Search: a057954 -id:a057954
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A046051
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Number of prime factors of Mersenne number M(n) = 2^n - 1 (counted with multiplicity).
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+10
45
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0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 5, 1, 3, 3, 4, 1, 6, 1, 6, 4, 4, 2, 7, 3, 3, 3, 6, 3, 7, 1, 5, 4, 3, 4, 10, 2, 3, 4, 8, 2, 8, 3, 7, 6, 4, 3, 10, 2, 7, 5, 7, 3, 9, 6, 8, 4, 6, 2, 13, 1, 3, 7, 7, 3, 9, 2, 7, 4, 9, 3, 14, 3, 5, 7, 7, 4, 8, 3, 10, 6, 5, 2, 14, 3, 5, 6, 10, 1, 13, 5, 9, 3, 6, 5, 13, 2, 5, 8
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OFFSET
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1,4
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COMMENTS
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LINKS
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J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
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FORMULA
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EXAMPLE
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a(4) = 2 because 2^4 - 1 = 15 = 3*5.
The sequence of Mersenne numbers together with their prime indices begins:
1: {}
3: {2}
7: {4}
15: {2,3}
31: {11}
63: {2,2,4}
127: {31}
255: {2,3,7}
511: {4,21}
1023: {2,5,11}
2047: {9,24}
4095: {2,2,3,4,6}
8191: {1028}
16383: {2,14,31}
32767: {4,11,36}
65535: {2,3,7,55}
131071: {12251}
262143: {2,2,2,4,8,21}
524287: {43390}
1048575: {2,3,3,5,11,13}
(End)
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MATHEMATICA
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a[q_] := Module[{x, n}, x=FactorInteger[2^n-1]; n=Length[x]; Sum[Table[x[i][2], {i, n}][j], {j, n}]]
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A057951
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Number of prime factors of 10^n - 1 (counted with multiplicity).
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+10
26
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2, 3, 4, 4, 4, 7, 4, 6, 6, 6, 4, 9, 5, 6, 8, 8, 4, 11, 3, 9, 9, 9, 3, 12, 7, 8, 9, 10, 7, 15, 5, 13, 8, 8, 9, 14, 5, 5, 8, 13, 6, 17, 6, 13, 12, 8, 4, 15, 6, 12, 10, 11, 6, 16, 10, 14, 8, 10, 4, 22, 9, 7, 16, 17, 9, 17, 5, 12, 8, 14, 4, 20, 5, 9, 14, 8, 10, 18
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OFFSET
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1,1
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LINKS
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S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
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FORMULA
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MATHEMATICA
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PROG
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CROSSREFS
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Cf. A001222, A002283, A046053, A085035, A003020, A046107, A005422, A061075, A102380, A095370, A147556, A081317, A081318, A102347, A112505.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A057958
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Number of prime factors of 3^n - 1 (counted with multiplicity).
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+10
22
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1, 3, 2, 5, 3, 5, 2, 7, 3, 6, 3, 8, 2, 5, 5, 10, 3, 8, 3, 10, 4, 7, 3, 11, 5, 5, 6, 9, 4, 11, 4, 12, 5, 8, 6, 12, 3, 7, 7, 13, 4, 11, 3, 11, 9, 6, 5, 17, 7, 10, 6, 9, 4, 13, 8, 13, 7, 9, 3, 17, 3, 8, 6, 14, 7, 12, 4, 12, 6, 11, 2, 16, 5, 8, 10, 11, 7, 15, 4, 18, 9, 8, 5, 18, 7, 6, 8, 16, 4, 19, 5
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OFFSET
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1,2
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LINKS
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FORMULA
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A057957
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Number of prime factors of 4^n - 1 (counted with multiplicity).
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+10
19
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1, 2, 3, 3, 3, 5, 3, 4, 6, 6, 4, 7, 3, 6, 7, 5, 3, 10, 3, 8, 8, 7, 4, 10, 7, 7, 9, 8, 6, 13, 3, 7, 9, 7, 9, 14, 5, 7, 8, 10, 5, 14, 5, 10, 13, 9, 6, 13, 5, 14, 11, 10, 6, 15, 12, 11, 9, 9, 6, 17, 3, 8, 14, 9, 9, 15, 5, 11, 9, 16, 6, 19, 6, 10, 14, 11, 10, 18, 5, 13, 16, 10, 8, 19, 7, 10, 11
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OFFSET
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1,2
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LINKS
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FORMULA
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A057953
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Number of prime factors of 8^n - 1 (counted with multiplicity).
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+10
18
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1, 3, 2, 5, 3, 6, 4, 7, 3, 7, 4, 10, 4, 8, 6, 10, 5, 9, 4, 13, 7, 9, 4, 14, 7, 8, 6, 14, 6, 13, 3, 13, 8, 11, 11, 15, 6, 9, 9, 17, 5, 14, 5, 15, 10, 9, 6, 19, 7, 14, 8, 18, 8, 16, 10, 19, 7, 11, 6, 24, 5, 8, 10, 16, 8, 17, 6, 20, 9, 22, 7, 21, 7, 13, 14, 17, 10, 16, 8, 23, 10, 12, 5, 24
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OFFSET
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1,2
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LINKS
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FORMULA
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MATHEMATICA
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PROG
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(Magma) f:=func<n|&+[p[2]: p in Factorization(n)]>; [f(8^n - 1):n in [1..90]]; // Marius A. Burtea, Feb 02 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A057952
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Number of prime factors of 9^n - 1 (counted with multiplicity).
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+10
17
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3, 5, 5, 7, 6, 8, 5, 10, 8, 10, 7, 11, 5, 9, 11, 12, 8, 12, 7, 13, 11, 11, 6, 17, 10, 9, 13, 13, 9, 17, 8, 14, 12, 12, 11, 16, 8, 11, 15, 18, 8, 18, 6, 16, 19, 10, 10, 21, 12, 18, 15, 13, 8, 18, 15, 19, 15, 13, 7, 24, 7, 13, 19, 16, 12, 18, 8, 17, 15, 20, 9, 24, 9, 13, 22, 17, 13, 22
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OFFSET
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1,1
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LINKS
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FORMULA
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MATHEMATICA
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PrimeOmega[Table[9^n - 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)
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CROSSREFS
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bigomega(b^n-1): A046051 (b=2), A057958 (b=3), A057957 (b=4), A057956 (b=5), A057955 (b=6), A057954 (b=7), A057953 (b=8), this sequence (b=9), A057951 (b=10), A366682 (b=11), A366708 (b=12).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A057955
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Number of prime factors of 6^n - 1 (counted with multiplicity).
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+10
17
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1, 2, 2, 3, 3, 4, 2, 4, 4, 6, 3, 7, 3, 6, 6, 6, 5, 7, 3, 9, 4, 5, 5, 9, 6, 7, 6, 9, 2, 11, 3, 9, 6, 8, 7, 13, 6, 6, 6, 12, 3, 10, 5, 9, 11, 8, 4, 13, 5, 10, 9, 11, 4, 11, 7, 14, 7, 6, 4, 20, 4, 5, 10, 12, 9, 12, 3, 11, 8, 18, 2, 18, 5, 10, 12, 9, 6, 15, 4, 17, 8, 7, 8, 17, 10, 7, 7, 12, 4, 18, 6
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OFFSET
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1,2
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LINKS
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FORMULA
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EXAMPLE
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6^10 - 1 = 60466175 = 5^2 * 7 * 11 * 101 * 311 and a(10) = bigomega(60466175) = 2+1+1+1+1 = 6. - Bernard Schott, Feb 02 2020
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MATHEMATICA
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PROG
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(Magma) f:=func<n|&+[p[2]: p in Factorization(n)]>; [f(6^n - 1):n in [1..90]]; // Marius A. Burtea, Feb 02 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A057956
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Number of prime factors of 5^n - 1 (counted with multiplicity).
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+10
17
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2, 4, 3, 6, 4, 7, 3, 8, 5, 7, 3, 10, 3, 7, 7, 11, 4, 11, 5, 11, 6, 8, 4, 13, 8, 7, 9, 10, 5, 14, 4, 14, 6, 8, 9, 16, 5, 10, 6, 15, 4, 16, 4, 12, 12, 8, 3, 17, 4, 13, 8, 12, 5, 19, 10, 13, 7, 9, 4, 21, 5, 9, 11, 18, 8, 15, 7, 14, 9, 16, 4, 22, 5, 10, 16, 14, 7, 14, 5, 20, 11, 10, 5, 22, 9, 10
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OFFSET
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1,1
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LINKS
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FORMULA
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A059889
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a(n) = |{m : multiplicative order of 7 mod m=n}|.
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+10
16
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4, 6, 8, 26, 4, 42, 12, 48, 52, 66, 12, 778, 4, 138, 80, 300, 12, 528, 12, 1430, 72, 138, 28, 15216, 24, 66, 1216, 966, 28, 3630, 28, 1344, 360, 58, 108, 16988, 28, 138, 176, 12752, 28, 7398, 12, 4422, 1900, 122, 12, 131028, 240, 536, 744, 1046, 28, 23744, 44
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OFFSET
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1,1
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COMMENTS
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The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).
a(n) = number of orders of degree n monic irreducible polynomials over GF(7).
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LINKS
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FORMULA
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a(n) = Sum_{d|n} mu(n/d)*tau(7^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).
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MAPLE
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with(numtheory):
a:= n-> add(mobius(n/d)*tau(7^d-1), d=divisors(n)):
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A057937
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Number of prime factors of 7^n + 1 (counted with multiplicity).
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+10
15
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3, 3, 4, 2, 5, 5, 5, 3, 5, 6, 5, 5, 5, 4, 7, 3, 4, 6, 5, 4, 8, 6, 4, 4, 6, 5, 6, 6, 4, 10, 6, 4, 8, 6, 11, 8, 5, 4, 9, 6, 7, 7, 10, 5, 12, 7, 4, 6, 12, 11, 8, 5, 7, 9, 13, 8, 9, 7, 7, 9, 4, 6, 14, 4, 9, 12, 6, 5, 8, 10, 5, 9, 6, 4, 11, 7, 12, 10, 7, 6, 10, 6, 5, 12, 8, 6, 8, 5, 8, 14, 11, 6, 9, 6, 11
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OFFSET
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1,1
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LINKS
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FORMULA
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MATHEMATICA
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PrimeOmega[Table[7^n + 1, {n, 1, 30}]] (* Amiram Eldar, Feb 02 2020 *)
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PROG
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(Magma) f:=func<n|&+[p[2]: p in Factorization(n)]>; [f(7^n + 1):n in [1..110]]; // Marius A. Burtea, Feb 02 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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