Yaroslav Deryavko - cp's OEIS frontend

User: Yaroslav Deryavko

Yaroslav Deryavko has authored 3 sequences.

A387364 Least number which is not a prime power and whose prime factors are equal modulo n.

6, 15, 10, 21, 14, 55, 46, 33, 22, 39, 26, 85, 82, 51, 34, 57, 38, 115, 118, 69, 46, 141, 142, 145, 159, 87, 58, 93, 62, 259, 314, 185, 202, 111, 74, 205, 226, 123, 82, 129, 86, 235, 262, 141, 94, 371, 291, 265, 298, 159, 106, 321, 327, 295, 334, 177, 118, 183

Offset: 1

Yaroslav Deryavko, Aug 27 2025

			For n = 2, factors of a(2) = 15 are 3 and 5, they both have a residue of 1 mod 2.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

First term of A380758 when n = 10.

a[n_]:=Module[{k=1},Until[!PrimePowerQ[k]&&Min[Mod[First/@FactorInteger[k],n]]==Max[Mod[First/@FactorInteger[k],n]],k++];k];Array[a,58] (* James C. McMahon, Sep 03 2025 *)

f(k, n) = if (!isprimepower(k), my(f=factor(k)[,1]); #Set(apply(x->Mod(x, n), f)) == 1);
a(n) = my(k=1); while (!f(k,n), k++); k; \\ Michel Marcus, Aug 27 2025

A380758 Numbers which are not prime powers and their prime factors share a last digit in base 10.

Original entry on oeis.org

39, 69, 117, 119, 129, 159, 207, 219, 249, 259, 299, 309, 329, 339, 341, 351, 387, 451, 469, 477, 489, 507, 519, 551, 559, 579, 621, 629, 657, 669, 671, 679, 689, 699, 747, 749, 781, 789, 799, 833, 849, 879, 889, 897, 927, 939, 949, 959, 989, 1017, 1053, 1059

Offset: 1

Yaroslav Deryavko, Feb 01 2025

Also called the one-sided numbers.

They can end only in either 1, 3, 7 or 9.

			39 = 3*13.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
VOS 2025 Math Region stage, Problem 10.9 (in Russian).

Cf. A004615.

q:= n-> (l-> nops(l)>1 and nops({map(i-> irem(i[1], 10), l)[]})=1)(ifactors(n)[2]):
select(q, [$1..2000])[];  # Alois P. Heinz, Feb 18 2025

Sort[Times@@@Cases[Subsets[Prime[Range[100]],{2}],?(Mod[#[[1]]-#[[2]],10]==0&)]][[;;100]] (* _Shenghui Yang, Feb 18 2025 *)

isok(k) = my(f=factor(k)); (#f~ != 1) && (#Set(vector(#f~, i, f[i,1] % 10)) == 1); \\ Michel Marcus, Feb 18 2025

from sympy import factorint
def ok(n): return len(f:=factorint(n)) > 1 and len(set(p%10 for p in f)) == 1
print([k for k in range(1, 1060) if ok(k)]) # Michael S. Branicky, Feb 18 2025

A380108 Number of distinct partitions of length n binary strings into maximal constant substrings up to permutation.

Original entry on oeis.org

1, 2, 3, 6, 10, 18, 29, 48, 75, 118, 179, 272, 403, 596, 865, 1252, 1786, 2538, 3566, 4990, 6918, 9552, 13086, 17856, 24205, 32684, 43881, 58698, 78125, 103618, 136820, 180064, 236031, 308432, 401585, 521340, 674579, 870446, 1119786, 1436798, 1838405, 2346480, 2987204

Offset: 0

Yaroslav Deryavko, Jan 12 2025

nonn

Equivalently, a(n) is the number of partitions of n into parts of two kinds where the number of parts of each kind differ by at most one.

			For n = 3, the partitions are (000), (111), (00, 1), (0, 11), (0, 0, 1), (0, 1, 1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000712, A114921, A342528.

g:= (n, i, t)-> `if`(t>1+n, 0, `if`(n=0, 1, b(n, i, t))):
b:= proc(n, i, t) option remember; add(add(g(n-i*j,
      min(n-i*j, i-1), abs(t+2*h-j)), h=0..j), j=`if`(i=1, n, 0..n/i))
    end:
a:= n-> g(n$2, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 15 2025

g[n_, i_, t_] := If[t > 1+n, 0, If[n == 0, 1, b[n, i, t]]];
b[n_, i_, t_] := b[n, i, t] = Sum[Sum[g[n-i*j,
   Min[n-i*j, i-1], Abs[t+2*h-j]], {h, 0, j}], {j, If[i == 1, n, 0], n/i}];
a[n_] := g[n, n, 0];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 02 2025, after Alois P. Heinz *)

seq(n)={my(p=1/prod(k=1, n, 1-y*x^k + O(x*x^n))); Vec(sum(k=0, n\2, polcoef(p, k, y)*(2*polcoef(p, k+1, y) + polcoef(p, k, y))))} \\ Andrew Howroyd, Jan 12 2025

n = 0
while True:
    m = set()
    for i in range(2**n):
        t = bin(i)[2:]
        t = '0' * (n - len(t)) + t + '2'
        l = []
        s = 0
        for j in range(1, n + 1):
            if t[j] != t[j - 1]:
                l.append(t[s:j])
                s = j
        l.sort()
        l = tuple(l)
        m.add(l)
    print(len(m), end=' ')
    n += 1

G.f.: Sum_{k>=0} ([y^k] P(x,y))*([y^k] (1 + 2*y)*P(x,y)), where P(x,y) = Product_{k>=1} 1/(1 - y*x^k). - Andrew Howroyd, Jan 12 2025

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User: Yaroslav Deryavko

A387364 Least number which is not a prime power and whose prime factors are equal modulo n.

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A380758 Numbers which are not prime powers and their prime factors share a last digit in base 10.

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Maple

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A380108 Number of distinct partitions of length n binary strings into maximal constant substrings up to permutation.

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Maple

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