A229207
Numbers k such that Sum_{j=1..k} tau(j)^j == 0 (mod k), where tau(j) = A000005(j), the number of divisors of j.
Original entry on oeis.org
1, 46, 135, 600, 1165, 1649, 5733, 6788, 6828, 9734, 29686, 363141, 1542049
Offset: 1
tau(1)^1 + tau(2)^2 + ... + tau(45)^45 + tau(46)^46 = 1^1 + 2^2 + ... + 6^45 + 4^46 = 86543618042218910328339719795268200166 and 86543618042218910328339719795268200166 / 46 = 1881383000917802398442167821636265221.
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with(numtheory); P:=proc(q) local n, t; t:=0;
for n from 1 to q do t:=t+tau(n)^n; if t mod n=0 then print(n);
fi; od; end: P(10^6);
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Module[{nn=30000,ac},ac=Accumulate[Table[DivisorSigma[0,i]^i,{i,nn}]];Select[ Thread[{ac,Range[nn]}],Divisible[#[[1]],#[[2]]]&]][[All,2]](* Harvey P. Dale, Dec 13 2018 *)
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isok(n) = sum(i=1, n, Mod(numdiv(i), n)^i) == 0; \\ Michel Marcus, Feb 25 2016
A229209
Numbers k such that Sum_{j=1..k} phi(j)^j == 0 (mod k).
Original entry on oeis.org
1, 2, 5, 7, 11, 39, 126, 266, 683, 2514, 12929
Offset: 1
phi(1)^1 + phi(2)^2 + phi(3)^3 + phi(4)^4 + phi(5)^5 = 1^1 + 1^2 + 2^3 + 2^4 + 4^5 = 1050 and 1050/5 = 210.
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with(numtheory); P:=proc(q) local n, t; t:=0;
for n from 1 to q do t:=t+phi(n)^n; if t mod n=0 then print(n);
fi; od; end: P(10^6);
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is(k) = sum(i=1, k, Mod(eulerphi(i), k)^i) == 0; \\ Jinyuan Wang, Feb 19 2021
A260654
Numbers k such that Sum_{i=1..k} sigma(i)^d(i) == 0 (mod k), where sigma = A000203 and d = A000005.
Original entry on oeis.org
1, 2, 5, 56, 59, 60, 75, 122, 743, 2835, 3951, 5712, 6866, 7884, 14754, 18751, 292123, 465289, 1921892, 3902477, 7609760, 21855984, 22013406, 358753359, 570535294, 582046711, 1846338478, 13691385818
Offset: 1
sigma(1)^tau(1) + sigma(2)^tau(2) + sigma(3)^tau(3) + sigma(4)^tau(4) + sigma(5)^tau(5) = 1^1 + 3^2 + 4^2 + 7^3 + 6^2 = 1 + 9 + 16 + 343 + 36 = 405 and 405 / 5 = 81.
Cf.
A000005,
A000203,
A227427,
A227429,
A227502,
A227848,
A229095,
A229207,
A229208,
A229209,
A229210,
A229211.
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with(numtheory): P:=proc(q) local a,n; a:=0;
for n from 1 to q do a:=a+sigma(n)^tau(n);
if a mod n=0 then print(n); fi; od; end: P(10^6);
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for(n=1, 1e4, if(sum(k=1, n, sigma(k)^numdiv(k))%n==0, print1(n", "))) \\ Altug Alkan, Nov 13 2015
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list(lim) = {my(s = 0, f); for(k = 1, lim, f = factor(k); s += sigma(f)^numdiv(f); if(!(s % k), print1(k, ", ")));} \\ Amiram Eldar, Dec 29 2024
Incorrect terms removed by and more terms from
Jinyuan Wang, Feb 18 2021
Showing 1-3 of 3 results.
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