A178193 -id:A178193 - cp's OEIS frontend

A174460 Smith numbers of order 2.

56, 58, 810, 822, 1075, 1519, 1752, 2145, 2227, 2260, 2483, 2618, 2620, 3078, 3576, 3653, 3962, 4336, 4823, 4974, 5216, 5242, 5386, 5636, 5719, 5762, 5935, 5998, 6220, 6424, 6622, 6845, 7015, 7251, 7339, 7705, 7756, 8460, 9254, 9303, 9355, 10481, 10626, 10659

Offset: 1

Paul Weisenhorn, Dec 20 2010

Composite numbers a(n) such that the sum of digits^2 equals the sum of digits^2 of its prime factors without the numbers of A176670 that have the same digits as its prime factors (without the zero digit).

It seems as though as the order n approaches infinity, the sequence of n-order Smith numbers approaches A176670. Is there a value of n where the only n-order Smith numbers are members of A176670? - Ely Golden, Dec 07 2016

			a(2) = 58 = 2*29 is a Smith number of order 2 because 5^2 + 8^2 = 2^2 + 2^2 + 9^2 = 89.

Ely Golden and Donovan Johnson, Table of n, a(n) for n = 1..10000 (terms 1 to 1000 by Donovan Johnson)
Patrick Costello, A new largest Smith number, Fibonacci Quarterly 40(4) (2002), 369-371.
Underwood Dudley, Smith numbers, Mathematics Magazine 67(1) (1994), 62-65.
Ely Golden, Smith number sequence generator optimized for order 2.
S. S. Gupta, Smith Numbers, Mathematical Spectrum 37(1) (2004/5), 27-29.
S. S. Gupta, Smith Numbers.
Eric Weisstein's World of Mathematics, Smith number.
Wikipedia, Smith number.
A. Wilansky, Smith Numbers, Two-Year College Math. J. 13(1) (1982), p. 21.
Amin Witno, Another simple construction of Smith numbers, Missouri J. Math. Sci. 22(2) (2010), 97-101.
Amin Witno, Smith multiples of a class of primes with small digital sum, Thai Journal of Mathematics 14(2) (2016), 491-495.

Cf. A006753 (Smith numbers), A176670, A178213, A178193, A178203, A178204.

for s from 2 to 10000 do g:=nops(ifactors(s)[2]): qsp:=0: for u from 1 to g do z:=ifactors(s)[2,u][1]: h:=0: while (z>0) do z:=iquo(z,10,'r'): h:=h+r^2: end do: h:=h*ifactors(s)[2,u][2]: qsp:=qsp+h: end do: z:=s: qs:=0: while (z>0) do z:=iquo(z,10,'r'): qs:=qs+r^2: end do: if (qsp=qs) then print(s): end if: end do:

With[{k = 2},Select[Range[12000], Function[n, And[Total@ Map[#^k &, IntegerDigits@ n] == Total@ Map[#^k &, Flatten@ IntegerDigits[#]], Not[Sort@ DeleteCases[#, 0] &@ IntegerDigits@ n == Sort@ DeleteCases[#, 0] &@ #]] &@ Flatten@ Map[IntegerDigits@ ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* Michael De Vlieger, Dec 10 2016 *)

A178184 Sum 2^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

4, 36, 548, 16932, 1065508, 135283236, 34495021604, 17626681066020, 18032025190548004, 36911520172609651236, 151152638972001256489508, 1238091191924352276155613732, 20283647694843594776223406899748

Offset: 1

Artur Jasinski, May 21 2010

nonn

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Cf. A178184-A178193.

aa = {}; m = 2; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, sum], {n, 1, 20}]; aa (*Artur Jasinski*)
Accumulate[Table[2^((k^2+3k)/2),{k,20}]] (* Harvey P. Dale, Aug 17 2021 *)

A178213 Smith numbers of order 3.

Original entry on oeis.org

6606, 8540, 13086, 16866, 21080, 26637, 27468, 33387, 34790, 35364, 35377, 40908, 44652, 48154, 48860, 52798, 54814, 55055, 57726, 57894, 66438, 67297, 67356, 67594, 69549, 72465, 72598, 73026, 74371, 74785, 77485, 78745, 81546, 83175, 85927, 90174, 91208

Offset: 1

Paul Weisenhorn, Dec 19 2010

Composite numbers n not in A176670 such that the sum of the cubes of the digits of n equals the sum of the cubes of the digits of the prime factors of n (with multiplicity). A176670 lists composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.

			6606 = 2*3*3*367 is composite; sum of cubes of the digits is 6^3+6^3+0^3+6^3 = 648. Sum of cubes of the digits of the prime factors 2, 3, 3, 367 (with multiplicity) is 2^3+3^3+3^3+3^3+6^3+7^3 = 648. The sums are equal, so 6606 is in the sequence.
21080 = 2*2*2*5*17*31 is composite; sum of cubes of the digits is 2^3+1^3+0^3+8^3+0^3 = 521. Sum of cubes of the digits of the prime factors 2, 2, 2, 5, 17, 31 (with multiplicity) is 2^3+2^3+2^3+5^3+1^3+7^3+3^3+1^3 = 521. The sums are equal, so 21080 is in the sequence.

Ely Golden and Donovan Johnson, Table of n, a(n) for n = 1..10000 (terms 1 to 1000 by Donovan Johnson)
Ely Golden, Smith number sequence generator optimized for order 3
Eric W. Weisstein, Smith Number

Cf. A006753 (Smith numbers), A174460, A176670, A178193, A178203, A178204.

fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, fid = Sort@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ fid[[1]] == 0, fid = Drop[fid, 1]]; !PrimeQ@ n && id != fid && Plus @@ (id^3) == Plus @@ (fid^3)]; k = 2; lst = {}; While[k < 22002, If[fQ@ k, AppendTo[ lst, k]; Print@ k]; k++]; lst
With[{k = 3}, Select[Range[10^5], Function[n, And[Total@ Map[#^k &, IntegerDigits@ n] == Total@ Map[#^k &, Flatten@ IntegerDigits[#]], Not[Sort@ DeleteCases[#, 0] &@ IntegerDigits@ n == Sort@ DeleteCases[#, 0] &@#]] &@ Flatten@ Map[IntegerDigits@ ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* Michael De Vlieger, Dec 10 2016 *)

A178203 Smith numbers of order 5; composite numbers n such that sum of digits^5 equal sum of digits^5 of its prime factors without the numbers in A176670 that have the same digits as its prime factors (without the zero digits).

Original entry on oeis.org

414966, 443166, 454266, 1274664, 1371372, 1701856, 1713732, 1734616, 1771248, 1858436, 1858616, 2075664, 2624976, 3606691, 3771031, 3771301, 4266914, 4414866, 4461786, 4605146, 4670576, 4710739, 5209663, 5281767, 5434572, 5836565, 5861712, 5871968, 6046357

Offset: 1

Paul Weisenhorn, Dec 19 2010

			a(10) = 1858436 = 2*2*29*37*433;
1^5 + 3^5 + 4^5 + 5^5 + 6^5 + 2*8^5 = 3*2^5 + 3*3^5 + 4^5 + 7^5 + 9^5 = 77705.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Patrick Costello, A new largest Smith number, Fibonacci Quarterly 40(4) (2002), 369-371.
Underwood Dudley, Smith numbers, Mathematics Magazine 67(1) (1994), 62-65.
S. S. Gupta, Smith Numbers, Mathematical Spectrum 37(1) (2004/5), 27-29.
S. S. Gupta, Smith Numbers.
Eric Weisstein's World of Mathematics, Smith number.
Wikipedia, Smith number.
A. Wilansky, Smith Numbers, Two-Year College Math. J. 13(1) (1982), p. 21.
Amin Witno, Another simple construction of Smith numbers, Missouri J. Math. Sci. 22(2) (2010), 97-101.
Amin Witno, Smith multiples of a class of primes with small digital sum, Thai Journal of Mathematics 14(2) (2016), 491-495.

Cf. A006753 (Smith numbers), A176670, A174460, A178213, A178193, A178204.

a(21) corrected by Donovan Johnson, Jan 02 2013

A178204 Smith numbers of order 6; composite numbers n such that sum of digits^6 equal sum of digits^6 of its prime factors without the numbers in A176670 that have the same digits as its prime factors (without the zero digits).

Original entry on oeis.org

40844882, 113986781, 130852098, 141176320, 168137185, 170774472, 178180163, 181681157, 181693781, 183161897, 187117638, 215149451, 261666000, 284804842, 294557945, 307711074, 335524949, 337194240, 344552927, 347391040, 355318188, 358831104, 368657536

Offset: 1

Paul Weisenhorn, Dec 19 2010

			a(4) = 141176320 = 2^9*5*55147;
3*1^6+2^6+3^6+4^6+6^6+7^6 = 1^6+9*2^6+4^6+3*5^6+7^6 = 169197

Donovan Johnson, Table of n, a(n) for n = 1..1000
Eric W. Weisstein, Smith Number

Cf. A006753 (Smith numbers), A176670, A174460, A178213, A178193, A178203.

fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, fid = Sort@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ fid[[1]] == 0, fid = Drop[fid, 1]]; id != fid && Plus @@ (id^6) == Plus @@ (fid^6)]; k = 2; lst = {}; While[k < 50000001, If[fQ@ k, AppendTo[ lst, k]; Print@ k]; k++]; lst

Example corrected by Donovan Johnson, Jan 02 2013

A178185 Numerator of Sum_{k=1..n} 1/2^((k^2 + 3*k)/2).

Original entry on oeis.org

1, 9, 145, 4641, 297025, 38019201, 9732915457, 4983252713985, 5102850779120641, 10450638395639072769, 42805814868537642061825, 350665235403060363770470401, 5745299216843741000015387049985

Offset: 1

Artur Jasinski, May 21 2010

Series of the kind m^((k^2 + 3*k)/2) from k=1 to n were studied by Bernoulli and Euler.

Vincenzo Librandi, Table of n, a(n) for n = 1..80

Cf. A178184-A178193.

Cf. A036442 (denominators).

aa = {}; m = 1/2; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa
Numerator[Table[Sum[1/2^((k^2 + 3*k)/2), {k, 1, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Apr 10 2024 *)

a(n) = numerator(sum(k=1, n, (1/2)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013

a(n) = 2^(n+1)*a(n-1) + 1, a(1) = 1. - Alexandre Herrera, Mar 23 2024

a(n) ~ c * A036442(n+1) = c * 2^(n*(n+3)/2), where c = 2^(1/8) * EllipticTheta[2, 0, 1/Sqrt[2]] - 3 [in Mathematica] = 2^(1/8) * JacobiTheta2(0, 1/sqrt(2)) - 3 [in Maple] = 0.2832651213103077325876855404508588684521230759134794956... - Vaclav Kotesovec, Apr 10 2024

A178186 Sum 3^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

9, 252, 19935, 4802904, 3491587305, 7629089072292, 50039174188071999, 984820941357799304880, 58150721823981417489695049, 10301109611599361435391036962892, 5474411390529830981438591324606714655

Offset: 1

Artur Jasinski, May 21 2010

nonn

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Vincenzo Librandi, Table of n, a(n) for n = 1..60

Cf. A178184-A178193.

aa = {}; m = 3; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, sum], {n, 1, 20}]; aa (*Artur Jasinski*)
Table[Sum[3^((k^2+3k)/2),{k,n}],{n,20}] (* Harvey P. Dale, Jul 10 2020 *)
Accumulate[Table[3^((k^2+3k)/2),{k,15}]] (* Harvey P. Dale, Mar 25 2023 *)

a(n) = sum(k=1, n, 3^((k^2+3*k)/2)); \\ Michel Marcus, Sep 09 2013

A178187 Numerators of sum (1/3)^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

1, 28, 2269, 551368, 401947273, 879058686052, 5767504039187173, 113521782003321126160, 6703347705514109178621841, 1187477935988707898665323267628, 631074461779774914374598062671491949

Offset: 1

Artur Jasinski, May 21 2010

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Vincenzo Librandi, Table of n, a(n) for n = 1..60

Cf. A178184-A178193.

Cf. A217628 (denominators).

aa = {}; m = 1/3; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa (*Artur Jasinski*)
Accumulate[Table[(1/3)^((k^2+3k)/2),{k,20}]]//Numerator (* Harvey P. Dale, May 29 2020 *)

a(n) = numerator(sum(k=1, n, (1/3)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013

A178188 a(n) = Sum_{k=1..n} 5^((k^2+3k)/2).

Original entry on oeis.org

25, 3150, 1956275, 6105471900, 95373537112525, 7450675970460940650, 2910390496349340822268775, 5684344796471297836309816409400, 55511156915602623492479419714357425025

Offset: 1

Artur Jasinski, May 21 2010

nonn

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Vincenzo Librandi, Table of n, a(n) for n = 1..50

Cf. A178184-A178193.

aa = {}; m = 5; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, sum], {n, 1, 20}]; aa (*Artur Jasinski*)
Table[Sum[5^((k^2+3k)/2),{k,n}],{n,10}] (* Harvey P. Dale, Jan 17 2015 *)

a(n) = sum(k=1, n, 5^((k^2+3*k)/2)); \\ Michel Marcus, Sep 09 2013

A178189 Numerators of sum (1/5)^((k^2+3k)/2) from k=1 to n.

Original entry on oeis.org

1, 126, 78751, 246096876, 3845263687501, 300411225586015626, 117348134994537353906251, 229195576161205769348146484376, 2238238048449275091290493011484375001

Offset: 1

Artur Jasinski, May 21 2010

Series of the kind m^((k^2+3k)/2) from k=1 to n was studied by Bernoulli and Euler.

Vincenzo Librandi, Table of n, a(n) for n = 1..50

Cf. A178184-A178193.

aa = {}; m = 1/5; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa (*Artur Jasinski*)

a(n) = numerator(sum(k=1, n, (1/5)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013

cp's OEIS Frontend

A174460 Smith numbers of order 2.

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A178184 Sum 2^((k^2+3k)/2) from k=1 to n.

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A178213 Smith numbers of order 3.

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A178203 Smith numbers of order 5; composite numbers n such that sum of digits^5 equal sum of digits^5 of its prime factors without the numbers in A176670 that have the same digits as its prime factors (without the zero digits).

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A178204 Smith numbers of order 6; composite numbers n such that sum of digits^6 equal sum of digits^6 of its prime factors without the numbers in A176670 that have the same digits as its prime factors (without the zero digits).

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A178185 Numerator of Sum_{k=1..n} 1/2^((k^2 + 3*k)/2).

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A178186 Sum 3^((k^2+3k)/2) from k=1 to n.

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A178187 Numerators of sum (1/3)^((k^2+3k)/2) from k=1 to n.

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A178188 a(n) = Sum_{k=1..n} 5^((k^2+3k)/2).