Search: a127837 -id:a127837
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A060073
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a(n) = (n^(n-1)-1)/(n-1)^2.
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+10
12
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1, 2, 7, 39, 311, 3268, 42799, 672605, 12345679, 259374246, 6140565047, 161792257795, 4696537119847, 148943500129544, 5124095576030431, 190082780764323705, 7563707819165039903, 321380710796022350410, 14523213296398891966759, 695546073617378871592991
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OFFSET
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2,2
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COMMENTS
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Written in base n, a(n) has n-2 digits and looks like 12345... except that the final digit is n-1 rather than n-2.
Note that 2^m-1 divides a(m+1) = ((m+1)^m-1)/m^2 if and only if m = 2^k-1 with gcd(k,m) = 1. Mersenne numbers M = 2^p-1 such that a(M+1)/(2^M-1) is prime are Mersenne primes 2^3-1 = 7 and 2^7-1 = 127. - Thomas Ordowski, Sep 19 2021
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LINKS
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FORMULA
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EXAMPLE
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a(10) = 999999999/81 = 111111111/9 = 12345679.
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MATHEMATICA
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PROG
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(PARI) { for (n=2, 200, write("b060073.txt", n, " ", (n^(n - 1) - 1)/(n - 1)^2); ) } \\ Harry J. Smith, Jul 01 2009
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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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OFFSET
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1,1
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COMMENTS
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Corresponding numbers k are listed in A127837.
Next term has 15850 = 1 + floor((4357*log(4358) - 2*log(4357))/log(10)) digits and is too large to include. - M. F. Hasler, May 22 2007
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LINKS
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FORMULA
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MATHEMATICA
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Select[Table[((n+1)^n-1)/n^2, {n, 500}], PrimeQ] (* Harvey P. Dale, Apr 30 2011 *)
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PROG
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(PARI) A128466(n)=A060073(A127837(n)+1) /* see there. --- or: */ forprime(p=1, 10^5, if(ispseudoprime(n=((p+1)^p-1)/p^2), print1(n, ", "))); \\ M. F. Hasler, May 22 2007
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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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A137666
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Largest prime factor of A137664(n) = (p + 1)^p - 1 for p = prime(n).
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+10
3
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2, 7, 311, 337, 266981089, 29914249171, 7563707819165039903, 192696104561, 58769065453824529, 847499019384726257346113954958447091, 18158209813151, 138233050898929517126243814850350442620694127
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OFFSET
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1,1
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COMMENTS
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a(n) is also the largest prime factor of A137665(n) = A137664(n)/prime(n)^2. p^2 divides A137664(n) = (p + 1)^p - 1, p = prime(n). Least prime factors of A137664(n) are listed in A128456.
a(n) = A128456(n) = A137665(n) = ((p + 1)^p - 1)/p^2 for n = {1,2,3,7,595,...} corresponding to p = prime(n) = {2,3,5,17,4357,...} = A127837.
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LINKS
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MATHEMATICA
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FactorInteger[#][[-1, 1]]&/@((#+1)^#-1&/@Prime[Range[12]]) (* Harvey P. Dale, Apr 07 2018 *)
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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STATUS
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approved
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A137665
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Quotients ((p+1)^p - 1)/p^2 for p = prime(n).
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+10
2
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2, 7, 311, 42799, 6140565047, 4696537119847, 7563707819165039903, 14523213296398891966759, 105051652240885643072548950287, 8160568057655529131985731272294887039239, 47525417447024678661670292427038339608998847, 20681861558186805237407813095538883147812221153173966103
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OFFSET
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1,1
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COMMENTS
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p^2 divides a(n) = (p+1)^p - 1, p = prime(n). (p+1)^p - 1 = A137664(n) = {8, 63, 7775, 2097151, 743008370687, 793714773254143, 2185911559738696531967, ...}.
Least prime factors of a(n) are listed in A128456(n) = {2, 7, 311, 127, 23, 157, 7563707819165039903, ...}.
Largest prime factors a(n) are listed in A137666.
a(n) is prime for n = {1, 2, 3, 7, 595, ...} corresponding to p = prime(n) = {2, 3, 5, 17, 4357, ...} = A127837.
Primes in this sequence are A128466.
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LINKS
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FORMULA
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a(n) = ((prime(n) + 1)^prime(n) - 1)/prime(n)^2;
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MATHEMATICA
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Table[ ((Prime[n] + 1)^Prime[n] - 1)/Prime[n]^2, {n, 1, 15} ]
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PROG
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(PARI) a(n) = my(p=prime(n)); polcyclo(p, p+1)/p \\ Hugo Pfoertner, Jul 21 2024
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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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