A067175 -id:A067175 - cp's OEIS frontend

A387173 First differences of A067175.

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2

Offset: 0

Paolo Xausa, Aug 20 2025

From n = 2, run lengths of A385863.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A067175, A385863.

Differences[Join[{1}, IntegerLength[FoldList[Times, Prime[Range[100]]]]]]

A383677 Irregular triangle read by rows: T(n,k), 2 <= n , 3 <= k <= largest k such that A067175(k) <= n , is the smallest n-digit number m such that omega(m) = A001221(m) = k, and its largest prime factor equals the sum of its remaining prime factors. or -1 if no such number exists.

Original entry on oeis.org

30, 120, -1, 1080, 3135, 3570, 10336, 10695, 10626, -1, 100672, 102695, 103730, 844305, -1, 1003520, 1005039, 1003450, 1218945, 1231230, -1, 10017286, 10000295, 10003390, 10064145, 10314150, -1, 100216924, 100019275, 100017216, 100367745, 100327920, 463798335, -1

Offset: 2

Jean-Marc Rebert, May 11 2025

For n ranging from 2 to 20, the corresponding maximum values of k are as follows: [3, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16].

			T(4,3) = 1080 is the smallest 4-digit number having 3 distinct prime factors (namely 2, 3, and 5) such that the largest one is the sum of the others (2 + 3 = 5).
T(5,4) = 10695 is the smallest 5-digit number having 4 distinct prime factors (namely 3, 5, 23 and 31) such that the largest one is the sum of the others (3 + 5 + 23 = 31).
Triangle begins:
       30;
      120,      -1;
     1080,    3135,    3570;
    10336,   10695,   10626,      -1;
   100672,  102695,  103730,  844305,      -1;
  1003520, 1005039, 1003450, 1218945, 1231230, -1;
  ...

Cf. A001221, A002110, A067175, A365795, A382469, A383725, A383726, A383728, A383729.

A079099 Number of 0's in n primorial or 0's in A002110(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 1, 2, 1, 5, 2, 3, 4, 1, 3, 4, 3, 5, 4, 5, 4, 3, 5, 4, 8, 2, 3, 3, 6, 6, 7, 12, 12, 6, 6, 5, 9, 7, 8, 7, 7, 10, 9, 5, 9, 7, 9, 10, 17, 12, 11, 16, 16, 13, 10, 10, 14, 11, 14, 9, 16, 11, 14, 8, 13, 16, 14, 4, 15, 15, 17, 9, 19, 15, 17, 17, 22, 15, 12, 19, 14, 18, 11, 21, 19, 18, 19, 22, 15

Offset: 0

Cino Hilliard, Feb 03 2003

The sum of the reciprocals appear to converge.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A002110, A067175.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
a:= n-> nops([StringTools[SearchAll]("0", ""||(p(n)))]):
seq(a(n), n=0..90);  # Alois P. Heinz, Jul 21 2025

DigitCount[#,10,0]&/@FoldList[Times,Prime[Range[90]]] (* Harvey P. Dale, Mar 05 2022 *)

a(n,dg=0) = { my(y=vecprod(primes(n))); #select(x->x==dg, digits(y)) }

Offset corrected by Alois P. Heinz, Jul 21 2025

A384502 Maximum number of distinct prime factors in an n-digit number, n > 3, where its set of distinct prime factors can be partitioned into two equal-sum subsets, each containing at least two elements.

Original entry on oeis.org

5, 5, 7, 7, 7, 9, 9, 9, 11, 11, 11, 13, 13, 13, 15, 15, 15, 16, 17, 17, 17, 19, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 23, 25, 25, 25, 25, 27, 27, 27, 27, 29, 29, 29, 29, 31, 31, 31, 31, 33, 33, 33, 33, 34, 35, 35, 35, 35, 37, 37, 37, 37, 39, 39, 39, 39, 39, 41

Offset: 4

Jean-Marc Rebert, May 31 2025

			a(4) = 5, since 2310 = 2 * 3 * 5 * 7 * 11 is a 4-digit number with omega(2310) = 5, and its prime factors can be split into two equal-sum parts: 2 + 5 + 7 = 3 + 11. No 4-digit number that meets this partitioning criterion has an omega value exceeding 5.

David A. Corneth, Table of n, a(n) for n = 4..704

Cf. A067175, A001221, A221054, A383858.

a(n) <= (largest m such that A067175(m) <= n).

a(11)-a(59) from Sean A. Irvine, Jun 23 2025

More terms from David A. Corneth, Aug 15 2025

A385863 a(n) is the largest number of distinct prime factors a number with at most n digits can have.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 39, 40

Offset: 1

David A. Corneth, Aug 20 2025

Also the largest k such that primorial(k) < 10^n.
"at most" in name could also be "exactly" and it gives the same data.
a(n) is the number of distinct prime factors of A091800(n).

			a(5) = 6 as primorial(6) = 30030 < 10^5 < 510510 = primorial(6 + 1) = primorial(7).

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001221, A002110, A002283, A067175, A091800, A387173.

Table[First[FirstPosition[#, ?(# > n &)]] - 1, {n, Last[#] - 1}] & [IntegerLength[FoldList[Times, Prime[Range[50]]]]] (* _Paolo Xausa, Aug 20 2025 *)

a(n) = my(ulim=10^n-1, pp=1, t=0); forprime(p=2, oo, pp*=p; if(pp > ulim, return(t)); t++)

a(n) = A001221(A091800(n)).

A048856 Number of digits of prime(n)# + 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 56, 58, 60, 62, 64, 66, 69, 71, 73, 76, 78, 80, 82, 85, 87, 89, 92, 94, 97, 99, 101, 104, 106, 109, 111, 113, 116, 118, 121, 123, 126, 128

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com)

H. Ibstedt, A Few Smarandache Integer Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, pp. 171-183.

Essentially the same as A067175.

Cf. A002110.

prod = 1; Table[prod = prod*Prime[n]; Length[IntegerDigits[prod + 1]], {n, 100}] (* T. D. Noe, May 05 2013 *)
IntegerLength/@(FoldList[Times,Prime[Range[70]]]+1) (* Harvey P. Dale, Nov 17 2020 *)

Corrected and extended by Arkadiusz Wesolowski, May 04 2013

cp's OEIS Frontend

A387173 First differences of A067175.

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A383677 Irregular triangle read by rows: T(n,k), 2 <= n , 3 <= k <= largest k such that A067175(k) <= n , is the smallest n-digit number m such that omega(m) = A001221(m) = k, and its largest prime factor equals the sum of its remaining prime factors. or -1 if no such number exists.

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A079099 Number of 0's in n primorial or 0's in A002110(n).

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A384502 Maximum number of distinct prime factors in an n-digit number, n > 3, where its set of distinct prime factors can be partitioned into two equal-sum subsets, each containing at least two elements.

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A385863 a(n) is the largest number of distinct prime factors a number with at most n digits can have.

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A048856 Number of digits of prime(n)# + 1.

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