A085629 -id:A085629 - cp's OEIS frontend

A353699 Heinz numbers of integer partitions whose product equals their length.

2, 6, 20, 36, 56, 176, 240, 416, 864, 1088, 1344, 2432, 3200, 5888, 8448, 14848, 23040, 31744, 35840, 39936, 75776, 167936, 208896, 331776, 352256, 450560, 516096, 770048, 802816, 933888, 1736704, 2457600, 3866624, 4259840, 4521984, 7995392, 12976128, 17563648

Offset: 1

Gus Wiseman, May 19 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      2: {1}
      6: {1,2}
     20: {1,1,3}
     36: {1,1,2,2}
     56: {1,1,1,4}
    176: {1,1,1,1,5}
    240: {1,1,1,1,2,3}
    416: {1,1,1,1,1,6}
    864: {1,1,1,1,1,2,2,2}
   1088: {1,1,1,1,1,1,7}
   1344: {1,1,1,1,1,1,2,4}
   2432: {1,1,1,1,1,1,1,8}
   3200: {1,1,1,1,1,1,1,3,3}
   5888: {1,1,1,1,1,1,1,1,9}
   8448: {1,1,1,1,1,1,1,1,2,5}
  14848: {1,1,1,1,1,1,1,1,1,10}
  23040: {1,1,1,1,1,1,1,1,1,2,2,3}
  31744: {1,1,1,1,1,1,1,1,1,1,11}
  35840: {1,1,1,1,1,1,1,1,1,1,3,4}
  39936: {1,1,1,1,1,1,1,1,1,1,2,6}
  75776: {1,1,1,1,1,1,1,1,1,1,1,12}

Length is A001222, counted by A008284, distinct A001221.
Product is A003963, counted by A339095, firsts A318871.
A similar sequence is A353503, counted by A353506.
These partitions are counted by A353698.
A005361 gives product of signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353394 gives product of shadows of prime indices, firsts A353397.
Cf. A000720, A003586, A114640, A182850, A320325, A325131, A325755, A353399, A353507.

Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==PrimeOmega[#]&]

A354233 Least number with n runs in ordered prime signature.

Original entry on oeis.org

1, 2, 12, 90, 2100, 48510, 3303300, 139369230, 18138420300, 1157182716690, 278261505822300, 30168910606824990, 9894144362523521100, 1693350783450479863710, 715178436956287675671300, 147157263134197051595990130, 83730945863531292204568790100

Offset: 0

Gus Wiseman, May 20 2022

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The prime indices of 90 are {1,2,2,3}, with multiplicities {1,2,1}, with runs {{1},{2},{1}}, and this is the first case of 3 runs, so a(3) = 90.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Positions of first appearances in A353745.
A001222 counts prime factors with multiplicity, distinct A001221.
A005361 gives product of signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A182850 gives frequency depth of prime indices, counted by A225485.
A323014 gives adjusted frequency depth of prime indices, counted by A325280.
Cf. A005811, A097318, A304678, A325755, A353507, A353742.

Table[Product[Prime[i]^If[EvenQ[n-i],1,2],{i,n}],{n,0,15}]

A361418 a(n) is the least number with exactly n noninfinitary divisors.

Original entry on oeis.org

1, 4, 12, 16, 60, 36, 48, 256, 360, 4096, 180, 144, 240, 576, 768, 65536, 2520, 1048576, 12288, 900, 1260, 1296, 720, 2304, 1680, 9216, 2880, 5184, 3840, 147456, 196608, 36864, 27720, 46656, 3145728, 4398046511104, 61440, 3600, 6300, 18014398509481984, 10080, 20736

Offset: 0

Amiram Eldar, Mar 11 2023

nonn

a(n) is the least number k such that A348341(k) = n.

Since A348341(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

			a(1) = 4 since 4 is the least number with exactly one noninfinitary divisor, 2.

Amiram Eldar, Table of n, a(n) for n = 0..50

Cf. A025487, A348341, A348342.

Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A130279 (square), A187941 (even), A309181 (non-unitary), A340232 (bi-unitary), A340233 (exponential), A357450 (odd square), A358252 (non-unitary square).

f[1] = 0; f[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]];
seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s];
seq[35, 10^7]

s(n) = {my(f = factor(n)); numdiv(f) - prod(i = 1, #f~, 2^hammingweight(f[i,2]));}
lista(len, nmax) = {my(v = vector(len), c = 0, n = 1, i); while(c < len && n < nmax, i = s(n) + 1; if(i <= len && v[i] == 0, c++; v[i] = n); n++); v};

A372505 a(n) = log_2(A368473(n)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 04 2024

The first position of k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005361, A085629, A138302, A368473.

Cf. A369934, A370078, A372467, A372469.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]], e}, e = IntegerExponent[p, 2]; If[p == 2^e, e, Nothing]]; Array[f, 150]

lista(kmax) = {my(p, e); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); e = valuation(p, 2); if(p >> e == 1, print1(e, ", ")));}

a(n) = log_2(A005361(A138302(n))).

cp's OEIS Frontend

A353699 Heinz numbers of integer partitions whose product equals their length.

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A354233 Least number with n runs in ordered prime signature.

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A361418 a(n) is the least number with exactly n noninfinitary divisors.

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A372505 a(n) = log_2(A368473(n)).

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