A055482 -id:A055482 - cp's OEIS frontend

A115227 Numbers k such that k = (d_1+7)(d_2+7)...(d_m+7) where d_1 d_2 ... d_m is the decimal expansion of k.

8314460009856000, 31746120037632000, 92632873013093597184000000, 1108240107492643314063114240000, 25425833233403394290952098021376000000000, 67839608823081187608930897100800000000000, 1064102035269050218905833606927320350720000000

Offset: 1

Giovanni Resta, Jan 16 2006

			a(1) = 8314460009856000 = (7+8)*(7+3)*(7+1)*...*(7+6)*(7+0)*(7+0)*(7+0).

Max Alekseyev, Table of n, a(n) for n = 1..18 (contains all terms below 10^100)
Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 377.

Cf. A055482, A098113, A098114, A097371, A097372, A113756

a(6)-a(18) from Max Alekseyev, Jan 25 2015

A055481 Numbers k for which there exists some m such that k = Sum_{i=1..1+floor(log_10(k))} binomial(m, d_i), where d_i is the i-th digit of k.

Original entry on oeis.org

1, 10, 18, 21, 72, 100, 101, 111, 134, 231, 246, 505, 682, 1000, 1010, 1100, 1122, 2210, 3103, 4006, 6008, 10000, 10001, 10012, 11101, 15453, 20101, 29358, 34698, 56576, 84304, 100000, 100010, 100011, 100100, 100101, 100110, 100303, 101000, 101001, 101010

Offset: 1

Erich Friedman, Jun 27 2000

Contains numbers of the form 10^k, k >= 0 so the sequence is infinite. - David A. Corneth, Oct 30 2018

			3103 = C(22, 3) + C(22, 1) + C(22, 0) + C(22, 3).
C(k, 1) + C(k, 1) + C(k, 1) + C(k, 0) + C(k, 0) + C(k, 0) = 3k + 3 so all 6-digit numbers with 3 ones and 3 zeros are in the sequence. - _David A. Corneth_, Oct 30 2018

Giovanni Resta, Table of n, a(n) for n = 1..500

Cf. A007088, A009994, A055482.

ok[n_] := Block[{d = IntegerDigits@n, k=1, v, x}, If[ Max@d <= 3, False =!= Reduce[ Total@ Binomial[x, d] == n && x>0, x, Integers], While[(v = Total@ Binomial[k, d]) < n, k++]; v == n]]; Select[ Range[10^5], ok] (* Giovanni Resta, Oct 30 2018 *)

is(n) = my(d = digits(n)); for(i = 1, n, s = sum(j = 1, #d, binomial(i, d[j])); if(s >= n, return(s == n))) \\ David A. Corneth, Oct 30 2018

a(31)-a(41) from Giovanni Resta, Oct 30 2018

A113756 Numbers n>9 such that n=Abs[(c+d_1)(c+d_2)...*(c+d_k)] where d_1 d_2 ... d_k is the decimal expansion of n and c is an integer constant.

Original entry on oeis.org

12, 18, 24, 35, 50, 56, 90, 100, 120, 180, 210, 315, 350, 450, 500, 672, 728, 780, 840, 910, 1500, 1800, 3150, 3500, 3920, 4320, 4752, 5000, 7056, 7200, 7744, 8960, 16500, 18000, 19008, 24960, 31500, 35000, 50000, 57915, 59400, 60480, 67392, 91728

Offset: 1

Giovanni Resta, Jan 17 2006

Some entries, namely 12, 18, 24, 35, 50, 56, 90, 120, 210, 315, 840, 4752, 7744, 917280 (up to 10^20), have 2 representations, e.g., 840 for c=6 and c=-14 or 917280 for c=6 and c=-15. Sequence is infinite since contains 35*10^k, 315*10^k, 18*10^k and 5*10^k, for k>=0 and c=-10.

			315 belongs since 315=|(4+3)(4+1)(4+5)|
728 belongs since 728=|(-15+7)(-15+2)(-15+8)|

Cf. A055482, A115227, A098113, A098114, A097371, A097372.

L = {}; Do[d = IntegerDigits@n; Do[If[n == Abs[Times @@ (d + c)], AppendTo[L, {n, c}]; Print[{n, c}]], {c, -19, 10}], {n, 10, 1000000}]; Print[Union[Transpose[L][[1]]]]; L

A260213 Numbers j such that j = (c_1 + k)(c_2 + k)...*(c_m + k) for some k > 0 where c_1, c_2, ..., c_m is the centesimal expansion of j.

Original entry on oeis.org

114, 120, 147, 198, 264, 420, 500, 506, 513, 525, 533, 550, 558, 568, 581, 648, 1102, 1116, 1168, 1302, 1320, 1377, 1680, 1692, 1710, 1720, 1734, 1755, 1771, 1872, 2106, 2132, 2310, 2332, 2380, 2664, 2714, 2736, 2790, 2914, 2940, 3312

Offset: 1

Pieter Post, Jul 19 2015

k cannot be larger than 99, because in that case the product of the terms c_i+k is larger than the number j itself. For j up to 10^12, the highest value for k is 71, for 910602 = (71+91)*(71+6)*(71+2).

All terms j < 10000 have the following property. j = c_1//c_2, so j = (c_1 + k)*(c_2 + k). Let kk = c_1 + c_2 + k then j = (kk - c_1)*(kk - c_2). For example, 513 = (5 + 14)*(13 + 14), kk = 5 + 13 + 14 = 32, so 513 = (32 - 5)*(32 - 13).

			114 = (1 + 5)*(14 + 5) and 114 = (20 - 1)*(20 - 14).
1710 = (17 + 28)*(10 + 28) and 1710 = (55 - 17)*(55 - 10).

Pieter Post and Giovanni Resta, Table of n, a(n) for n = 1..295 (terms < 10^12, first 141 terms from Pieter Post)

Cf. A055482.

is(n)=my(d=digits(n,100),t); while((t=vecprod(d))99 \\ Charles R Greathouse IV, Aug 28 2015

def pod(n,m,a):
    kk = 1
    while n > 0:
        kk= kk*(n%m+a)
        n =int(n//m)
    return kk
for c in range (1,10000):
    for a in range (1,100):
        if c==pod(c,100,a):
            print (c)

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A115227 Numbers k such that k = (d_1+7)(d_2+7)*...*(d_m+7) where d_1 d_2 ... d_m is the decimal expansion of k.

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A055481 Numbers k for which there exists some m such that k = Sum_{i=1..1+floor(log_10(k))} binomial(m, d_i), where d_i is the i-th digit of k.

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A113756 Numbers n>9 such that n=Abs[(c+d_1)*(c+d_2)*...*(c+d_k)] where d_1 d_2 ... d_k is the decimal expansion of n and c is an integer constant.

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A260213 Numbers j such that j = (c_1 + k)*(c_2 + k)*...*(c_m + k) for some k > 0 where c_1, c_2, ..., c_m is the centesimal expansion of j.

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A115227 Numbers k such that k = (d_1+7)(d_2+7)...(d_m+7) where d_1 d_2 ... d_m is the decimal expansion of k.

A113756 Numbers n>9 such that n=Abs[(c+d_1)(c+d_2)...*(c+d_k)] where d_1 d_2 ... d_k is the decimal expansion of n and c is an integer constant.

A260213 Numbers j such that j = (c_1 + k)(c_2 + k)...*(c_m + k) for some k > 0 where c_1, c_2, ..., c_m is the centesimal expansion of j.