A342433
a(n) = Sum_{k=1..n} gcd(k,n)^(n-1).
Original entry on oeis.org
1, 3, 11, 74, 629, 8085, 117655, 2113796, 43059849, 1001955177, 25937424611, 743379914746, 23298085122493, 793811662313709, 29192938251553759, 1152956691126550536, 48661191875666868497, 2185928270773974154773
Offset: 1
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a[n_] := Sum[GCD[k, n]^(n - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 12 2021 *)
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a(n) = sum(k=1, n, gcd(k, n)^(n-1));
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a(n) = sumdiv(n, d, eulerphi(n/d)*d^(n-1));
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a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n-2));
A332620
a(n) = Sum_{k=1..n} n^(n/gcd(n, k)).
Original entry on oeis.org
1, 6, 57, 532, 12505, 93786, 4941265, 67117128, 2324524401, 40000400110, 2853116706121, 35664407810076, 3634501279107049, 66672041585829330, 3503151123049919265, 147573952606856413456, 13235844190181388226849, 236078448452969449231206, 35611553801885644604231641
Offset: 1
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[&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
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Table[Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]
Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^j x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]
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a(n) = sum(k=1, n, n^(n/gcd(n, k))); \\ Michel Marcus, Mar 10 2021
A332653
a(n) = (1/n) * Sum_{k=1..n} n^(k/gcd(n, k)).
Original entry on oeis.org
1, 2, 5, 19, 157, 1306, 19609, 266372, 5321721, 101001214, 2593742461, 61920391842, 1941507093541, 56984643437138, 2076518238897649, 72340172854919941, 3041324492229179281, 121440691499123469858, 5784852794328402307381, 262799364106291328009626
Offset: 1
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[(1/n)*&+[n^(k div Gcd(n,k)):k in [1..n]]:n in [1..21]]; // Marius A. Burtea, Feb 18 2020
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Table[(1/n) Sum[n^(k/GCD[n, k]), {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, n^(k - 1), 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]
A332655
a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.
Original entry on oeis.org
1, 2, 10, 84, 1301, 15693, 376762, 6168552, 176787631, 3770427352, 142364319626, 3152758480715, 154718778284149, 4340093860950619, 210971170836848270, 7281694486114555088, 435659030617933827137, 14181121059071691716406, 1052864393300587929716722, 41673907052879908244100770
Offset: 1
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[&+[(k div Gcd(n,k))^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 18 2020
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Table[Sum[(k/GCD[n, k])^n, {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, k^n, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]
A332652
a(n) = Sum_{k=1..n} n^(k/gcd(n, k)).
Original entry on oeis.org
1, 4, 15, 76, 785, 7836, 137263, 2130976, 47895489, 1010012140, 28531167071, 743044702104, 25239592216033, 797785008119932, 31147773583464735, 1157442765678719056, 51702516367896047777, 2185932446984222457444, 109912203092239643840239, 5255987282125826560192520
Offset: 1
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[&+[n^(k div Gcd(n,k)):k in [1..n]]:n in [1..21]]; // Marius A. Burtea, Feb 18 2020
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Table[Sum[n^(k/GCD[n, k]), {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, n^k, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]
A342396
a(n) = Sum_{k=1..n} k^(n/gcd(k,n) - 1).
Original entry on oeis.org
1, 2, 6, 31, 355, 3150, 67172, 904085, 22998481, 427799450, 14914341926, 287337926355, 13421957361111, 339940911160914, 15434209582905140, 493467700905592777, 28101527071305611529, 836396358233559195382
Offset: 1
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a[n_] := Sum[k^(n/GCD[k, n] - 1), {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Mar 10 2021 *)
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a(n) = sum(k=1, n, k^(n/gcd(k, n)-1));
Showing 1-6 of 6 results.