A278981 -id:A278981 - cp's OEIS frontend

A280270 Numbers n such that A278981(n) < 2*n^2.

6, 8, 12, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 70, 72, 80, 84, 88, 90, 96, 108, 112, 116, 120, 126, 132, 140, 144, 148, 150, 156, 160, 162, 168, 174, 176, 180, 186, 188, 192, 196, 198, 200, 204, 210, 216, 220, 224, 230, 232, 234, 240

Offset: 1

Ely Golden, Dec 30 2016

All members of this sequence are even, as for any odd number m A278981(m) > m^3 + m^2 + m + 1 > 2*m^2.

It appears that, apart from 2, all members of A280236 appear in this sequence.

Ely Golden, Table of n, a(n) for n = 1..1889
Ely Golden, Table of n, a(n), A278981(a(n)) for n = 1..1889 (a-file)

Cf. A278981, A280236.

def nonZeroDigits(x, n):
    if(x<=0|n<2):
        return []
    li=[]
    while(x>0):
        d=divmod(x, n)
        if(d[1]!=0):
            li.append(d[1])
        x=d[0]
    li.sort()
    return li;
def nonZeroFactorDigits(x, n):
    if(x<=0|n<2):
        return []
    li=[]
    f=list(factor(x))
    #ensures inequality of nonZeroFactorDigits(x, n) and nonZeroDigits(x, n) if x is prime
    if((len(f)==1)&(f[0][1]==1)):
        return [];
    for c in range(len(f)):
        for d in range(f[c][1]):
            ld=nonZeroDigits(f[c][0], n)
            li+=ld
    li.sort()
    return li;
#the actual function
def a(n):
    c=n**2+n+1
    limit=2*(n**2)
    if(n%2!=0):
        return -1
    while((nonZeroFactorDigits(c, n)!=nonZeroDigits(c, n))&(c=limit):
        return -1
    return c;
index=1
value=2
while(index<=1000):
    result=a(value)
    if(result!=-1):
        print(str(index)+" "+str(value)+" "+str(result))
        index+=1
    value+=1
print("complete")

A280236 Numbers n such that A278981(n) < A278981(m) for all m > n (excluding values of m where A278981(m) = 0).

Original entry on oeis.org

2, 6, 8, 12, 20, 24, 28, 30, 32, 36, 40, 44, 48, 50, 54, 56, 60, 64, 66, 70, 72, 80, 88, 90, 96

Offset: 1

Ely Golden, Dec 29 2016

It is not necessary to check A278981(m) for all values of m > n (of which there are infinitely many). One need check only values of a(m) where m^2 + m + 1 <= A278981(n), due to the lower bound of A278981(m).
It appears that all members in this sequence are even, although it is possible that some members could be odd.
It appears that, apart from 2, all members in this sequence appear in A280270. If this is the case, all members in this sequence must be even.

			6 is a member of this sequence as A278981(6) = 57, which is which is less than all the terms in A278981 which succeed it.
One need check only values of a(m) where m^2 + m + 1 <= 57. In this case, only m=7 needs to be checked, and A278981(7) = 906, with 906 >= 57. Thus A278981(6) < A278981(m) for all m > 6.

Cf. A278981, A280270.

A280110 Numbers n such that A278981(n) > A278981(m) for 2 <= m < n.

Original entry on oeis.org

2, 3, 7, 9, 11, 13, 25, 29, 37

Offset: 1

Ely Golden, Dec 26 2016

The density of this sequence, and the pattern of A278981(a(n)), seem to suggest that computing the terms of A278981(n) gets exponentially more difficult as n tends to infinity.

			a(3) = 7 since A278981(7) = 906, which is greater than all the terms in A278981 which precede it.

Cf. A278981.

A281336 a(n) is the smallest composite number having the same base-n digits (both type and quantity) as its prime factors (with multiplicity).

Original entry on oeis.org

159, 7847, 1135, 83494, 57, 30057, 85, 72646, 1255, 18193, 185, 101212405, 4119, 63791, 88357, 31054582, 489, 4196517, 451, 4598494, 13315, 1012985, 679, 7009758597, 26533, 2884373, 985, 646372334, 1057, 989775393, 1285, 1395750166, 179503, 73294351, 1387

Offset: 2

Hans Havermann, Jan 20 2017

A278981(n) <= A281336(n)

			In base 2: The digits of 159 (10011111) are the same type and quantity as the total of those in its prime factors, 3 (11) and 53 (110101); 2 zeros, 6 ones.
In base 26: The digits of 26533 (1,13,6,13) are the same type and quantity as the total of those in its prime factors, 13 (13), 13 (13), and 157 (6,1); 1 one, 1 six, 2 thirteens.

Cf. A278981, A280928, A281335.

A281189 a(n) is the first composite number having the same base-(2n) digits as its prime factors (with multiplicity), excluding zero digits (or 0 if no such composite number exists).

Original entry on oeis.org

15, 85, 57, 85, 1111, 185, 4119, 4369, 489, 451, 13315, 679, 26533, 985, 1057, 1285, 179503, 1387, 82311, 2005, 2649, 2047, 4663957, 2509, 2761, 3385, 3097, 3277, 243895, 4207, 16246817, 4369, 4577, 471651, 5401, 5629, 607839, 466429, 483731, 6817, 1009273, 10587, 1132547, 8119, 8401, 798731, 990583, 9809, 1411791, 1062517

Offset: 1

Ely Golden and Robert G. Wilson v, Jan 16 2017

Bisection of A278981.
Conjecture: a(n) always exceeds 0.
If a(n) = 0 then it must be the case that there exists no more than one prime of the form (2n)^m + 1. Otherwise, the product of two such primes would satisfy the condition of A278981 in base 2n.
Records: 15, 85, 1111, 4119, 4369, 13315, 26533, 179503, 4663957, 16246817, 75927167, 120872069, 335192766, ..., .
a(76) > 2^27.

			a(2) = A278981(4) since 85 is the least composite number which satisfies the criterion of A278981.

Ely Golden and Robert G. Wilson v, Table of n, a(n) for n = 1..75

Cf. A278981, A280270.

g[n_] := g[n] = Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger@ n]; f = Compile[{{b, _Integer}}, Block[{c = b^2}, While[ PrimeQ@ c || DeleteCases[ Sort[ IntegerDigits[c, b]], 0] != DeleteCases[ Sort[ Flatten[ IntegerDigits[ g[c], b]]], 0], c++]; c]]; Table[ f[b], {b, 2, 80, 2}]

a(n) = A278981(2n).

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A280270 Numbers n such that A278981(n) < 2*n^2.

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A280236 Numbers n such that A278981(n) < A278981(m) for all m > n (excluding values of m where A278981(m) = 0).

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A280110 Numbers n such that A278981(n) > A278981(m) for 2 <= m < n.

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A281336 a(n) is the smallest composite number having the same base-n digits (both type and quantity) as its prime factors (with multiplicity).

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A281189 a(n) is the first composite number having the same base-(2n) digits as its prime factors (with multiplicity), excluding zero digits (or 0 if no such composite number exists).

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