A380888
Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = -1.
Original entry on oeis.org
2, 9, 75, 625, 1029, 1365, 8575, 11375, 24843, 32955, 73815, 117649, 156065, 207025, 274625, 483153, 599781, 615125, 866481, 1008273, 1252815, 1337505, 1343433, 1553937, 1782105, 1955085, 2061345, 2840383, 3051015, 3432165, 3737085, 3767855, 4026275, 4998175
Offset: 1
73815 = 3*5*7*19*37 = 73815/(3-1) + 73815/(5-1) + 73815/(7-1) + 73815/(19-1) + 73815/(37-1);
599781 = 3*7*13^4 = 599781/(3-1) + 599781/(7-1) + 599781*4/(13-1).
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with(numtheory): P:=proc(q,h) local k,n,v; v:=[];
for n from 1 to q do if n=add(n*k[2]/(k[1]+h),k=ifactors(n)[2]) then v:=[op(v),n]; fi;
od; op(v); end: P(4998175,-1);
A380901
Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = 2.
Original entry on oeis.org
16, 243, 78125, 120393, 166725, 177957, 316953, 792585, 1478925, 40353607, 55883275, 59648043, 77389375, 82602975, 88167807, 106237047, 107171875, 114391875, 122098275, 128153375, 130323843, 147121275, 157032603, 177471875, 189427875, 190142667, 203739375, 217464975
Offset: 1
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with(numtheory): P:=proc(q,h) local k,n,v; v:=[];
for n from 1 to q do if n=add(n*k[2]/(k[1]+h),k=ifactors(n)[2]) then v:=[op(v),n]; fi;
od; op(v); end: P(79350,2);
A380900
Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = -2.
Original entry on oeis.org
3, 125, 16807, 29155, 33275, 50575, 90475, 7761061, 8857805, 11796113, 13463065, 20462645, 21102389, 24084445, 35496425, 36606185, 63500525, 65485805, 73776725, 99798725, 113597825, 117779585, 178056445, 193155305, 200599525, 203878325, 204311525, 251218345
Offset: 1
29155 = 5*7^3*17 = 29155/(5-2) + 29155*3/(7-2) + 29155/(17-2)
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with(numtheory): P:=proc(q,h) local k,n,v; v:=[];
for n from 1 by 2 to q do if n=add(n*k[2]/(k[1]+h),k=ifactors(n)[2]) then v:=[op(v),n]; fi;
od; op(v); end: P(251218345,-2);
A380923
Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = -3.
Original entry on oeis.org
25, 245, 1250, 2401, 4235, 12250, 41503, 62500, 73205, 120050, 136045, 138985, 211750, 215215, 612500, 717409, 1176490, 1333241, 1362053, 1856465, 2075150, 2109107, 2351635, 2402455, 3125000, 3660250, 3720145, 4561235, 5330605, 5535985, 6002500, 6802250, 6949250
Offset: 1
138985 = 5*7*11*19^2 = 138985/(5-3) +138985/(7-3) +138985/(11-3) +138985*2/(19-3)
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with(numtheory): P:=proc(q,h) global k,n,v; v:=[];
for n from 1 to q do if n mod 3>0 then if n=add(n*k[2]/(k[1]+h),k=ifactors(n)[2]) then v:=[op(v),n];
print(n); fi; fi; od; op(v); end: P(6949250,-3);
A380928
Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = 5.
Original entry on oeis.org
128, 6561, 6624, 250047, 252448, 253125, 264627, 267168, 290871, 293664, 342792, 377622, 381248, 557424, 648432, 762696, 841824, 1109052, 2198208, 2374464, 2472384, 5018304, 9529569, 9646875, 9765625, 10085229, 10209375, 10673289, 10775776, 11085417, 11211291
Offset: 1
648432 = 2^4*3^3*19*79 = 648432*4/(2+5) + 648432*3/(3+5) + 648432/(19+5) + 648432/(79+5).
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with(numtheory): P:=proc(q, h) local k, n, v; v:=[];
for n from 1 to q do if n=add(n*k[2]/(k[1]+h), k=ifactors(n)[2]) then v:=[op(v), n];
fi; od; op(v); end: P(11211291, 5);
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s={};j=5;f[{a_,b_}]:=Table[a,b];Do[pf=f/@FactorInteger[k]//Flatten;L=Length[pf];If[Sum[k/(pf[[i]]+j),{i,L}]==k,AppendTo[s,k]],{k,3*10^6}];s (* James C. McMahon, Mar 04 2025 *)
A380924
Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = 3.
Original entry on oeis.org
32, 729, 756, 784, 16875, 17500, 18522, 19208, 22950, 23800, 31212, 32368, 37000, 50320, 243760, 390625, 428750, 453789, 470596, 531250, 562275, 570375, 583100, 591500, 722500, 764694, 775710, 793016, 804440, 874125, 906500, 982600, 1188810, 1232840, 1250600
Offset: 1
562275 = 3^3*5^2*7^2*17 = 562275*3/(3+3) + 562275*2/(5+3) + 562275*2/(7+3) + 562275/(17+3)
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with(numtheory): P:=proc(q, h) local k, n, v; v:=[];
for n from 1 by 2 to q do if n=add(n*k[2]/(k[1]+h), k=ifactors(n)[2]) then v:=[op(v), n]; fi;
od; op(v); end: P(1250600, 3);
A380925
Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = -4.
Original entry on oeis.org
5, 75, 100, 343, 1125, 1500, 2000, 5145, 6860, 16875, 22500, 30000, 40000, 77175, 102900, 107653, 137200, 253125, 337500, 352947, 450000, 470596, 600000, 800000, 1157625, 1543500, 1614795, 2058000, 2153060, 2744000, 3796875, 5062500, 5294205, 6750000, 7058940
Offset: 1
337500 = 2^2*3^3*5^5 = 337500*2/(2-4) + 337500*3/(3-4) + 337500*5/(5-4)
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with(numtheory): P:=proc(q, h) local k, n, v; v:=[];
for n from 1 to q do if n=add(n*k[2]/(k[1]+h), k=ifactors(n)[2]) then v:=[op(v), n];
fi; od; op(v); end: P(7058940, -4);
A380926
Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = 4.
Original entry on oeis.org
64, 2000, 2187, 2448, 62500, 76500, 93636, 110484, 159300, 514836, 1953125, 2390625, 2576816, 2926125, 3452625, 3581577, 4009008, 4226013, 4365680, 4615408, 4730352, 4866800, 4978125, 5581488, 6084477, 6093225, 6810608, 6820400, 7396400, 8047600, 8909109, 9456240
Offset: 1
514836 = 2^2*3^4*7*227 = 514836*2/(2+4) + 514836*4/(3+4) + 514836/(7+4) + 514836/(227+4)
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with(numtheory): P:=proc(q, h) local k, n, v; v:=[];
for n from 1 to q do if n=add(n*k[2]/(k[1]+h), k=ifactors(n)[2]) then v:=[op(v), n];
fi; od; op(v); end: P(9456240, 4);
A380927
Integers k such that k = Sum k/(p_i + j), where p_i are the prime factors of k (with multiplicity). Case j = -5.
Original entry on oeis.org
49, 1029, 9317, 11858, 15092, 19208, 21609, 195657, 199927, 221221, 244783, 249018, 281554, 311542, 316932, 319319, 396508, 403368, 406406, 453789, 517244, 1771561, 2254714, 2869636, 3652264, 4108797, 4198467, 4645641, 4648336, 5140443, 5229378, 5812079, 5912634
Offset: 1
517244 = 2^2*7^3*13*29 = 517244*2/(2-5) +517244*3/(7-5) + 517244/(13-5) + 517244/(29-5)
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with(numtheory): P:=proc(q, h) local k, n, v; v:=[];
for n from 1 to q do if n mod 5>0 then if n=add(n*k[2]/(k[1]+h), k=ifactors(n)[2]) then v:=[op(v), n];
fi; fi; od; op(v); end: P(6949250, -5);
Showing 1-9 of 9 results.
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