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 1-4 of 4 results.

A372046 Composite numbers that divide the concatenation of the reverse of their ascending order prime factors, with repetition.

Original entry on oeis.org

998, 1636, 9998, 15584, 49447, 99998, 1639964, 2794612, 9999998, 15842836, 1639360636, 1968390098, 27879461212, 65226742928
Offset: 1

Views

Author

Scott R. Shannon, Apr 17 2024

Keywords

Comments

A number 999...9998 will be a term if it has two prime factors 2 and 4999...999. Therefore 999999999999998 and 999...9998 (with 54 9's) are both terms. See A056712.
100000000000 < a(15) <= 999999999999998. Robert P. P. McKone, May 07 2024

Examples

			998 is a term as 998 = 2 * 499 = "2" * "994" when each prime factor is reversed. This gives "2994", and 2994 is divisible by 998.
15584 is a term as 15584 = 2 * 2 * 2 * 2 * 2 * 487 = "2" * "2" * "2" * "2" * "2" * "784" when each prime factor is reversed. This gives "22222784", and 22222784 is divisible by 15584.

Crossrefs

Cf. A371696, A371695, A371641, A027746, A056712.

Programs

  • Mathematica
    a[n_Integer] := Module[{f}, f = Flatten[ConstantArray @@@ FactorInteger[n]]; If[Length[f] < 2, Return[False]]; Mod[FromDigits[StringJoin[StringReverse[IntegerString[#, 10]] & /@ f], 10], n] == 0];
Select[Range[2, 10^5], a] (* Robert P. P. McKone, May 03 2024 *)
  • Python
    from itertools import count, islice
from sympy import factorint
def A372046_gen(startvalue=4): # generator of terms >= startvalue
    for n in count(max(startvalue,4)):
        f = factorint(n)
        if sum(f.values()) > 1:
            c = 0
            for p in sorted(f):
                a = pow(10,len(s:=str(p)),n)
                q = int(s[::-1])
                for _ in range(f[p]):
                    c = (c*a+q)%n
            if not c:
                yield n
A372046_list = list(islice(A372046_gen(),5)) # Chai Wah Wu, Apr 24 2024

Extensions

a(13)-a(14) from Robert P. P. McKone, May 05 2024

A378893 Numbers that are a proper substring of the concatenation (with repetition) in increasing order of their prime factors.

Original entry on oeis.org

333, 22564, 113113, 210526, 252310, 1143241, 3331233, 3710027, 31373137, 217893044, 433100023, 2263178956
Offset: 1

Views

Author

Scott R. Shannon, Dec 10 2024

Keywords

Examples

			333 is a term as 333 = 3 * 3 * 37 = "3337" when concatenated, which contains "333" as a substring.
113113 is a term as 113113 = 7 * 11 * 13 * 113 = "71113113" when concatenated, which contains "113113" as a substring.
433100023 is a term as 433100023 = 433 * 1000231 = "4331000231" when concatenated, which contains "433100023" as a substring.

Crossrefs

Cf. A027746, A378894, A376078, A372309, A371958, A371696, A372046, A371641.

A378894 Numbers, when written in binary, that are a proper substring of the concatenation (with repetition) in increasing order of their prime factors, when written in binary.

Original entry on oeis.org

10, 57, 63, 355, 737, 921, 1526, 13803, 22008, 43364, 44016, 48895, 65151, 88032, 130545, 235929, 255987, 563207, 702460, 1456355, 2799617, 3020897, 3137557, 3774873, 4163463, 5697350, 5995862, 14176747, 42172441, 55933611, 87559273, 93206755, 108530173, 126474397, 180677710, 193337441, 249550095, 259779663, 533713761, 536378647, 715881440, 940339099, 1000732491
Offset: 1

Views

Author

Scott R. Shannon, Dec 10 2024

Keywords

Examples

			10 is a term as 10 = 1010_2 = 2 * 5 = 10_2 * 101_2 = "10101" when concatenated, which contains "1010" as a substring.
63 is a term as 63 = 111111_2 = 3 * 3 * 7 = 11_2 * 11_2 * 111_2 = "1111111" when concatenated, which contains "111111" as a substring.
1526 is a term as 1526 = 10111110110_2 = 2 * 7 * 109 = 10_2 * 111_2 * 1101101_2 = "101111101101" when concatenated, which contains "10111110110" as a substring.

Crossrefs

Cf. A027746, A378893, A376078, A372309, A371958, A371696, A372046, A371641.

A372277 Composite numbers that divide the concatenation of the reverse of their ascending order prime factors, with repetition, when written in binary.

Original entry on oeis.org

87339, 332403, 9813039
Offset: 1

Views

Author

Scott R. Shannon, Apr 25 2024

Keywords

Comments

The base-2 version of A372046.
a(4) > 120000000000. - Robert P. P. McKone, May 20 2024

Examples

			332403 is a term as 332403 = 3 * 179 * 619 = 11_2 * 10110011_2 * 1001101011_2 = "11"_2 * "11001101"_2 * "1101011001"_2 when each prime factor is reversed. This gives "11110011011101011001"_2 when concatenated, and 11110011011101011001_2 = 997209 which is divisible by 332403.

Crossrefs

Cf. A372046, A371821, A371696, A027746.

Programs

  • Mathematica
    a[n_Integer] := Module[{f}, f = Flatten[ConstantArray @@@ FactorInteger[n]]; If[Length[f] < 2, Return[False]]; Mod[FromDigits[StringJoin[StringReverse[IntegerString[#, 2]] & /@ f], 2], n] == 0];
Select[Range[2, 10^5], a] (* Robert P. P. McKone, May 02 2024 *)
  • Python
    from itertools import count, islice
from sympy import factorint
def A372277_gen(startvalue=4): # generator of terms >= startvalue
    for n in count(max(startvalue,4)):
        f = factorint(n)
        if sum(f.values()) > 1:
            c = 0
            for p in sorted(f):
                a = pow(2,len(s:=bin(p)[2:]),n)
                q = int(s[::-1],2)
                for _ in range(f[p]):
                    c = (c*a+q)%n
            if not c:
                yield n
A372277_list = list(islice(A372277_gen(),3)) # Chai Wah Wu, Apr 25 2024
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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