A227848
Numbers n such that Sum_{i=1..n} (i')^i == 0 (mod n), where i' is the arithmetic derivative of i.
Original entry on oeis.org
1, 9, 71, 120, 331, 393, 728, 1223, 3697, 4123, 6791, 7391, 23911, 25099, 35287, 86442, 86716, 118034, 292411, 352970, 527255, 606425
Offset: 1
1'^1 + 2'^2 + 3'^3 + 4'^4 + 5'^5 + 6'^6 + 7'^7 + 8'^8 + 9'^9 = 0^1 + 1^2 + 1^3 + 4^4 + 1^5 + 5^6 + 1^7 + 12^8 + 6^9 = 440075277 and 440075277 / 9 = 48897253.
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with(numtheory); ListA227848:=proc(q) local a, n, p; a:=0;
for n from 1 to q do a:=a+(n*add(op(2, p)/op(1, p), p=ifactors(n)[2]))^n;
if a mod n=0 then print(n); fi; od; end: ListA227848(10^6);
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d[n_] := d[n] = n* Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; Reap[For[n = 1, n <= 2*10^5, n++, If[Mod[Sum[d[k]^k, {k, 1, n}], n] == 0, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 21 2014 *)
A229095
Numbers k such that Sum_{i=1..k} i^tau(i) == 0 (mod k), where tau(i) = A000005(i), the number of divisors of i.
Original entry on oeis.org
1, 8, 9, 67, 72, 467, 801, 1071, 5141, 7193, 25688, 68488, 97768, 111816, 381061, 7829505, 17079937, 25615576, 44582211, 91110856, 639359784, 3492789629
Offset: 1
1^tau(1) + 2^tau(2) + ... + 8^tau(8) + 9^tau(9) = 1^1 + 2^2 + 3^2 + 4^3 + 5^2 + 6^4 + 7^2 + 8^4 + 9^3 = 6273 and 6273 / 9 = 697.
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with(numtheory); P:=proc(q) local n, t; t:=0;
for n from 1 to q do t:=t+n^tau(n); if t mod n=0 then print(n);
fi; od; end: P(10^6);
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list(lim) = {my(s = 0, f); for(k = 1, lim, s += k^numdiv(k); if(!(s % k), print1(k, ", ")));} \\ Amiram Eldar, Dec 29 2024
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
A229211
Numbers k such that Sum_{j=1..k} (j*(j+1)/2 - sigma(j))^j == 0 (mod k), where sigma(j) = A000203(j) and j*(j+1)/2 - sigma(j) = A024816(j).
Original entry on oeis.org
1, 2, 9, 78, 3205, 5589, 14153, 246123
Offset: 1
(1*2 / 2 - sigma(1))^1 + (2*3 / 2 - sigma(2))^2 + ... + (9*10 / 2 - sigma(10))^9 = 35223475538772 and 35223475538772 / 9 = 3913719504308.
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with(numtheory); P:=proc(q) local n, t; t:=0;
for n from 1 to q do t:=t+(n*(n+1)/2-sigma(n))^n; if t mod n=0 then print(n); fi; od; end: P(10^6);
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isok(n) = sum(i=1, n, (i*(i+1)/2 - sigma(i))^i) % n == 0; \\ Michel Marcus, Nov 09 2014
Typo in name and crossref corrected by
Michel Marcus, Nov 09 2014
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
A229210
Numbers k such that Sum_{i=1..k} (i-tau(i))^i == 0 (mod k), where tau(i) = A000005(i), the number of divisors of i, and i-tau(i) = A049820(i).
Original entry on oeis.org
1, 2, 5, 24, 36, 371, 445, 1578, 3616, 9292, 38123, 142815, 184097
Offset: 1
(1 - tau(1))^1 + (2 - tau(2))^2 + ... + (5 - tau(5))^5 = 245 and 245 / 5 = 49.
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with(numtheory); P:=proc(q) local n, t; t:=0;
for n from 1 to q do t:=t+(n-tau(n))^n; if t mod n=0 then print(n);
fi; od; end: P(10^6);
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isok(n) = sum(i=1, n, Mod(i-numdiv(i), n)^i) == 0; \\ Michel Marcus, Feb 25 2016
A229208
Numbers k such that Sum_{j=1..k} sigma(j)^j == 0 (mod k).
Original entry on oeis.org
1, 2, 9, 55, 758, 16685, 29224, 84293, 87018, 98122
Offset: 1
sigma(1)^1 + sigma(2)^2 + ... + sigma(9)^9 = 13172483385 and 13172483385 / 9 = 1463609265.
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with(numtheory); P:=proc(q) local n, t; t:=0;
for n from 1 to q do t:=t+sigma(n)^n; if t mod n=0 then print(n);
fi; od; end: P(10^6);
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Module[{nn=100000},Select[Thread[{Accumulate[Table[DivisorSigma[1,n]^n,{n,nn}]],Range[nn]}],Divisible[#[[1]],#[[2]]]&]][[All,2]] (* Harvey P. Dale, Dec 06 2018 *)
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lista(nn) = {v = vector(nn, i, sigma(i)); for (n=1, nn, if (! sum(i=1, n, Mod(v[i], n)^i), print1(n, ", ");););} \\ Michel Marcus, Sep 21 2013
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 1-8 of 8 results.
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