cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A133185 Conjectured list of numbers that are 7-smooth (A002473) but are not in A133184.

Original entry on oeis.org

14, 21, 24, 30, 42, 60, 63, 70, 75, 84, 90, 100, 105, 108, 112, 120, 126, 135, 147, 150, 168, 175, 189, 210, 224, 225, 240, 250, 252, 280, 294, 300, 315, 336, 350, 360, 375, 378, 405, 420, 448, 450, 490, 504, 525, 560, 567, 588, 600, 630, 640, 672, 675, 686, 700, 735, 756
Offset: 1

Views

Author

J. H. Conway and N. J. A. Sloane, Jan 01 2008

Keywords

Links

A248794 a(n) = the smallest number k for which the sum of digits (A007953(k)) and the product of digits (A007954(k)) are both equal to A002473(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11125, 111126, 1111127, 111111135, 11111128, 111111129, 1111111111145, 1111111111137, 111111111111138, 11111111111111155, 11111111111111139, 1111111111111111147, 111111111111111111156, 1111111111111111111148, 1111111111111111111111157
Offset: 1

Views

Author

Jaroslav Krizek, Nov 02 2014

Keywords

Comments

See comment in A034710 (positive numbers for which the sum of digits equals the product of digits).

Examples

			For n = 11; a(11) = 111126 because A002473(11) = 12, A007953(111126) = A007954(111126) = 12.

Crossrefs

Cf. A002473, A034710, A007953, A007954.

Programs

Extensions

Name clarified by Andrew Howroyd, Sep 20 2024

A351809 a(0) = 32; then, for n >= 1, a(n) is the smallest positive integer k such that pod(k^2)/pod(k) = A002473(n) where pod = product of digits = A007954.

Original entry on oeis.org

32, 1, 2, 3, 15, 381, 25, 61, 12, 27, 16, 41, 28, 23, 336, 13, 1766, 26, 43, 2675, 118, 278, 74, 22, 76, 128, 392, 343, 228, 121, 418, 976, 258, 193, 116, 194, 93, 218, 441, 1231, 112, 63, 219, 984, 136, 4165, 2271, 1894, 183, 615, 434, 22831, 523, 1592, 2435
Offset: 0

Views

Author

Bernard Schott, Feb 24 2022

Keywords

Comments

As pod(m) is a 7-smooth number and pod(m^2) can be 0, all terms of A351808 are in {0} union A002473. See example section for why a(0) = 32.

Examples

			pod(32) = 3*2 = 6, pod(32^2) = pod(1024) = 1*0*2*4 = 0, and k = 32 is the smallest positive integer k such that pod(k^2) = 0 while pod(k) <> 0, so a(0) = 32.
A002473(5) = 5; pod(381) = 3*8*1 = 24, pod(381^2) = pod(145161) = 1*4*5*1*6*1 = 120; as 120/24 = 5, and 381 is the smallest positive integer k such that pod(k^2)/pod(k) = 5 then a(5) = 381.
A002473(11) = 12; pod(41)= 4*1 = 4, pod(41^2) = pod(1681) = 1*6*8*1 = 48; as 48/4 = 12 and 41 is the smallest positive integer k such that pod(k^2)/pod(k) = 12, then a(11) = 41.

Crossrefs

Cf. A002473, A007954, A280012, A351807, A351808.

Programs

  • Mathematica
    sevenSmooths = Select[Range[150], Max[FactorInteger[#][[;; , 1]]] <= 7 &]; pod[n_] := Times @@ IntegerDigits[n]; r[n_] := If[(p = pod[n]) > 0, pod[n^2]/p, -1]; s = Array[r, 3*10^4]; TakeWhile[FirstPosition[s, #] & /@ Join[{0}, sevenSmooths] // Flatten, NumberQ] (* Amiram Eldar, Feb 24 2022 *)
  • PARI
    pod(k) = vecprod(digits(k)); \\ A007954
smp(m) = my(k=1); while (!pod(k) || (pod(k^2)/pod(k) != m), k++); k;
isss(n) = (n<11) || (vecmax(factor(n, 7)[, 1])<8); \\ A002473
lista(nn) = apply(smp, select(isss, [0..nn]));
lista(200) \\ Michel Marcus, Feb 24 2022

Extensions

More terms from Amiram Eldar, Feb 24 2022

A085123 Smallest member of A002473 whose digital product = A002473(n), or 0 if no such number exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 126, 27, 35, 28, 36, 45, 0, 64
Offset: 1

Views

Author

Amarnath Murthy, Jul 06 2003

Keywords

Comments

Conjecture: No entry is zero.
Next term (A002473(n) = 21) is either 0 or > 10^100. - David Wasserman, Jan 27 2005
a(17) = 0, because A002473(17) = 21 = 3*7 and the last two digits of every member of A002473 greater than 7 contain a digit other than 1, 3, and 7. - Robert Israel, Nov 23 2019
Similarly, a(n) = 0 whenever n >=17 and A002473(n) is coprime to 10. - Robert Israel, Nov 24 2019

Examples

			20 is a member of A002473 and the corresponding term is 45 which is also a member of A002473.

Crossrefs

Cf. A002473, A068129.
Cf. A068189.

Extensions

a(17) and a(18) from Robert Israel, Nov 23 2019

A085889 7-smooth numbers (A002473) using digits in increasing order. 9 is followed by a zero.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 45, 56, 90, 567, 3456
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 10 2003

Keywords

Comments

No more terms < 10^1000. Probably no more terms. - David Wasserman, Feb 10 2005

Examples

			567 is a member as 567 = 3^4*7.

Crossrefs

Intersection of A002473 and A059043.
Cf. A085890.

Programs

  • PARI
    hcn(n) = while (!(n%2), n \=2); while (!(n%3), n \=3); while (!(n%5), n \=5); while (!(n%7), n \=7); n == 1;
for (i = 0, 1000, for (j = 1, 9, my(n = sum(k = j, i + j, (k%10)*10^(i + j - k))); if (hcn(n), print1(n, ", ")))); /* David Wasserman, Feb 10 2005 */

Extensions

3456 from David Wasserman, Feb 10 2005

A085890 7-smooth numbers (A002473) using digits in descending order. Zero is followed by a 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21, 32, 54, 98, 210, 432
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 10 2003

Keywords

Comments

No more terms < 10^1000. Probably no more terms. - David Wasserman, Feb 10 2005

Examples

			432 is a member as 432= 2^4*3^3.

Crossrefs

Intersection of A002473 and A158699.
Cf. A085889.

Programs

  • PARI
    hcn(n) = while (!(n%2), n \=2); while (!(n%3), n \=3); while (!(n%5), n \=5); while (!(n%7), n \=7); n == 1;
for (i = 0, 1000, for (j = 1, 9, my(n = sum(k = j - i, j, (k%10)*10^(i - j + k))); if (hcn(n), print1(n, ", ")))); /* David Wasserman, Feb 10 2005 */

Extensions

More terms from David Wasserman, Feb 10 2005

A085911 Least common multiple of the first n terms of A002473 (7-smooth numbers).

Original entry on oeis.org

1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 2520, 2520, 2520, 5040, 5040, 5040, 5040, 5040, 25200, 75600, 75600, 75600, 151200, 151200, 151200, 151200, 151200, 151200, 151200, 1058400, 1058400, 1058400, 1058400, 1058400, 1058400, 2116800, 2116800
Offset: 1

Views

Author

Amarnath Murthy, Jul 09 2003

Keywords

Examples

			a(4) = 12 as the first four 7-smooth numbers are 1, 2, 3 and 4 and their lcm is lcm(1, 2, 3, 4) = 12. - _David A. Corneth_, Apr 02 2021

Links

Crossrefs

Cf. A002473.

Extensions

More terms from David Wasserman, Feb 11 2005
Offset changed to 1 by David A. Corneth, Apr 02 2021

A232093 Position of 7^n among 7-smooth numbers (A002473).

Original entry on oeis.org

1, 7, 30, 87, 202, 403, 726, 1214, 1911, 2874, 4158, 5832, 7968, 10640, 13933, 17937, 22747, 28464, 35195, 43054, 52162, 62644, 74630, 88257, 103671, 121020, 140462, 162155, 186267, 212973, 242453, 274894, 310483, 349420, 391909, 438161, 488388, 542814, 601667, 665181, 733594, 807154, 886109, 970720, 1061252, 1157972, 1261156, 1371084, 1488047, 1612341
Offset: 0

Views

Author

Zak Seidov, Nov 18 2013

Keywords

Comments

Note that all powers of 7 are terms in A002473.
Polynomial of fourth order is sufficient for very accurate approximation of a(n).

Examples

			A002473(a(1)) = A002473(7) = 7.
A002473(a(2)) = A002473(30) = 49 = 7^2.
A002473(a(200)) = A002473(411921660) = 7^200.

Links

Crossrefs

Cf. A000420, A002473.

Programs

  • Mathematica
    ss7 = {}; Do[m = 7^n; s = Sum[1 + Floor[Log[2, 7^(n - k)/5^i/3^j]], {k, 0, n}, {i, 0, Log[5, 7^(n - k)]}, {j, 0, Log[3, 7^(n - k)/5^i]}]; AppendTo[ss7, {n, s}], {n, 0, 50}]; ss7

Formula

a(n) ~ c * n^4, where c = log(7)^3/(24*log(2)*log(3)*log(5)) = 0.250503020417439... - Vaclav Kotesovec and Amiram Eldar, Sep 22 2024

A341634 Smallest prime whose product of digits (A007954) is the n-th 7-smooth number = A002473(n), with a(0) = 101.

Original entry on oeis.org

101, 11, 2, 3, 41, 5, 23, 7, 181, 19, 251, 43, 127, 53, 281, 29, 541, 37, 83, 11551, 139, 47, 523, 1481, 157, 149, 12451, 67, 59, 283, 11177, 2551, 239, 1187, 1453, 79, 881, 257, 89, 1553, 2851, 199, 347, 563, 1483, 277, 14551, 1753, 269, 827, 853, 15551, 367
Offset: 0

Views

Author

Bernard Schott, Feb 16 2021

Keywords

Comments

For n>=1, equals A107698 without the zeros.
101 is the smallest prime with the digit 0, so A007954(101) = 0 but as 0 is not a 7-smooth number, it is chosen a(0) = 101.

Examples

			83 is prime, A007954(83) = 8*3 = 24 that is the 18th 7-smooth number, and as no prime < 83 has a product of digits = 24, a(18) = 83.

Links

Crossrefs

Cf. A002473, A004954, A007954, A053666, A107698, A335331.

Programs

  • Mathematica
    pod[n_] := Times @@ IntegerDigits[n]; seq[max_] := Module[{sm7 = Join[{0}, Select[Range[max], Max[FactorInteger[#][[;; , 1]]] <= 7 &]], m, s, n, c, i, ind}, m = Length[sm7]; s = Table[0, {m}]; n = 1; c = 0; While[c < m, n = NextPrime[n]; i = pod[n]; If[MemberQ[sm7, i], ind = Position[sm7, i][[1, 1]]]; If[s[[ind]] == 0, c++; s[[ind]] = n]]; s]; seq[150] (* Amiram Eldar, Feb 16 2021 *)

Formula

a(n) = A107698(A002473(n)) for n>=1. - Amiram Eldar, Feb 17 2021

Extensions

More terms from Amiram Eldar, Feb 16 2021

A085134 Palindromes in A002473, that is, palindromes with the largest prime divisor < 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 252, 343, 525, 686, 48384, 65856
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 06 2003

Keywords

Comments

No more terms < 10^100; probably no more terms. - David Wasserman, Feb 22 2005

Examples

			252 = 2^2*3^2*7.

Crossrefs

Intersection of A002113 and A002473.

Programs

  • Mathematica
    Select[Range[66000],PalindromeQ[#]&&FactorInteger[#][[-1,1]]<10&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 07 2018 *)
  • PARI
    isok(n) = if (n==1, return(1)); my(d=digits(n)); (Vecrev(d)==d) && (vecmax(factor(n, 7)[, 1]) < 8); \\ Michel Marcus, Dec 08 2021

Extensions

2 more terms from David Wasserman, Feb 22 2005
Name edited by Michel Marcus, Dec 07 2021
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