A357975 -id:A357975 - cp's OEIS frontend

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

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]&]

A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 4, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 4, 4, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 4, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Oct 25 2022

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. This sequence gives the number of ways to choose a strict partition of each prime index of n.

The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - Ivan N. Ianakiev, Oct 27 2022

			The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The non-strict version is A299200.
A horizontal version is A357978, non-strict A357977.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.
Cf. A003586, A059485.

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsQ[PrimePi[p]]^k],{n,100}]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f9(primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Oct 26 2022

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]&]

A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 17, 12, 31, 22, 15, 16, 47, 18, 79, 20, 33, 34, 113, 24, 25, 62, 27, 44, 181, 30, 263, 32, 51, 94, 55, 36, 389, 158, 93, 40, 547, 66, 761, 68, 45, 226, 1049, 48, 121, 50, 141, 124, 1453, 54, 85, 88, 237, 362, 1951, 60, 2659, 526

Offset: 1

Gus Wiseman, Oct 23 2022

In the definition, taking A000041(k) instead of prime(A000041(k)) gives A299200.

			We have 35 = prime(3) * prime(4), so a(35) = prime(A000041(3)) * prime(A000041(4)) = prime(3) * prime(5) = 55.

Applying the same transformation again gives A357979.
The strict version is A357978.
Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PartitionsP],100]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(numbpart(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

A357980 Replace prime(k) with prime(A000720(k)) in the prime factorization of n, assuming prime(0) = 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 6, 1, 7, 4, 7, 3, 6, 5, 7, 2, 9, 5, 8, 3, 7, 6, 11, 1, 10, 7, 9, 4, 11, 7, 10, 3, 13, 6, 13, 5, 12, 7, 13, 2, 9, 9, 14, 5, 13, 8, 15, 3, 14, 7, 17, 6, 17, 11, 12, 1, 15, 10, 19, 7, 14, 9, 19, 4, 19, 11, 18, 7, 15, 10

Offset: 1

Gus Wiseman, Oct 24 2022

In the definition, taking A000720(k) in place of prime(A000720(k)) gives A357984.

			We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(0) * prime(1)^2 * prime(2) = 12.

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The version for p instead of pi is A357977, strict A357978.
The triangular version is A357984.
A000040 lists the prime numbers.
A000720 = PrimePi.
A056239 adds up prime indices, row-sums of A112798.
Cf. A063834, A066207, A076610, A215366, A296150, A357975, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PrimePi],100]

myprime(n) = if (n==0, 1, prime(n));
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = myprime(primepi(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

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.

Original entry on oeis.org

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

A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 8, 4, 6, 5, 8, 7, 6, 6, 16, 11, 8, 13, 12, 6, 10, 19, 16, 9, 14, 8, 12, 29, 12, 37, 32, 10, 22, 9, 16, 47, 26, 14, 24, 61, 12, 79, 20, 12, 38, 103, 32, 9, 18, 22, 28, 131, 16, 15, 24, 26, 58, 163, 24, 199, 74, 12, 64, 21, 20, 251, 44, 38

Offset: 1

Gus Wiseman, Oct 24 2022

In the definition, taking A000009(k) instead of prime(A000009(k)) gives A357982.

			We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(1) * prime(1)^2 * prime(2) = 24.

The non-strict version is A357977.
Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357979, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PartitionsQ],100]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(f9(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 5, 4, 11, 10, 23, 8, 25, 22, 31, 20, 47, 46, 55, 16, 59, 50, 103, 44, 115, 62, 97, 40, 121, 94, 125, 92, 137, 110, 127, 32, 155, 118, 253, 100, 197, 206, 235, 88, 179, 230, 233, 124, 275, 194, 257, 80, 529, 242, 295, 188, 419, 250, 341, 184, 515, 274

Offset: 1

Gus Wiseman, Oct 24 2022

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. We define the MTF-transform as shifting a number's prime indices along a function; see the Mathematica program.

			First, we have
- 4 = prime(1) * prime(1),
- A000040(1) = 2,
- A064988(4) = prime(2) * prime(2) = 9.
Similarly, A064988(3) = 5. Next,
- 35 = prime(3) * prime(4),
- A064988(3) = 5,
- A064988(4) = 9,
- a(35) = prime(5) * prime(9) = 253.

Other multiplicative sequences: A003961, A357852, A064989, A357977, A357980.
Applying the transformation only once gives A064988.
The union is A076610 (numbers whose prime indices are themselves prime).
For partition numbers instead of primes we have A357979.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A003964, A063834, A215366, A296150, A299201, A299202, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[mtf[Prime]],100]

A357979 Second MTF-transform of A000041. Replace prime(k) with prime(A357977(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 31, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 59, 32, 33, 62, 35, 36, 37, 38, 39, 40, 127, 42, 79, 44, 45, 46, 47, 48, 49, 50, 93, 52, 53, 54, 55, 56, 57, 58, 211, 60, 61, 118, 63, 64, 65, 66

Offset: 1

Gus Wiseman, Oct 24 2022

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. We define the MTF-transform as applying a function horizontally along a number's prime indices; see the Mathematica program.

			We have:
- 51 = prime(2) * prime(7),
- A357977(2) = 2,
- A357977(7) = 11,
- a(51) = prime(2) * prime(11) = 93.

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
Applying the transformation only once gives A357977, strict A357978.
For primes instead of partition numbers we have A357983.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[mtf[PartitionsP]],100]

A357984 Replace prime(k) with A000720(k) in the prime factorization of n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 3, 0, 2, 0, 4, 0, 4, 0, 2, 0, 4, 0, 4, 0, 1, 0, 4, 0, 5, 0, 3, 0, 4, 0, 5, 0, 3, 0, 6, 0, 6, 0, 2, 0, 6, 0, 4, 0, 4, 0, 6, 0, 6, 0, 4, 0, 7, 0, 7, 0, 2, 0, 6, 0, 8, 0, 4, 0, 8, 0, 8, 0, 4, 0, 6, 0, 8, 0, 1, 0, 9, 0, 8, 0, 4

Offset: 1

Gus Wiseman, Oct 25 2022

			We have 91 = prime(4) * prime(6), so a(91) = pi(4) * pi(6) = 6.

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357983.
The version for p instead of pi is A299200, horz A357977, strict A357982.
The version for nu is A355741.
The version for bigomega is A355742.
The horizontal version is A357980.
A000040 lists the prime numbers.
A000720 is PrimePi.
A056239 adds up prime indices, row-sums of A112798.
Cf. A033879, A033880, A063834, A066207, A076610, A296150, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@PrimePi/@primeMS[n],{n,100}]

cp's OEIS Frontend

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

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A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

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

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A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.

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A357980 Replace prime(k) with prime(A000720(k)) in the prime factorization of n, assuming prime(0) = 1.

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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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A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

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A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

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