A095411 -id:A095411 - cp's OEIS frontend

A095409 Numbers k such that total number of decimal digits of all distinct prime factors of k is smaller than number of digits of k.

1, 16, 25, 27, 32, 49, 64, 81, 100, 108, 112, 121, 125, 128, 135, 144, 147, 160, 162, 169, 175, 189, 192, 196, 200, 216, 224, 225, 243, 245, 250, 256, 288, 289, 320, 324, 343, 361, 375, 384, 392, 400, 405, 432, 441, 448, 486, 500, 512, 529, 567, 576, 625, 640

Offset: 1

Labos Elemer, Jun 21 2004

			k = 100: prime set = {2,5}, 3 digits and 2 digits of prime factors, so 100 is a term.
k = 147: prime set = {3,7}, 3 digits and 2 digits of prime factors, so 147 is a term.

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

Cf. A055642, A095407, A095408, A095410, A095411.

q[n_] := Total[IntegerLength /@ FactorInteger[n][[;; , 1]]] < IntegerLength[n]; q[1] = True; Select[Range[640], q] (* Amiram Eldar, Mar 25 2025 *)

Solutions to A095407(x) < A055642(x).

A095410 Numbers n such that total number of decimal digits of all distinct prime factors of n equals the number of digits of n.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 28, 29, 31, 35, 36, 37, 40, 41, 43, 45, 47, 48, 50, 53, 54, 56, 59, 61, 63, 67, 71, 72, 73, 75, 79, 80, 83, 89, 96, 97, 98, 101, 103, 104, 105, 106, 107, 109, 111, 113, 115, 116, 117, 118, 119, 120, 122

Offset: 1

Labos Elemer, Jun 21 2004

			n=184, 3 digits,prime set={2,23} also with 3 digits {2,2,3}.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A055642, A095407, A095408, A095409, A095411.

ffi[x_] :=Flatten[FactorInteger[x]] lf[x_] :=Length[FactorInteger[x]] ba[x_] :=Table[Part[ffi[x], 2*j-1], {j, 1, lf[x]}] tdp[x_] :=Flatten[Table[IntegerDigits[Part[ba[x], j]], {j, 1, lf[x]}], 1] pl[x_] :=Length[tdp[x]] nl[x_] :=Length[IntegerDigits[x]] t1=Table[nl[w], {w, 1, 1000}];t2=Table[pl[w], {w, 1, 1000}];t2-t1 Flatten[Position[t2-t1, 0]]
Rest[Select[Range[200],Length[Flatten[IntegerDigits/@Transpose[ FactorInteger[ #]][[1]]]]==IntegerLength[#]&]] (* Harvey P. Dale, Oct 22 2011 *)

Solutions to A095407[x]=A055642[x].

A382364 a(n) is the smallest squarefree number k such that the sum of the digit counts of the prime factors of k equals the sum of n and the digit count of k.

Original entry on oeis.org

6, 66, 858, 72930, 6374082, 643782282, 66309575046

Offset: 1

Jean-Marc Rebert, Mar 24 2025

			a(1) = 6 = 2*3, because the total number of digits in its distinct prime factors (2 and 3) is 2. This equals the sum of n = 1 and the number of digits in 6, which is 1, and no lesser number has this property.
a(2) = 66 = 2*3*11, because the total number of digits in its distinct prime factors (2, 3 and 11) is 4. This equals the sum of n = 2 and the number of digits in 66, which is 2, and no lesser number has this property.
Table begins:
  1 6 = 2 * 3;
  2 66 = 2 * 3 * 11;
  3 858 = 2 * 3 * 11 * 13;
  4 72930 = 2 * 3 * 5 * 11 * 13 * 17;
  5 6374082 = 2 * 3 * 11 * 13 * 17 * 19 * 23;
  6 643782282 = 2 * 3 * 11 * 13 * 17 * 19 * 23 * 101;
  7 66309575046 = 2 * 3 * 11 * 13 * 17 * 19 * 23 * 101 * 103;

Cf. A055642, A095411.

isok(k, n) = if (issquarefree(k), my(f=factor(k)[,1]); sum(i=1, #f, #digits(f[i])) == n+#digits(k));
a(n) = my(k=2); while (!isok(k, n), k++); k; \\ Michel Marcus, Apr 02 2025

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A095409 Numbers k such that total number of decimal digits of all distinct prime factors of k is smaller than number of digits of k.

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A095410 Numbers n such that total number of decimal digits of all distinct prime factors of n equals the number of digits of n.

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Mathematica

Formula

A382364 a(n) is the smallest squarefree number k such that the sum of the digit counts of the prime factors of k equals the sum of n and the digit count of k.

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