Original entry on oeis.org
1, 4, 21, 40, 1555, 3906, 299593, 3280, 87381, 435848050, 67546215517, 61035156, 61054982558011, 328114698808274, 76861433640456465, 21523360, 128583032925805678351, 953674316406, 275941052631578947368421, 1743392200
Offset: 1
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A110737 := proc(n) local i,a ; if n = 1 then RETURN(1) ; else a := 2 ; while (1-a^n)/(1-a) mod n <> 0 do a := a+1 ; od ; RETURN( (1-a^n)/(1-a) ) ; fi ; end: for n from 1 to 25 do printf("%d, ",A110737(n)) : od : # R. J. Mathar, Mar 13 2007
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Block[{a = {1}, k, s}, Do[k = 2; While[Mod[Set[s, Total@ NestList[# k &, 1, i - 1]], i] != 0, k++]; AppendTo[a, s], {i, 2, 20}]; a] (* Michael De Vlieger, Dec 31 2020 *)
A125598
a(n) = ((n+1)^(n-1) - 1)/n.
Original entry on oeis.org
0, 1, 5, 31, 259, 2801, 37449, 597871, 11111111, 235794769, 5628851293, 149346699503, 4361070182715, 139013933454241, 4803839602528529, 178901440719363487, 7143501829211426575, 304465936543600121441
Offset: 1
Cf. other sequences of generalized repunits, such as
A125118,
A053696,
A055129,
A060072,
A031973,
A173468,
A023037,
A119598,
A085104, and
A162861.
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[((n+1)^(n-1) -1)/n: n in [1..25]]; // G. C. Greubel, Aug 15 2022
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Table[((n+1)^(n-1)-1)/n, {n,25}]
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[gaussian_binomial(n,1,n+2) for n in range(0,18)] # Zerinvary Lajos, May 31 2009
A125120
Sum of values of repunits of length n in base b representation with 1
Original entry on oeis.org
1, 7, 41, 296, 2829, 34637, 519049, 9197344, 188039787, 4356087231, 112754069273, 3224945523736, 100999970565337, 3437517630509497, 126332966608699441, 4986057436997869696, 210331809309776028055, 9443995455881145458715
Offset: 1
a(4) = [1111]_2 + [1111]_3 + [1111]_4 + [1111]_5 = ((2+1)*2+1)*2+1 + ((3+1)*3+1)*3+1 + ((4+1)*4+1)*4+1 + ((5+1)*5+1)*5+1 = 296.
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[(&+[(k^n -1)/(k-1): k in [2..n+1]]) : n in [1..30]]; // G. C. Greubel, Aug 14 2022
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Table[Sum[(k^n -1)/(k-1), {k, 2, n+1}], {n, 30}] (* G. C. Greubel, Aug 14 2022 *)
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a(n) = sum(k=1, n, sum(i=0, n-1, (k+1)^i)); \\ Michel Marcus, Dec 14 2020
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[sum(((k+1)^n -1)/k for k in (1..n)) for n in (1..30)] # G. C. Greubel, Aug 14 2022
A125119
Values of repunits with odd length L in base (L+3)/2 representation.
Original entry on oeis.org
1, 13, 341, 19531, 2015539, 329554457, 78536544841, 25736391511831, 11111111111111111, 6115909044841454629, 4182283628124518315101, 3479492117784426363920483, 3461445831219105624193478971
Offset: 1
a(4) = [1111111]_5 = (((((5+1)*5+1)*5+1)*5+1)*5+1)*5+1 = 19531.
- Eric Weisstein's World of Mathematics, Repunit
Comments