cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Views

Author

Paolo P. Lava, Mar 03 2025

Keywords

Links

Crossrefs

Cf. A380888, A380889, A380900, A380923 - A380928.

Programs

  • Maple
    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);

Extensions

a(22) - a(28) from Giovanni Resta, Mar 03 2025

A380889 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

8, 81, 90, 100, 132, 1125, 1250, 1323, 1470, 1485, 1650, 2156, 2178, 2420, 2898, 3220, 6348, 6612, 12948, 15625, 18375, 20625, 21609, 24010, 24255, 26950, 27225, 30250, 35574, 35937, 36225, 39930, 40250, 47334, 47817, 53130, 58564, 71415, 74385, 77924, 79350
Offset: 1

Views

Author

Paolo P. Lava, Feb 07 2025

Keywords

Comments

Are there other squarefree integers besides 53130?

Examples

			53130 = 2*3*5*7*11*23 = 53130/(2+1) + 53130/(3+1) + 53130/(5+1) + 53130/(7+1) + 53130/(11+1) + 53130/(23+1);
124722 = 2*3^2*13^2*41 = 124722/(2+1) + 124722*2/(3+1) + 124722*2/(13+1) + 124722/(41+1).

Crossrefs

Cf. A380888, A380900, A380901, A380923-A380928.

Programs

  • Maple
    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(79350,1);

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

Views

Author

Paolo P. Lava, Feb 09 2025

Keywords

Examples

			29155 = 5*7^3*17 = 29155/(5-2) + 29155*3/(7-2) + 29155/(17-2)

Crossrefs

Cf. A380888, A380889, A380901, A380923-A380928.

Programs

  • Maple
    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

Views

Author

Paolo P. Lava, Mar 03 2025

Keywords

Examples

			138985 = 5*7*11*19^2 = 138985/(5-3) +138985/(7-3) +138985/(11-3) +138985*2/(19-3)

Crossrefs

Cf. A380888, A380889, A380900, A380901, A380924 - A380928.

Programs

  • Maple
    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

Views

Author

Paolo P. Lava, Mar 04 2025

Keywords

Examples

			648432 = 2^4*3^3*19*79 = 648432*4/(2+5) + 648432*3/(3+5) + 648432/(19+5) + 648432/(79+5).

Crossrefs

Cf. A380888, A380889, A380900, A380901, A380923, A380924, A380925, A380926, A380927.

Programs

  • Maple
    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);
  • Mathematica
    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

Views

Author

Paolo P. Lava, Mar 03 2025

Keywords

Examples

			562275 = 3^3*5^2*7^2*17 = 562275*3/(3+3) + 562275*2/(5+3) + 562275*2/(7+3) + 562275/(17+3)

Crossrefs

Cf. A380888, A380889, A380900, A380901, A380923, A380925 - A380928.

Programs

  • Maple
    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

Views

Author

Paolo P. Lava, Mar 03 2025

Keywords

Examples

			337500 = 2^2*3^3*5^5 = 337500*2/(2-4) + 337500*3/(3-4) + 337500*5/(5-4)

Crossrefs

Cf. A380888, A380889, A380900, A380901, A380923, A380924, A380926 - A380928.

Programs

  • Maple
    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

Views

Author

Paolo P. Lava, Mar 03 2025

Keywords

Examples

			514836 = 2^2*3^4*7*227 = 514836*2/(2+4) + 514836*4/(3+4) + 514836/(7+4) + 514836/(227+4)

Crossrefs

Cf. A380888, A380889, A380900, A380901, A380923-A380925, A380927, A380928.

Programs

  • Maple
    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

Views

Author

Paolo P. Lava, Mar 03 2025

Keywords

Examples

			517244 = 2^2*7^3*13*29 = 517244*2/(2-5) +517244*3/(7-5) + 517244/(13-5) + 517244/(29-5)

Crossrefs

Cf. A380888, A380889, A380900, A380901, A380923-A380926, A380928.

Programs

  • Maple
    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);

A381507 Squarefree numbers k such that the sum of 1/(p-1) over the prime divisors p of k is 1.

Original entry on oeis.org

2, 1365, 73815, 6702045, 8788065, 26241285, 32426205, 237539445, 269409855, 445317015, 475231515, 709296105, 1085962395, 1329722835, 1447857915, 2403281595, 3255993615, 5145721455, 5254163355, 5824953435, 6560751435, 7176232455, 7703697855, 8332635255, 8542035645
Offset: 1

Views

Author

Robert Israel, Apr 23 2025

Keywords

Comments

Squarefree terms of A380888.
All terms > 2 are odd.

Examples

			1365 is a term because 1365 = 3 * 5 * 7 * 13 and 1/(3-1) + 1/(5-1) + 1/(7-1) + 1/(13-1) = 1/2 + 1/4 + 1/6 + 1/12 = 1.

Crossrefs

Intersection of A005117 and A380888.

Programs

  • Maple
    filter:= proc(n) local F,t;
   F:=ifactors(n)[2];
   if F[..,2] <> [1$nops(F)] then return false fi;
   add(1/(t-1),t=F[..,1]) = 1
end proc:
select(filter, [2, seq(i,i=1..10^8,2)]);

Extensions

More terms from Giorgos Kalogeropoulos, Apr 27 2025
Showing 1-10 of 10 results.

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