A334527 -id:A334527 - cp's OEIS frontend

A334528 Palindromic numbers that are also Niven numbers and Smith numbers.

4, 666, 28182, 45054, 51315, 82628, 239932, 454454, 864468, 2594952, 2976792, 3189813, 3355533, 4172714, 4890984, 5319135, 5367635, 5777775, 7149417, 7247427, 8068608, 8079708, 8100018, 8280828, 8627268, 9227229, 9423249, 21699612, 22544522, 24166142, 27677672

Offset: 1

Amiram Eldar, May 05 2020

Witno (2014) proved that this sequence is infinite.

			666 is a term since it is palindromic, a Niven number (6 + 6 + 6 = 18 is a divisor of 666) and a Smith number (666 = 2 * 3 * 3 * 37 and 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7).

Amiram Eldar, Table of n, a(n) for n = 1..512
Amin Witno, Smith Numbers With Extra Digital Features, Integers, Vol. 14 (2014), Article A66.

Intersection of A002113, A005349 and A006753.

Intersection of any two of the sequences A082232, A098834 and A334527.

digSum[n_] := Plus @@ IntegerDigits[n]; palNivenSmithQ[n_] := PalindromeQ[n] && Divisible[n, (ds = digSum[n])] && CompositeQ[n] && Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == ds; Select[Range[10^5], palNivenSmithQ]

A334531 Numbers that are both binary Niven numbers and binary Smith numbers.

Original entry on oeis.org

55, 185, 205, 222, 246, 438, 623, 822, 973, 1503, 1939, 2359, 2471, 3126, 3205, 3462, 3573, 3661, 3771, 3846, 4711, 5877, 5949, 6093, 6198, 6655, 6918, 7083, 7550, 7931, 8151, 8170, 9567, 9863, 10265, 10683, 11241, 12280, 12318, 12486, 12678, 13695, 13790, 13820

Offset: 1

Amiram Eldar, May 05 2020

			The binary representation of 55 is 110111. It is a binary Niven number since 1 + 1 + 0 + 1 + 1 + 1 = 5 is a divisor of 55. It is also a binary Smith number since its prime factorization, 5 * 11, is 101 * 1011 in binary representation, and 1 + 1 + 0 + 1 + 1 + 1 = (1 + 0 + 1) + (1 + 0 + 1 + 1). Thus 55 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wayne L. McDaniel, On the Intersection of the Sets of Base b Smith Numbers and Niven Numbers, Missouri Journal of Mathematical Sciences, Vol. 2, No. 3 (1990), pp. 132-136.

Intersection of A049445 and A278909.

Cf. A334527.

binWt[n_] := DigitCount[n, 2, 1]; binNivenSmithQ[n_] := Divisible[n, (bw = binWt[n])] && CompositeQ[n] && Plus @@ (Last@# * binWt[First@#] & /@ FactorInteger[n]) == bw; Select[Range[10^4], binNivenSmithQ]

A351618 Numbers that are both Zuckerman numbers and Smith numbers.

Original entry on oeis.org

4, 1111, 3168, 7119, 31488, 141184, 698112, 1169316, 1621248, 1687392, 1938816, 1967112, 12469248, 12822912, 14112672, 16616448, 41484288, 79817472, 116149248, 121911264, 128894976, 163319328, 166491936, 193916916, 218431488, 247984128, 798142464, 817883136

Offset: 1

Bernard Schott, Feb 15 2022

			3168 is a term since it is a Zuckerman number (3*1*6*8) = 144 is a divisor of 3168 and a Smith number (3168 = 2*2*2*2*2*3*3*11 and 2+2+2+2+2+3+3+1+1 = 3+1+6+8).

Giovanni Resta, Smith numbers, Numbers Aplenty.
Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Intersection of A007602 and A006753.

Cf. A334527.

digSum[n_] := Plus @@ IntegerDigits[n]; smithQ[n_] := CompositeQ[n] && Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == digSum[n]; zuckQ[n_] := (prodig = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prodig]; Select[Range[10^6], zuckQ[#] && smithQ[#] &] (* Amiram Eldar, Feb 15 2022 *)

isok(m) = my(d=digits(m)); if (vecmin(d) && !(m % vecprod(d)) && !isprime(m) , my(f=factor(m)); sum(k=1, #f~, sumdigits(f[k,1])*f[k,2]) == vecsum(d)); \\ Michel Marcus, Feb 15 2022

More terms from Amiram Eldar, Feb 15 2022

cp's OEIS Frontend

A334528 Palindromic numbers that are also Niven numbers and Smith numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A334531 Numbers that are both binary Niven numbers and binary Smith numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A351618 Numbers that are both Zuckerman numbers and Smith numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions