A349190 -id:A349190 - cp's OEIS frontend

A349194 a(n) is the product of the sum of the first i digits of n, as i goes from 1 to the total number of digits of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 49, 56, 63, 70, 77

Offset: 1

Malo David, Nov 10 2021

The only primes in the sequence are 2, 3, 5 and 7. - Bernard Schott, Nov 23 2021

			For n=256, a(256) = 2*(2+5)*(2+5+6) = 182.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Malo David, About the Product Sum Digit Sequence

Cf. A055642, A284001 (binary analog), A349190 (fixed points).
Cf. A007953 (sum of digits), A059995 (floor(n/10)).
Cf. A349278 (similar, with the last digits).
Cf. A349898, A349899.

f:=func; [f(n):n in [1..100]]; // Marius A. Burtea, Nov 23 2021

Table[Product[Sum[Part[IntegerDigits[n],j],{j,i}],{i,Length[IntegerDigits[n]]}],{n,74}] (* Stefano Spezia, Nov 10 2021 *)

a(n) = my(d=digits(n)); prod(i=1, #d, sum(j=1, i, d[j])); \\ Michel Marcus, Nov 10 2021

first(n)=if(n<9,return([1..n])); my(v=vector(n)); for(i=1,9,v[i]=i); for(i=10,n, v[i]=sumdigits(i)*v[i\10]); v \\ Charles R Greathouse IV, Dec 04 2021

from math import prod
from itertools import accumulate
def a(n): return prod(accumulate(map(int, str(n))))
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Nov 10 2021

For n>10: a(n) = a(A059995(n))*A007953(n) where A059995(n) = floor(n/10).
In particular, for n<100: a(n) = floor(n/10)*A007953(n)
From Bernard Schott, Nov 23 2021: (Start)
a(n) = 1 iff n = 10^k, k >= 0 (A011557).
a(n) = 2 iff n = 10^k + 1, k >= 0 (A000533 \ {1}).
a(n) = 3 iff n = 10^k + 2, k >= 0 (A133384).
a(n) = 5 iff n = 10^k + 4, k >= 0.
a(n) = 7 iff n = 10^k + 6, k >= 0. (End)
From Marius A. Burtea, Nov 23 2021: (Start)
a(A002275(n)) = n! = A000142(n), n >= 1.
a(A090843(n - 1)) = (2*n - 1)!! = A001147(n), n >= 1.
a(A097166(n)) = (3*n - 2)!!! = A007559(n).
a(A093136(n)) = 2^n = A000079(n).
a(A093138(n)) = 3^n = A000244(n). (End)

A349279 Fixed points of A349278.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 45, 8643024

Offset: 1

Michel Marcus, Nov 13 2021

This is similar to A349190 but with digits taken in reversed order.

If it exists, a(13) > 10^18. - Max Alekseyev, Jan 19 2025

			24 is a term because A349278(24) = 4*(4+2) = 4*6 = 24.

Cf. A349190, A349278.

f[n_] := Times @@ Accumulate @ Reverse @ IntegerDigits[n]; Select[Range[100], f[#] == # &] (* Amiram Eldar, Nov 13 2021 *)

from math import prod
from itertools import accumulate
def ok(n):
  return n == (0 if n%10==0 else prod(accumulate(map(int, str(n)[::-1]))))
print([k for k in range(1, 10**7) if ok(k)]) # Michael S. Branicky, Nov 13 2021

A349251 a(n) is the integer reached after repeated application of the map x->A349194(x) or -1 if this process does not terminate.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 4, 6, 8, 1, 3, 5, 7, 9, 4, 8, 9, 3, 6, 9, 6, 3, 9, 9, 9, 9, 7, 4, 3, 4, 6, 9, 7, 6, 48, 3, 5, 9, 3, 7, 9, 5, 5, 9, 9, 3, 9, 3, 48, 9, 9, 9, 9, 6, 9, 9, 3, 5, 9, 3, 9, 9, 9, 9, 6, 8, 9, 9, 9, 9, 9, 5, 8, 9, 9, 7, 9

Offset: 1

Michel Marcus, Nov 12 2021

Heuristics suggest that numbers n such that a(n) = -1 have density 1 and may be quite dense by 10^10. - Charles R Greathouse IV, Nov 15 2021

			For n=19, A349194(19) = 10 and A349194(10) = 1 and 1 is a fixed point of A349194 (see A349190), so a(19)=1.

Michel Marcus, Table of n, a(n) for n = 1..378

Cf. A349190, A349194.

f[n_] := Times @@ Accumulate @ IntegerDigits[n]; a[n_, itermax_] := Module[{m = FixedPoint[f, n, itermax]}, If[f[m] == m, m, 0]]; itermax = 100; Table[a[k, itermax], {k, 1, 100}] (* returns 0 if the number of iterations exceeds itermax, Amiram Eldar, Nov 12 2021 *)

f(n) = my(d=digits(n)); prod(i=1, #d, sum(j=1, i, d[j])); \\ A349194
a(n) = {my(nb=0); while (1, my(m=f(n)); nb++; if (m==n, return (m)); if (nb > 100, return (0)); n = m;);}

cp's OEIS Frontend

A349194 a(n) is the product of the sum of the first i digits of n, as i goes from 1 to the total number of digits of n.

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A349279 Fixed points of A349278.

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A349251 a(n) is the integer reached after repeated application of the map x->A349194(x) or -1 if this process does not terminate.

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