A349194 -id:A349194 - cp's OEIS frontend

A349733 Composite numbers that are missing from A349194.

26, 34, 38, 39, 46, 51, 57, 58, 62, 68, 69, 74, 76, 82, 85, 86, 87, 92, 93, 94, 95, 102, 106, 111, 114, 115, 116, 118, 119, 121, 122, 123, 124, 129, 133, 134, 138, 141, 142, 143, 145, 146, 148, 152, 155, 158, 159, 161, 164, 166, 169, 171, 172, 174, 177, 178

Offset: 1

Bernard Schott, Nov 28 2021

Since no prime >= 11 is a term in A349194, only composite numbers are listed here.

			There does not exist an integer 'du' such that 26 = d*(d+u), so 26 is a term.

Rémy Sigrist, PARI program for A349733

Cf. A349194, A349732.

Equals A349865 \ A350061.

PARI
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More terms from Rémy Sigrist, Nov 28 2021

A349732 Smallest k such that A349194(k) = n, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 0, 24, 0, 25, 32, 26, 0, 27, 0, 28, 34, 29, 0, 35, 50, 0, 36, 43, 0, 37, 0, 44, 38, 0, 52, 39, 0, 0, 0, 46, 0, 61, 0, 47, 54, 0, 0, 48, 70, 55, 0, 49, 0, 63, 56, 71, 0, 0, 0, 57, 0, 0, 72, 80, 58, 65, 0, 0, 0, 59, 0, 66, 0, 0, 0, 0, 74, 67, 0, 82, 90

Offset: 1

Bernard Schott, Nov 28 2021

Composite numbers m for which a(m) = 0 are in A349733.

			a(10) = 19 since 1*(1+9) = 10 and no integer du < 19 gives d*(d+u) = 10.

Cf. A349194, A349733.

If p prime >= 11, a(p) = 0.

A349898 Numbers that appear in A349194.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 104, 105, 108, 110, 112, 117, 120

Offset: 1

Charles R Greathouse IV, Dec 04 2021

Cf. A349194, A349899.

is(n)=if(n<11,return(n>0)); my(v=apply(k->[k,k],select(k->n%k==0,[2..10]))); while(#v, my(u=List()); for(i=1,#v, my(sd=v[i][1],a=v[i][2]); for(k=0,9, my(nsd=sd+k,t=nsd*a); if(n%t==0, if(n==t, return(1)); listput(u,[nsd,t])))); v=Set(u)); 0

A350061 Numbers k for which there exists a preimage m_1 such that A349194(m_1) = k but there is no preimage m_2 such that A349278(m_2) = k.

Original entry on oeis.org

25, 49, 75, 125, 147, 242, 245, 343, 363, 375, 484, 605, 625, 676, 726, 845, 847, 968, 1014, 1029, 1089, 1183, 1210, 1225, 1352, 1452, 1521, 1690, 1694, 1715, 1815, 1875, 1936, 2028, 2178, 2312, 2366, 2401, 2420, 2535, 2541, 2601, 2662, 2704, 2890, 3025, 3042, 3125, 3267, 3380

Offset: 1

Bernard Schott, Dec 12 2021

Numbers that can be expressed as the product of the sum of the first i digits of k, as i goes from 1 to the total number of digits of k for some k, but not as the product of the sum of the last i digits of m, with i going from 1 to the total number of digits of m, for any m.
The preimages m_1 are necessarily multiples of 10; the first few are 50, 70, 320, 500, 340, ...
As A349733 is a subsequence of A349865, there are no numbers t for which there exists a preimage m_4 such that A349278(m_4) = t but there is no preimage m_3 such that A349194(m_3) = t.

			A349194(122) = 1*(1+2)*(1+2+2) = 15 and A349278(23) = 3*(3+2) = 15, hence, 15 is not a term.
A349194(50) = 5*(5+0) = 25 but there is no m_2 such that A349278(m_2) = 25, because 25 = A349865(1), hence 25 is a term.
A349194(340) = 3*(3+4)*(3+4+0) = 147 but there is no m_2 such that A349278(m_2) = 340, because 147 = A349865(47), hence 147 is a term.

Cf. A349194, A349278.

Equals A349865 \ A349733.

a(6)-a(50) from Michel Marcus, Dec 12 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;);}

A349278 a(n) is the product of the sum of the last i digits of n, with i going from 1 to the total number of digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 0, 4, 10, 18, 28, 40, 54, 70, 88, 108, 0, 5, 12, 21, 32, 45, 60, 77, 96, 117, 0, 6, 14, 24, 36, 50, 66, 84, 104, 126, 0, 7, 16, 27, 40, 55, 72, 91, 112, 135, 0

Offset: 1

Michel Marcus, Nov 13 2021

This is similar to A349194 but with digits taken in reversed order.
The only primes in the sequence are 2, 3, 5 and 7. - Bernard Schott, Dec 04 2021
The positive terms form a subsequence of A349194. - Bernard Schott, Dec 19 2021

			For n=256, a(256) = 6*(6+5)*(6+5+2) = 858.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A349194, A349279 (fixed points).

Cf. A349864, A349865.

a[n_] := Times @@ Accumulate @ Reverse @ IntegerDigits[n]; Array[a, 70] (* Amiram Eldar, Nov 13 2021 *)

a(n) = my(d=Vecrev(digits(n))); prod(i=1, #d, sum(j=1, i, d[j]));

from math import prod
from itertools import accumulate
def a(n): return 0 if n%10==0 else prod(accumulate(map(int, str(n)[::-1])))
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Nov 13 2021

From Bernard Schott, Dec 04 2021: (Start)
a(n) = 0 iff n is a multiple of 10 (A008592).
a(n) = 1 iff n = 1.
a(n) = 2 (resp. 3, 4, 5, 7, 9) iff n = 10^k+1 (A000533) (resp. 2*10^k+1 (A199682), 3*10^k+1 (A199683), 4*10^k+1 (A199684), 6*10^k+1 (A199686), 8*10^k+1 (A199689)).
a(R_n) = n! where R_n = A002275(n) is repunit > 0, and n! = A000142(n).
a(n) = A349194(n) if n is palindrome (A002113). (End)

A349190 Numbers k such that k equals the product of the sum of its first i digits, with i going from 1 to the total number of digits of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 48, 24192

Offset: 1

Malo David, Nov 09 2021

a(12) > 10^22 if it exists. - Max Alekseyev, Jan 19 2025

			24192 is a term since 24192 = 2*(2+4)*(2+4+1)*(2+4+1+9)*(2+4+1+9+2).

Cf. A055642, A349194, A349279.

Select[Range[10^5],Times@@Total/@Table[IntegerDigits[#][[;;k]],{k,IntegerLength@#}]==#&] (* Giorgos Kalogeropoulos, Nov 10 2021 *)

isok(k) = {my(d=digits(k)); prod(i=1, #d, sum(j=1, i, d[j])) == k;} \\ Michel Marcus, Nov 10 2021

def main(N): # prints all terms <= N
  for k in range(1,N+1):
    n1=str(k)
    n2 = 1
    for i in range(1,len(n1)+1):
      sum1 = 0
      for j in range(0,i):
        sum1 += int(n1[j])
      n2 = n2*sum1
    if n2 == k:
       print(k, end=", ")

from itertools import islice, accumulate, count
from math import prod
def A349190gen(): return filter(lambda n:prod(accumulate(int(d) for d in str(n))) == n,count(1)) # generator of terms
A349190_list = list(islice(A349190gen(),11)) # Chai Wah Wu, Dec 02 2021

A349899 Least number in A349898 divisible by the n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 22, 52, 136, 190, 1610, 12760, 35464, 196840, 2112320, 4093600, 22789360, 288608320

Offset: 1

Charles R Greathouse IV, Dec 04 2021

Least value of A349194(k) divisible by the n-th prime for some k.

Does a(n) exist for each n? Is the sequence increasing?

Cf. A349194, A349898.

smooth(P,lim)=my(v=List([1]), nxt=vector(#P,i,1), indx, t); while(1, t=vecmin(vector(#P, i, v[nxt[i]]*P[i]), &indx); if(t>lim, break); if(t>v[#v], listput(v, t)); nxt[indx]++); Vec(v);
has(n)=my(v=apply(k->[k,k],select(k->n%k==0,[2..10]))); while(#v, my(u=List()); for(i=1,#v, my(sd=v[i][1],a=v[i][2]); for(k=0,9, my(nsd=sd+k,t=nsd*a); if(n%t==0, if(n==t, return(1)); listput(u,[nsd,t])))); v=Set(u)); 0
a(n)=if(n<5,return(prime(n))); my(P=primes(n+1),p=P[n],L=p,v=smooth(P,L),x); while(1, for(i=x+1,#v, if(has(p*v[i]), return(p*v[i]))); x=#v; v=smooth(P,L*=2))
\\ This program is heuristic and fails if a(n) is divisible by a prime >= prime(n+2). If this sequence is strictly increasing, "primes(n+1)" can be replaced with "primes(n)" for better speed and memory use.

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A349733 Composite numbers that are missing from A349194.

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A349732 Smallest k such that A349194(k) = n, or 0 if no such k exists.

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A349898 Numbers that appear in A349194.

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A350061 Numbers k for which there exists a preimage m_1 such that A349194(m_1) = k but there is no preimage m_2 such that A349278(m_2) = k.

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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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A349278 a(n) is the product of the sum of the last i digits of n, with i going from 1 to the total number of digits of n.

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A349190 Numbers k such that k equals the product of the sum of its first i digits, with i going from 1 to the total number of digits of k.

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A349899 Least number in A349898 divisible by the n-th prime.

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