Original entry on oeis.org
1, 2, 3, 6, 7, 23, 19, 66, 95, 255, 187, 1059, 631, 3227, 5243, 11426, 7711, 51887, 27595, 184911, 232887, 606627, 364723, 2807935, 2405183, 8671943, 10368079, 36873651, 18512791, 167268639, 69273667, 496472226, 551130063, 1856103039
Offset: 1
Row 6 of Pascal's triangle is 1,6,15,20,15,6,1. The greatest common divisors of n and each integer from 1 to 6 are gcd(1,6)=1, gcd(2,6)=2, gcd(3,6)=3, gcd(4,6)=2, gcd(5,6)=1, and gcd(6,6)=6. So a(6) = (1/6)*( 6*1 + 15*2 + 20*3 + 15*2 + 6*1 + 1*6) = 138/6 = 23. Note that each term of the sum in parentheses is a multiple of 6, so 138 is a multiple of 6.
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A159068 := proc(n) add(binomial(n,k)*gcd(k,n),k=1..n) ; end: A159069 := proc(n) A159068(n)/n ; end: seq(A159069(n),n=1..80) ; # R. J. Mathar, Apr 06 2009
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Table[Sum[Binomial[n, k] GCD[k, n], {k, n}]/n, {n, 34}] (* Michael De Vlieger, Aug 29 2017 *)
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a(n) = sum(k=1, n, binomial(n,k) * gcd(k,n))/n; \\ Michel Marcus, Aug 30 2017
A159458
Numbers m such that m^2 divides A159068(m), where A159068(m) = Sum_{k=1..m} binomial(m,k) * gcd(m,k).
Original entry on oeis.org
1, 2, 3, 11, 33, 309, 665, 1461, 2323, 6789
Offset: 1
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is(m) = sum(k=1, m, binomial(m, k)*gcd(k, m))%m^2 == 0; \\ Jinyuan Wang, Aug 10 2021
A159553
a(n) = Sum_{k=0..n} binomial(n,k) * gcd(n,k).
Original entry on oeis.org
2, 6, 12, 28, 40, 144, 140, 536, 864, 2560, 2068, 12720, 8216, 45192, 78660, 182832, 131104, 933984, 524324, 3698240, 4890648, 13345816, 8388652, 67390464, 60129600, 225470544, 279938160, 1032462256, 536870968, 5018059200
Offset: 1
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A159553 := proc(n) add(binomial(n, k)*gcd(k, n), k=0..n) ; end: seq(A159553(n),n=1..40) ; # R. J. Mathar, Apr 29 2009
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Table[Sum[Binomial[n, k] GCD[n, k], {k, 0, n}], {n, 30}] (* Michael De Vlieger, Oct 30 2017 *)
A159555
Numbers m where m^2 divides A159553(m), where A159553(m) = Sum_{k=0..m} binomial(m,k) * gcd(m,k).
Original entry on oeis.org
1, 6, 22, 72, 114, 148, 164, 260, 261, 780, 1078, 1184, 1266, 2952, 4674, 21868
Offset: 1
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A159068 := proc(n) option remember; add(binomial(n, k)*gcd(k, n), k=1..n) ; end: A159553 := proc(n) option remember ; A159068(n)+n; end: isA159555 := proc(n) if A159553(n) mod ( n^2) = 0 then true; else false; fi; end: for n from 1 do if isA159555(n) then printf("%d,\n",n) ; fi; od: # R. J. Mathar, Apr 29 2009
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f(n) = sum(k=0, n, binomial(n,k) * gcd(n,k)); \\ A159553
isok(n) = !(f(n) % n^2); \\ Michel Marcus, Sep 05 2019
A329969
a(n) = Sum_{k=1..n} Stirling2(n,k) * gcd(n,k).
Original entry on oeis.org
1, 3, 7, 25, 56, 484, 883, 9643, 32497, 344277, 678580, 13364806, 27644449, 582379763, 3257283507, 28924878925, 82864869820, 2320283069804, 5832742205075, 159175049844787, 1283043441049111, 10423261367890565, 44152005855084368, 1678303577387465300, 9478188244330160181
Offset: 1
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a[n_] := Sum[StirlingS2[n, k] GCD[n, k], {k, 1, n}]; Table[a[n], {n, 1, 25}]
Showing 1-5 of 5 results.
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