David A. Corneth - cp's OEIS frontend

A387100 a(n) is the least number that can be written in exactly n ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t}.

4, 545, 23506, 331979, 5260225, 10630307

Offset: 1

David A. Corneth, Aug 16 2025

			a(5) = 5260225 via
5260225 = 2^22 + 3^8 + 4^5 + 5^2 + 7^7 + 8^3 + 22^4
        = 2^21 + 3^8 + 4^10 + 7^3 + 8^7 + 10^4 + 21^2
        = 2^7 + 3^14 + 4^5 + 5^8 + 6^6 + 7^3 + 8^2 + 14^4
        = 2^15 + 3^10 + 4^9 + 5^5 + 6^4 + 7^6 + 9^7 + 10^3 + 15^2
        = 2^11 + 3^7 + 4^10 + 5^9 + 6^8 + 7^3 + 8^5 + 9^6 + 10^4 + 11^2,
and no positive integer smaller than 5260225 can be written as such in exactly five ways.

Cf. A385969, A386968.

A387099 Numbers that can be written in exactly five ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

Original entry on oeis.org

5260225, 7923882, 11054875, 11224211, 11870046, 15466174, 16859617, 16911017, 17276523, 17326946, 18664520, 18668302, 18908170, 19375153, 19706896, 19854394, 20050965, 20757873, 21468249, 24723272, 26689657, 26925803, 26974782, 27214122, 27336893, 28055974

Offset: 1

David A. Corneth, Aug 16 2025

			5260225 = 2^22 + 3^8 + 4^5 + 5^2 + 7^7 + 8^3 + 22^4
        = 2^21 + 3^8 + 4^10 + 7^3 + 8^7 + 10^4 + 21^2
        = 2^7 + 3^14 + 4^5 + 5^8 + 6^6 + 7^3 + 8^2 + 14^4
        = 2^15 + 3^10 + 4^9 + 5^5 + 6^4 + 7^6 + 9^7 + 10^3 + 15^2
        = 2^11 + 3^7 + 4^10 + 5^9 + 6^8 + 7^3 + 8^5 + 9^6 + 10^4 + 11^2.
7923882 = 2^8 + 3^5 + 4^11 + 5^9 + 6^3 + 8^4 + 9^2 + 11^6
        = 2^12 + 3^9 + 4^5 + 5^6 + 6^2 + 7^8 + 8^7 + 9^3 + 12^4
        = 2^14 + 3^13 + 4^4 + 5^5 + 6^7 + 7^8 + 8^6 + 13^2 + 14^3
        = 2^18 + 3^14 + 4^6 + 5^9 + 6^3 + 7^7 + 9^5 + 14^4 + 18^2
        = 2^19 + 3^14 + 4^8 + 5^9 + 6^6 + 8^2 + 9^4 + 14^5 + 19^3.
11054875 = 2^3 + 3^6 + 4^10 + 5^5 + 6^2 + 7^4 + 10^7
         = 2^15 + 3^8 + 4^6 + 5^2 + 6^9 + 7^7 + 8^3 + 9^5 + 15^4
         = 2^22 + 3^12 + 4^2 + 5^6 + 6^7 + 7^8 + 8^5 + 12^3 + 22^4
         = 2^9 + 3^13 + 4^10 + 5^3 + 6^5 + 7^8 + 8^7 + 9^6 + 10^4 + 13^2
         = 2^11 + 3^12 + 4^8 + 5^7 + 6^9 + 7^6 + 8^3 + 9^2 + 11^5 + 12^4.

David A. Corneth, Table of n, a(n) for n = 1..152

Cf. A385969, A386968.

A387215 a(n) is the smallest k such that, for any m >= k, m is a sum of exactly n distinct primes.

Original entry on oeis.org

18, 31, 42, 61, 84, 103, 138, 163, 204, 245, 294, 335, 390, 449, 516, 575, 648, 725, 804, 885, 978, 1067, 1164, 1277, 1374, 1493, 1608, 1739, 1866, 2003, 2142, 2291, 2436, 2603, 2760, 2933, 3096, 3281, 3468, 3647, 3858, 4055, 4248, 4457, 4684, 4913, 5142, 5375, 5604

Offset: 3

David A. Corneth, Aug 22 2025

In computation it is assumed that if for any m where a(n) = k <= m <= k + 3*n we have m is the sum of n distinct positive integers then a(n) = k.

			a(3) = 18 as 17 is not the sum of 3 distinct primes but any integer m where 18 <= m <= 27 is the sum of 3 distinct primes. It is therefore assumed that a(3) = 18.

Cf. A002375, A229219, A387198.

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)).

A387151 a(n) = n*n! / Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

Original entry on oeis.org

0, 1, 2, 3, 8, 10, 12, 14, 64, 216, 240, 264, 576, 624, 672, 720, 6144, 6528, 20736, 21888, 46080, 48384, 50688, 52992, 221184, 1152000, 1198080, 11197440, 23224320, 24053760, 24883200, 25712640, 424673280, 437944320, 451215360, 464486400, 2866544640

Offset: 0

David A. Corneth and Peter Luschny, Aug 18 2025

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..3167

Cf. A387140.

A387151 := n -> n*n! / mul(NumberTheory:-Radical(k), k = 1..n): seq(A387151(n), n = 0..36);

k = 1; {0}~Join~Reap[Do[k *= Times @@ FactorInteger[n][[;; , 1]]; Sow[n*n!/k], {n, 36}] ][[-1, 1]] (* Michael De Vlieger, Aug 18 2025 *)

a(n) = n! / A387140(n) for n >= 1.

A386521 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 with GCD(x,y,z)=1 has no positive integer solutions.

Original entry on oeis.org

1, 3, 4, 5, 6, 10, 13, 22, 27, 34, 36, 42, 43, 47, 62, 72, 76, 87, 95, 102, 111, 183, 251, 279, 315, 322, 327, 344, 483, 490, 528, 615, 708, 762, 1170, 1302, 2295, 2526, 3282, 3382, 6012

Offset: 1

David A. Corneth and Zhining Yang, Jul 24 2025

a(42) > 6500. - Giovanni Resta, Aug 12 2025

			9 is not a term because 9^5 = x^2 + y^3 + z^4 where GCD(x,y,z)=1 has 5 positive integer solutions: {220,22,1}, {64,38,3}, {241,7,5}, {9,38,8}, {118,29,12}.

Cf. A386373, A386377.

f[w_]:=(c=0;zz=w^5;Do[yy=zz-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,c++]],{y,Floor[yy^(1/3)]}],{z,Floor[zz^(1/4)]}];c);Select[Range@50,f@#==0&]

a(41) from Giovanni Resta, Aug 12 2025

A386377 a(n) is the number of solutions to the equation x^2 + y^3 + z^4 = w^5 where GCD(x, y, z)=1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 2, 2, 5, 0, 1, 1, 0, 1, 1, 1, 3, 2, 1, 2, 2, 0, 2, 2, 4, 1, 0, 2, 2, 1, 2, 1, 13, 0, 2, 0, 1, 3, 1, 1, 4, 0, 0, 7, 5, 3, 0, 2, 10, 1, 1, 2, 7, 2, 1, 1, 8, 1, 2, 1, 7, 0, 4, 3, 8, 4, 4, 1, 1, 5, 1, 0, 11, 1, 2, 0, 3, 1, 3, 5, 12, 7, 2, 2, 2, 2, 0, 1, 14, 2, 2, 1

Offset: 1

David A. Corneth and Zhining Yang, Jul 20 2025

nonn

			a(9) = 5 because x^2 + y^3 + z^4 = 9^5 where GCD(x,y,z)=1 has 5 positive integer solutions :{220,22,1},{64,38,3},{241,7,5},{9,38,8},{118,29,12}.

Zhining Yang, Table of n, a(n) for n = 1..4500 (terms 1..856 from David A. Corneth)
David A. Corneth, Tuples (x, y, z, w) that are solutions to the equation.
David A. Corneth, PARI program

Cf. A386373, A386521.

f[w_]:=(c=0;zz=w^5;Do[yy=zz-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,c++]],{y,Floor[yy^(1/3)]}],{z,Floor[zz^(1/4)]}];c);Array[f@#&, 30]

A387032 Numbers k with digits different from 0 and 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 222

Offset: 1

David A. Corneth, Aug 13 2025

A062998 contains numbers like 123, 124, 125,.. which are not in this sequence. - R. J. Mathar, Aug 14 2025

A037344 contains numbers like 2047 and 4095 which are not in this sequence. - R. J. Mathar, Aug 14 2025

			2 is in the sequence since it does not contain 0 nor 1.
12 is not in the sequence since it has digit 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Intersection of A052382 and A052383.

Cf. A037344, A172424.

isA387032 := proc(n)
    local d ;
    for d in convert(n,base,10) do
        if d <=1 then
            return false;
        end if;
    end do:
    true ;
end proc:
A387032 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA387032(a) then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(A387032(n),n=1..200) ; # R. J. Mathar, Aug 14 2025

Select[Range[222], Total@ DigitCount[#, 10, {0, 1}] == 0 &] (* Michael De Vlieger, Aug 13 2025 *)

is(n) = if(n <= 0, return(0)); Set(digits(n))[1] >= 2

def ok(n): return {"0","1"} & set(str(n)) == set()
print([k for k in range(223) if ok(k)]) # Michael S. Branicky, Aug 13 2025

def A387032(n):
    m = ((k:=7*n+1).bit_length()-1)//3
    return sum((2+((k-(1<<3*m))//(7<<3*j)&7))*10**j for j in range(m)) # Chai Wah Wu, Aug 13 2025

A380442 a(n) is the largest Frobenius number of three distinct relatively prime numbers that sum to n.

Original entry on oeis.org

1, 1, 3, 2, 5, 7, 7, 11, 13, 11, 17, 19, 23, 23, 29, 31, 35, 39, 43, 47, 51, 47, 59, 63, 67, 71, 79, 83, 89, 95, 101, 107, 113, 103, 125, 131, 139, 143, 153, 155, 167, 175, 181, 191, 199, 199, 215, 223, 233, 239, 251, 259, 269, 279, 289, 299, 309, 311, 329, 339

Offset: 9

David A. Corneth, Jul 25 2025

nonn

6 is the smallest number that can be partitioned into three distinct positive integers excluding n = 1 through 5.
a(6) would be -1 as the only partition of 6 into 3 distinct numbers that are relatively prime are (1, 2, 3) and the largest Frobenius number of that partition is -1.
Similarly a(7) and a(8) would be -1 as all partitions of these numbers into three distinct parts have a 1 in them.

			a(11) = 3 as the partitions of 11 into 3 distinct numbers that are relatively prime are (2,4,5) and (2,3,6) that have Frobenius number 3 and 1 respectively and their maximum is 3.

Brady Haran and David Eisenbud, The Frobenius Problem (and numerical semigroups) - Numberphile, Numberphile video, 2025

Cf. A386243.

A386905 Distinct terms in A385988.

Original entry on oeis.org

1, 2, 3, 7, 23, 11415, 56553873808298479843537432588765043239809588104012779518849306976361095532563320075112688420502361617080665176812695

Offset: 1

Paolo Xausa and David A. Corneth, Aug 07 2025

nonn

David A. Corneth, PARI program

Cf. A385988.

s = 0; k = 1; Reap[Do[s += k; If[IntegerQ@ Log2[s], k = n; Sow[k]], {n, 2^16}] ][[-1, 1]] (* Michael De Vlieger, Aug 07 2025 *)

PARI
```
\\ See Corneth link
```

cp's OEIS Frontend

User: David A. Corneth

A387100 a(n) is the least number that can be written in exactly n ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t}.

Views

Author

Keywords

Examples

Crossrefs

A387099 Numbers that can be written in exactly five ways as s_1^x_1 + ... + s_t^x_t, with 1 < s_1 < ... < s_t and {s_1,..., s_t} = {x_1,..., x_t} for some t > 0.

Views

Author

Keywords

Examples

Links

Crossrefs

A387215 a(n) is the smallest k such that, for any m >= k, m is a sum of exactly n distinct primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A387151 a(n) = n*n! / Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A386521 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 with GCD(x,y,z)=1 has no positive integer solutions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A386377 a(n) is the number of solutions to the equation x^2 + y^3 + z^4 = w^5 where GCD(x, y, z)=1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A387032 Numbers k with digits different from 0 and 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Python

A380442 a(n) is the largest Frobenius number of three distinct relatively prime numbers that sum to n.

Views

Author