A321144 -id:A321144 - cp's OEIS frontend

A357879 Number of divisors of n with the same sum of prime indices as their quotient. Central column of A321144, taking gaps as 0's.

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Oct 27 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(3600) = 5 divisors, their prime indices, and the prime indices of their quotients:
  45: {2,2,3} * {1,1,1,1,3}
  50: {1,3,3} * {1,1,1,2,2}
  60: {1,1,2,3} * {1,1,2,3}
  72: {1,1,1,2,2} * {1,3,3}
  80: {1,1,1,1,3} * {2,2,3}

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of nonzero terms are A357976, counted by A002219.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A213074, A235130, A237258, A276107, A300061, A321144, A357975.

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Table[Length[Select[Divisors[n],sumprix[#]==sumprix[n]/2&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
A357879(n) = sumdiv(n,d, A056239(d)==A056239(n/d)); \\ Antti Karttunen, Jan 20 2025

a(n) = Sum_{d|n} [A056239(d) = A056239(n/d)], where [ ] is the Iverson bracket. - Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

A357976 Numbers with a divisor having the same sum of prime indices as their quotient.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 144, 154, 160, 165, 169, 192, 196, 198, 210, 220, 225, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 351, 352, 360, 361, 364, 390, 400, 432, 441, 442, 448

Offset: 1

Gus Wiseman, Oct 26 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
   1: {}
   4: {1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  25: {3,3}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
For example, 40 has factorization 8*5, and both factors have the same sum of prime indices 3, so 40 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

The partitions with these Heinz numbers are counted by A002219.
A subset of A300061.
The squarefree case is A357854, counted by A237258.
Positions of nonzero terms in A357879.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A213086, A235130, A237194, A276107, A300273, A321144, A357975.

filter:= proc(n) local F,s,t,i,R;
  F:= ifactors(n)[2];
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], F);
  s:= add(t[1]*t[2],t=F)/2;
  if not s::integer then return false fi;
  try
  R:= Optimization:-Maximize(0, [add(F[i][1]*x[i],i=1..nops(F)) = s, seq(x[i]<= F[i][2],i=1..nops(F))], assume=nonnegint, depthlimit=20);
  catch "no feasible integer point found; use feasibilitytolerance option to adjust tolerance": return false;
  end try;
  true
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Oct 26 2023

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Select[Range[100],MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]

A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

Original entry on oeis.org

1, 30, 70, 154, 165, 210, 273, 286, 390, 442, 462, 561, 595, 646, 714, 741, 858, 874, 910, 1045, 1155, 1173, 1254, 1326, 1330, 1334, 1495, 1653, 1771, 1794, 1798, 1870, 1938, 2139, 2145, 2294, 2415, 2465, 2470, 2530, 2622, 2639, 2730, 2926, 2945, 2958, 3034

Offset: 1

Gus Wiseman, Oct 27 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     1: {}
    30: {1,2,3}
    70: {1,3,4}
   154: {1,4,5}
   165: {2,3,5}
   210: {1,2,3,4}
   273: {2,4,6}
   286: {1,5,6}
   390: {1,2,3,6}
For example, 210 has factorization 14*15, and both factors have the same sum of prime indices 5, so 210 is in the sequence.

The partitions with these Heinz numbers are counted by A237258.
A subset of A319241, squarefree case of A300061.
Squarefree positions of nonzero terms in A357879.
This is the squarefree case of A357976, counted by A002219.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A235130, A237194, A276107, A300273, A321144, A357975, A357976.

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Select[Range[1000],SquareFreeQ[#]&&MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]

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A357879 Number of divisors of n with the same sum of prime indices as their quotient. Central column of A321144, taking gaps as 0's.

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A357976 Numbers with a divisor having the same sum of prime indices as their quotient.

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A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

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