601, 8188, 131068, 524284, 5758678, 269332797, 2147483644, 60499757946, 322343434415, 5567835897839, 16557488261208, 7853427629182494, 426047939903614778, 2305843009213693948, 141920345591572240917

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 13 2007, Jul 20 2007

nonn

Mp is prime iff 3^(2^(p(n)-1)) is congruent to (-3) Mod Mp. Thus M7 = 127 is prime because 3^64 Mod 127 = 124 (=127-3) while M11 = 2047 is composite because 3^1024 Mod 2047 <> 2044.

			a(5) = 3^(2^(11-1)) Mod 2^11-1 = 3^1024 Mod 2047 = 601

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131460, A131461, A131462, A131463.

a(n) = 3^(2^(p(n)-1)) Mod 2^p(n)-1

A131460 Residues of 3^(2^(p(n)-1)+1) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 5, 22, 118, 1803, 8182, 131062, 524278, 498820, 271127480, 2147483638, 44060320367, 967030303245, 7907414671310, 49672464783624, 5545884378065500, 125222315103997360, 2305843009213693942, 130613131595363896897

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 13 2007, Jul 20 2007

nonn

Mp is prime iff 3^(2^(p(n)-1)+1) is congruent to (-9) Mod Mp. Thus M7 = 127 is prime because 3^65 Mod 127 = 118 (=127-9) while M11 = 2047 is composite because 3^1025 Mod 2047 <> 2038.

			a(5) = 3^(2^(11-1)+1) Mod 2^11-1 = 3^1025 Mod 2047 = 1803

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131461, A131462, A131463.

a(n) = 3^(2^(p(n)-1)+1) Mod 2^p(n)-1

A131461 Residues of 3^(2^p(n)-2) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 1, 1, 1, 1013, 1, 1, 1, 5884965, 65165529, 1, 103888408793, 474639880182, 4112907695371, 72685811469476, 5155089749987738, 440411515280180314, 1, 95591506202441271281, 69291880649932219827

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 20 2007

nonn

M_p is prime iff 3^(M_p-1) is congruent to 1 mod M_p. Thus M_7 = 127 is prime because 3^126 mod 127 = 1 while M_11 = 2047 is composite because 3^2046 mod 2047 <> 1.

			a(5) = 3^(2^11-2) mod 2^11-1 = 3^2046 mod 2047 = 1013

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131460, A131462, A131463.

a(n) = 3^(2^p(n)-2) mod 2^p(n)-1

A131462 Residues of 3^(2^p(n)-1) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 3, 3, 3, 992, 3, 3, 3, 877681, 195496587, 3, 36787319437, 1423919640546, 3542630063906, 77319946053101, 6458069995222223, 168313041233693968, 3, 139200566017647400916, 207875641949796659481

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 20 2007

nonn

M_p is prime iff 3 ^ M_p is congruent to 3 mod M_p. Thus M_7 = 127 is prime because 3^127 mod 127 = 3 while M_11 = 2047 is composite because 3^2047 mod 2047 <> 3.

			a(5) = 3^(2^11-1) mod 2^11-1 = 3^2047 mod 2047 = 992

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131460, A131461, A131463.

a(n) = 3^(2^p(n)-1) mod 2^p(n)-1

A131463 Residues of 3^(2^p(n)) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 2, 9, 9, 929, 9, 9, 9, 2633043, 49618850, 9, 110361958311, 2072735666087, 1831797169511, 91222349803976, 1359811476184687, 504939123701081904, 9, 122453792873589376894, 623626925849389978443

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 20 2007

nonn

M_p is prime iff 3^(M_p+1) is congruent to 9 mod M_p. Thus M_7 = 127 is prime because 3^128 mod 127 = 9 while M_11 = 2047 is composite because 3^2048 mod 2047 <> 9.

			a(5) = 3^(2^11) mod 2^11-1 = 3^2048 mod 2047 = 929

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131460, A131461, A131462.

a(n) = 3^(2^p(n)) mod 2^p(n)-1

cp's OEIS Frontend

A088863 Number of prime factors of n-th Mersenne number M(p_n).

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A303941 Triangle read by rows: T(0,0) = 1; T(n,k) = 3*T(n-1,k) - 2*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. Triangle of coefficients of Fermat polynomials.

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A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.

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Magma

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A108974 Sort the primes (except 2) according to the multiplicative order of 2 modulo that prime. If two primes have the same order of 2, they are arranged numerically.

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A131458 Residues of 3^(2^(p(n)-1)-1) for Mersenne numbers with prime indices.

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A131459 Residues of 3^(2^(p(n)-1)) for Mersenne numbers with prime indices.

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A131460 Residues of 3^(2^(p(n)-1)+1) for Mersenne numbers with prime indices.

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A131461 Residues of 3^(2^p(n)-2) for Mersenne numbers with prime indices.

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A303941 Triangle read by rows: T(0,0) = 1; T(n,k) = 3T(n-1,k) - 2T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. Triangle of coefficients of Fermat polynomials.