A034710 -id:A034710 - cp's OEIS frontend

A066306 Prime numbers such that sum of digits equals product of digits.

2, 3, 5, 7, 2141, 2411, 4211, 11251, 12511, 15121, 21221, 25111, 1112171, 1127111, 1172111, 1271111, 7112111, 11112811, 11128111, 11218111, 12111811, 12118111, 12181111, 18211111, 81111211, 81112111

Offset: 1

Labos Elemer, Dec 13 2001

			2141 = p[323], 2*1*4*1 = 8 = 2+1+4+1.

Chai Wah Wu, Table of n, a(n) for n = 1..8552

Primes from A034710.

Cf. A000040, A061672, A066307.

[NthPrime(n): n in [1..2*10^4] | &+Intseq(NthPrime(n)) eq &*Intseq(NthPrime(n))]; // Vincenzo Librandi, Nov 18 2015

f[n_] := IntegerDigits[ Prime[n]]; Prime[ Select[ Range[ PrimePi[10^10]], Apply[Plus, f[ # ]] == Apply[Times, f[ # ]] & ]]

More terms from Robert G. Wilson v, Dec 27 2001

A062035 Positive numbers whose product of digits is three times their sum.

Original entry on oeis.org

66, 159, 167, 176, 195, 235, 253, 325, 333, 352, 519, 523, 532, 591, 617, 671, 716, 761, 915, 951, 1168, 1186, 1236, 1263, 1326, 1362, 1618, 1623, 1632, 1681, 1816, 1861, 2136, 2163, 2316, 2361, 2613, 2631, 3126, 3162, 3216, 3261, 3612, 3621, 6118

Offset: 1

Amarnath Murthy, Jun 27 2001

			235 belongs to the sequence as (2*3*5)/(2+3+5) = 30/10 = 3.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A011540, A034710, A062034, A062036, A062382, A062037, A062384, A062040, A062041, A062043, A062045.

s3Q[n_]:=Module[{idn=IntegerDigits[n]},3Total[idn]==Times@@idn]; Select[Range[7000],s3Q]  (* Harvey P. Dale, Mar 20 2011 *)

isok(n) = my(d=digits(n)); vecprod(d)==3*vecsum(d) \\ Mohammed Yaseen, Jul 29 2022

from math import prod
def ok(n): d = list(map(int, str(n))); return prod(d) == 3*sum(d)
print([k for k in range(1, 7000) if ok(k)]) # Michael S. Branicky, Jul 29 2022

Corrected and extended by Harvey P. Dale and Larry Reeves (larryr(AT)acm.org), Jul 04 2001

Offset corrected by Mohammed Yaseen, Jul 29 2022

A062036 Positive numbers whose product of digits is four times their sum.

Original entry on oeis.org

88, 189, 198, 246, 264, 426, 462, 624, 642, 819, 891, 918, 981, 1247, 1274, 1344, 1427, 1434, 1443, 1472, 1724, 1742, 2147, 2174, 2226, 2262, 2417, 2471, 2622, 2714, 2741, 3144, 3414, 3441, 4127, 4134, 4143, 4172, 4217, 4271, 4314, 4341, 4413, 4431

Offset: 1

Amarnath Murthy, Jun 27 2001

			1344 belongs to the sequence as (1*3*4*4)/(1+3+4+4) = 48/12 = 4.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A011540, A034710, A062034, A062035, A062382, A062037, A062384, A062040, A062041, A062043, A062045.

p4sQ[n_]:=Module[{idn=IntegerDigits[n]},Times@@idn/Total[idn]==4]; Select[Range[5000],p4sQ]  (* Harvey P. Dale, Apr 26 2011 *)

isok(n) = my(d=digits(n)); vecprod(d)==4*vecsum(d) \\ Mohammed Yaseen, Jul 31 2022

from math import prod
def ok(n): d = list(map(int, str(n))); return prod(d) == 4*sum(d)
print([k for k in range(1, 5000) if ok(k)]) # Michael S. Branicky, Jul 31 2022

More terms from Larry Reeves (larryr(AT)acm.org), Jul 06 2001

Offset corrected by Mohammed Yaseen, Jul 31 2022

A062037 Positive numbers whose product of digits is 6 times their sum.

Original entry on oeis.org

268, 286, 347, 374, 437, 473, 628, 682, 734, 743, 826, 862, 1269, 1296, 1348, 1356, 1365, 1384, 1438, 1483, 1536, 1563, 1629, 1635, 1653, 1692, 1834, 1843, 1926, 1962, 2169, 2196, 2237, 2273, 2327, 2334, 2343, 2372, 2433, 2619, 2691, 2723, 2732, 2916

Offset: 1

Amarnath Murthy, Jun 27 2001

			2237 belongs to the sequence as (2*2*3*7)/(2+2+3+7) = 84/14 = 6.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A011540, A034710, A062034, A062035, A062036, A062382, A062384, A062040, A062041, A062043, A062045.

p6sQ[n_]:=Module[{idn=IntegerDigits[n]},Times@@idn==6*Total[idn]]; Select[ Range[ 3000],p6sQ] (* Harvey P. Dale, Aug 17 2014 *)

isok(n) = my(d=digits(n)); vecprod(d)==6*vecsum(d) \\ Mohammed Yaseen, Aug 02 2022

from math import prod
def ok(n): d = list(map(int, str(n))); return prod(d) == 6*sum(d)
print([k for k in range(1, 3000) if ok(k)]) # Michael S. Branicky, Aug 02 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001

Offset corrected by Mohammed Yaseen, Aug 02 2022

A062043 Positive numbers whose product of digits is 10 times their sum.

Original entry on oeis.org

459, 495, 549, 594, 945, 954, 1566, 1656, 1665, 2259, 2295, 2355, 2529, 2535, 2553, 2592, 2925, 2952, 3255, 3525, 3552, 5166, 5229, 5235, 5253, 5292, 5325, 5352, 5523, 5532, 5616, 5661, 5922, 6156, 6165, 6516, 6561, 6615, 6651, 9225, 9252, 9522

Offset: 1

Amarnath Murthy, Jun 28 2001

A subsequence of A011535 as each term must include the digit 5. - Chai Wah Wu, Dec 09 2015

			594 belongs to the sequence as (5*9*4)/(5+9+4) = 180/18 = 10.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A011540, A034710, A062034, A062035, A062036, A062382, A062037, A062384, A062040, A062041, A062045.

Cf. A011535, A124694.

filter:= proc(t) local L;
  L:= convert(t,base,10);
  convert(L,`*`) = 10*convert(L,`+`)
end proc:
select(filter, [$1..10^5]); # Robert Israel, Sep 11 2022

Select[Range[10000], Times @@ IntegerDigits[ # ] == 10 Plus @@ IntegerDigits[ # ] &] (* Tanya Khovanova, Dec 25 2006 *)

isok(n) = my(d=digits(n)); vecprod(d)==10*vecsum(d) \\ Mohammed Yaseen, Sep 11 2022

from math import prod
def ok(n): d = list(map(int, str(n))); return prod(d) == 10*sum(d)
print([k for k in range(1, 9999) if ok(k)]) # Michael S. Branicky, Sep 11 2022

More terms from Harvey P. Dale and Larry Reeves (larryr(AT)acm.org), Jul 06 2001

A062382 Positive numbers whose product of digits is 5 times their sum.

Original entry on oeis.org

257, 275, 345, 354, 435, 453, 527, 534, 543, 572, 725, 752, 1258, 1285, 1528, 1582, 1825, 1852, 2158, 2185, 2235, 2253, 2325, 2352, 2518, 2523, 2532, 2581, 2815, 2851, 3225, 3252, 3522, 5128, 5182, 5218, 5223, 5232, 5281, 5322, 5812, 5821, 8125, 8152

Offset: 1

Amarnath Murthy, Jun 27 2001

			2235 belongs to the sequence as (2*2*3*5)/(2+2+3+5) = 60/12 = 5.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A011540, A034710, A062034, A062035, A062036, A062037, A062384, A062040, A062041, A062043, A062045.

Select[Range[9000],Times@@IntegerDigits[#]==5*Total[IntegerDigits[#]]&] (* Harvey P. Dale, Mar 01 2012 *)

isok(n) = my(d=digits(n)); vecprod(d)==5*vecsum(d) \\ Mohammed Yaseen, Aug 02 2022

from math import prod
def ok(n): d = list(map(int, str(n))); return prod(d) == 5*sum(d)
print([k for k in range(1, 9000) if ok(k)]) # Michael S. Branicky, Aug 02 2022

More terms from Larry Reeves (larryr(AT)acm.org), Jul 06 2001

Offset corrected by Mohammed Yaseen, Aug 02 2022

A061672 Smallest positive number formed by a set of digits whose product = sum of the digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 1124, 11125, 11133, 11222, 111126, 1111127, 1111134, 11111128, 11111223, 111111129, 111111135, 1111111144, 11111111136, 11111111224, 111111112222, 1111111111137, 1111111111145, 1111111111233

Offset: 1

Amarnath Murthy, Jun 26 2001

From M. F. Hasler, Oct 29 2014: (Start)
This is the subsequence of terms of A034710 with digits in nondecreasing order, which is meant by "smallest": For example, 132 also has sum of digits = product of digits, but is already "represented" by 123. The word "set" in the definition actually means "multiset".
The sequence is infinite: for any number N whose digits form a nondecreasing sequence whose sum of digits S is not larger than the product of digits P (i.e., N in A062998), a term of the sequence is obtained by prefixing N with P-S digits '1'. (End)

			1124 is a term since 1 + 1 + 2 + 4 = 1*1*2*4 = 8.

Chai Wah Wu, Table of n, a(n) for n = 1..667 (terms n=1..134 from T. D. Noe).
Sean A. Irvine, Java program (github)

Cf. A034710, A249334.

is_A061672(n)={vecsort(n=digits(n))==n && normlp(n,1)==prod(i=1,#n,n[i])} \\ M. F. Hasler, Oct 29 2014

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 27 2001
Corrected by Franklin T. Adams-Watters, Oct 25 2006
Further corrections from T. D. Noe, Oct 12 2007

A062034 Positive numbers whose product of digits is twice the sum of the digits.

Original entry on oeis.org

36, 44, 63, 138, 145, 154, 183, 224, 242, 318, 381, 415, 422, 451, 514, 541, 813, 831, 1146, 1164, 1225, 1233, 1252, 1323, 1332, 1416, 1461, 1522, 1614, 1641, 2125, 2133, 2152, 2215, 2222, 2251, 2313, 2331, 2512, 2521, 3123, 3132, 3213, 3231, 3312, 3321

Offset: 1

Amarnath Murthy, Jun 27 2001

			1225 belongs to the sequence as (1*2*2*5)/(1+2+2+5) =20/10 = 2.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A011540, A034710, A062035, A062036, A062382, A062037, A062384, A062040, A062041, A062043, A062045.

Select[Range[4000],Times@@IntegerDigits[#]==2Total[IntegerDigits[#]]&] (* Harvey P. Dale, Dec 11 2016 *)

isok(n) = my(d=digits(n)); vecprod(d)==2*vecsum(d) \\ Mohammed Yaseen, Jul 28 2022

from math import prod
def ok(n): d = list(map(int, str(n))); return prod(d) == 2*sum(d)
print([k for k in range(1, 4000) if ok(k)]) # Michael S. Branicky, Jul 28 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001

Offset corrected by Mohammed Yaseen, Jul 28 2022

A062040 Positive numbers whose product of digits is 8 times their sum.

Original entry on oeis.org

448, 456, 465, 484, 546, 564, 645, 654, 844, 1368, 1386, 1449, 1494, 1638, 1683, 1836, 1863, 1944, 2248, 2256, 2265, 2284, 2428, 2482, 2526, 2562, 2625, 2652, 2824, 2842, 3168, 3186, 3618, 3681, 3816, 3861, 4149, 4194, 4228, 4282, 4419, 4491, 4822

Offset: 1

Amarnath Murthy, Jun 28 2001

			2248 belongs to the sequence as (2*2*4*8)/(2+2+4+8) = 128/16 = 8.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A011540, A034710, A062034, A062035, A062036, A062382, A062037, A062384, A062041, A062043, A062045.

isok(n) = my(d=digits(n)); vecprod(d)==8*vecsum(d) \\ Mohammed Yaseen, Sep 11 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001

Offset corrected by Mohammed Yaseen, Sep 11 2022

A062041 Positive numbers whose product of digits is 9 times their sum.

Original entry on oeis.org

369, 396, 466, 639, 646, 664, 693, 936, 963, 2266, 2338, 2383, 2626, 2662, 2833, 3238, 3283, 3328, 3382, 3823, 3832, 6226, 6262, 6622, 8233, 8323, 8332, 11379, 11397, 11459, 11495, 11549, 11594, 11739, 11793, 11937, 11945, 11954, 11973

Offset: 1

Amarnath Murthy, Jun 28 2001

			12339 belongs to the sequence as (1*2*3*3*9)/(1+2+3+3+9) = 162/18 = 9.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A011540, A034710, A062034, A062035, A062036, A062382, A062037, A062384, A062040, A062043, A062045.

isok(n) = my(d=digits(n)); vecprod(d)==9*vecsum(d) \\ Mohammed Yaseen, Sep 11 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001

Offset corrected by Mohammed Yaseen, Sep 11 2022

cp's OEIS Frontend

A066306 Prime numbers such that sum of digits equals product of digits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A062035 Positive numbers whose product of digits is three times their sum.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A062036 Positive numbers whose product of digits is four times their sum.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A062037 Positive numbers whose product of digits is 6 times their sum.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A062043 Positive numbers whose product of digits is 10 times their sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Extensions

A062382 Positive numbers whose product of digits is 5 times their sum.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A061672 Smallest positive number formed by a set of digits whose product = sum of the digits.

Views

Author

Keywords

Comments

Examples