cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A349865 Composite numbers that are missing from A349278.

Original entry on oeis.org

25, 26, 34, 38, 39, 46, 49, 51, 57, 58, 62, 68, 69, 74, 75, 76, 82, 85, 86, 87, 92, 93, 94, 95, 102, 106, 111, 114, 115, 116, 118, 119, 121, 122, 123, 124, 125, 129, 133, 134, 138, 141, 142, 143, 145, 146, 147, 148, 152, 155, 158, 159, 161, 164, 166, 169, 171, 172, 174, 177, 178
Offset: 1

Views

Author

Bernard Schott, Dec 03 2021

Keywords

Comments

Since no prime >= 11 is a term in A349278, only composite numbers are listed here.

Examples

			There does not exist an integer d.u, where . stands for concatenation, such that 26 = u*(u+d), so 26 is a term.
As 28 = A349278(34) = 4*(4+3), 28 is not a term.

Crossrefs

Cf. A349278, A349864.
Equals disjoint union of A349733 and A350061.

Extensions

More terms from Michel Marcus, Dec 04 2021

A349279 Fixed points of A349278.

Original entry on oeis.org

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

Views

Author

Michel Marcus, Nov 13 2021

Keywords

Comments

This is similar to A349190 but with digits taken in reversed order.
If it exists, a(13) > 10^18. - Max Alekseyev, Jan 19 2025

Examples

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

Crossrefs

Cf. A349190, A349278.

Programs

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

A349864 Smallest k such that A349278(k) = n, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 0, 13, 0, 52, 23, 62, 0, 33, 0, 14, 43, 92, 0, 24, 0, 0, 63, 34, 0, 15, 0, 44, 83, 0, 25, 54, 0, 0, 0, 35, 0, 16, 0, 74, 45, 0, 0, 26, 0, 55, 0, 94, 0, 36, 65, 17, 0, 0, 0, 46, 0, 0, 27, 422, 85, 56, 0, 0, 0, 37, 0, 18, 0, 0, 0, 0, 47, 76, 0, 28, 603, 0, 0, 57, 0, 0, 0, 38, 0, 19
Offset: 1

Views

Author

Bernard Schott, Dec 03 2021

Keywords

Comments

Composite numbers m for which a(m) = 0 are in A349865.

Examples

			a(10) = 32 since 2*(2+3) = 10 and no integer du < 32 gives u*(u+d) = 32.

Crossrefs

Cf. A349278, A349865.
Cf. A349732 (similar, with first digits).

Formula

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

Extensions

a(64) and a(81) corrected by Michel Marcus, Dec 03 2021.

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

Views

Author

Bernard Schott, Dec 12 2021

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A349194, A349278.
Equals A349865 \ A349733.

Extensions

a(6)-a(50) from Michel Marcus, Dec 12 2021

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.

Original entry on oeis.org

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

Views

Author

Malo David, Nov 10 2021

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

  • Magma
    f:=func; [f(n):n in [1..100]]; // Marius A. Burtea, Nov 23 2021
  • Mathematica
    Table[Product[Sum[Part[IntegerDigits[n],j],{j,i}],{i,Length[IntegerDigits[n]]}],{n,74}] (* Stefano Spezia, Nov 10 2021 *)
  • PARI
    a(n) = my(d=digits(n)); prod(i=1, #d, sum(j=1, i, d[j])); \\ Michel Marcus, Nov 10 2021
  • PARI
    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
  • Python
    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

Formula

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)
Showing 1-5 of 5 results.

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