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.

Showing 1-3 of 3 results.

A351651 a(n) is the quotient obtained when digsum(m^2) is divided by digsum(m), with digsum = sum of digits = A007953 and m = A351650(n).

Original entry on oeis.org

1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 3, 2, 3, 4, 3, 2, 3, 1, 1, 2, 1, 3, 2, 2, 2, 1, 2, 1, 1, 2, 3, 4, 2, 2, 3, 4, 5, 3, 3, 4, 5, 3, 3, 2, 4, 3, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 3, 4, 3, 3, 2, 3, 4, 5, 3, 2, 4, 5, 2, 2, 3, 3, 3, 3, 2, 2, 2, 3, 2, 1, 3, 4, 3, 4, 5
Offset: 1

Views

Author

Bernard Schott, Feb 16 2022

Keywords

Comments

All positive integers are terms of this sequence (see A280012).
a(n) = 1 iff m = A351650(n) is a term of A058369 \ {0}.
a(n) = digsum(n) if m = A351650(n) is a term of A061909 \ {0}.

Examples

			A351650(8) = 13, then digsum(13) = 1+3 = 4 while digsum(13^2) = digsum(169) = 1+6+9 = 16; hence, a(8) = 16/4 = 4.

Links

Crossrefs

Cf. A002283, A004159, A007953, A058369, A061909, A254066, A280012, A351650.

Programs

  • Mathematica
    Select[Total[IntegerDigits[#^2]]/Total[IntegerDigits[#]]& /@ Range[300], IntegerQ] (* Amiram Eldar, Feb 16 2022 *)
  • PARI
    lista(nn) = {my(list = List(), q); for (n=1, nn, if (denominator(q=sumdigits(n^2)/sumdigits(n))==1, listput(list, q));); Vec(list);} \\ Michel Marcus, Feb 16 2022

Formula

a(n) = A004159(A351650(n)) / A007953(A351650(n)).

Extensions

More terms from Michel Marcus, Feb 16 2022

A352084 Integers m such that wt(m) divides wt(m^2) where wt(m) = A000120(m) is the binary weight of m.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 21, 24, 28, 30, 31, 32, 37, 42, 45, 48, 53, 56, 60, 62, 63, 64, 69, 73, 74, 79, 81, 83, 84, 90, 91, 96, 106, 112, 120, 124, 126, 127, 128, 133, 137, 138, 141, 146, 148, 155, 157, 158, 159, 161, 162, 165, 166, 168, 177, 180
Offset: 1

Views

Author

Bernard Schott, Mar 03 2022

Keywords

Comments

Integers m such that A000120(m) divides A159918(m).
This is a problem proposed by the French site Diophante in the links section.
The first 18 terms are the same as A268415, then A268415(19) = 41 while a(19) = 42.
The corresponding quotients are in A352085.
The smallest term k such that the corresponding quotient = n is A352086(n).
Some subsequences:
-> wt(m^2) = wt(m) iff m is in A077436.
-> wt(m^2) / wt(m) = 2 iff m is in A083567.
-> When m is a power of 2 (A000079): wt(2^k) = wt((2^k)^2) = wt(2^(2k)) = 1.

Examples

			37_10 = 100101_2, digsum_2(37) = 1+1+1 = 3; then 37^2 = 1369_10 = 10101011001_2, digsum_2(1369) = 1+1+1+1+1+1 = 6; as 3 divides 6, 37 is a term.

Links

Crossrefs

Cf. A000120, A159918, A268415, A351650, A352085, A352086.
Cf. A351650 (similar for base 10).
Subsequences: A000079, A023758, A077436, A083567.

Programs

  • Mathematica
    Select[Range[180], Divisible[Total[IntegerDigits[#^2, 2]], Total[IntegerDigits[#, 2]]] &] (* Amiram Eldar, Mar 03 2022 *)
  • PARI
    isok(m) = !(hammingweight(m^2) % hammingweight(m)); \\ Michel Marcus, Mar 03 2022
  • Python
    def ok(n): return n > 0 and bin(n**2).count('1')%bin(n).count('1') == 0
print([m for m in range(1, 200) if ok(m)]) # Michael S. Branicky, Mar 03 2022

Extensions

More terms from Amiram Eldar, Mar 03 2022

A351807 Integers m such that pod(m) divides pod(m^2) where pod = product of digits = A007954.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 12, 13, 15, 16, 18, 19, 21, 22, 23, 25, 26, 27, 28, 31, 32, 33, 36, 41, 42, 43, 45, 47, 48, 49, 51, 52, 53, 55, 61, 62, 63, 64, 66, 68, 71, 74, 76, 78, 82, 83, 84, 93, 94, 95, 96, 97, 98, 99, 111, 112, 113, 114, 115, 116, 118, 121, 122, 123
Offset: 1

Views

Author

Bernard Schott, Feb 19 2022

Keywords

Comments

Inspired by A351650 where pod is replaced by sod.
All terms are zeroless (A052382).
Repunits form a subsequence (A002275).
Integers m without 0 and such that m^2 has a 0 form a subsequence (A134844).
The smallest term k such that the corresponding quotient = n is A351809(n).

Examples

			Product of digits of 27 = 2*7 = 14; then 27^2 = 729, product of digits of 729 = 7*2*9 = 81; as 81 divides 729, 27 is a term.

Crossrefs

Cf. A007954, A002473, A351808 (corresponding quotients), A351809.
Subsequences: A002275, A134844.

Programs

  • Mathematica
    pod[n_] := Times @@ IntegerDigits[n]; Select[Range[120], FreeQ[IntegerDigits[#], 0] && Divisible[pod[#^2], pod[#]] &] (* Amiram Eldar, Feb 19 2022 *)
  • PARI
    isok(m) = my(d=digits(m)); vecmin(d) && denominator(vecprod(digits(m^2))/vecprod(d)) == 1; \\ Michel Marcus, Feb 19 2022
  • Python
    from math import prod
def pod(n): return prod(map(int, str(n)))
def ok(m): pdm = pod(m); return pdm > 0 and pod(m*m)%pdm == 0
print([m for m in range(124) if ok(m)]) # Michael S. Branicky, Feb 19 2022

Extensions

More terms from Amiram Eldar, Feb 19 2022
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.