A351895 -id:A351895 - cp's OEIS frontend

A373505 Numbers k such that k and k+1 both have an equal number of odd and even digits in their factorial-base representations.

25, 29, 37, 41, 55, 67, 73, 77, 85, 89, 103, 115, 727, 739, 745, 749, 757, 761, 775, 787, 793, 797, 805, 809, 823, 835, 841, 845, 853, 857, 889, 893, 901, 905, 937, 941, 949, 953, 967, 979, 985, 989, 997, 1001, 1015, 1027, 1033, 1037, 1045, 1049, 1063, 1075, 1081

Offset: 1

Amiram Eldar, Jun 07 2024

If m is the sum of the first k odd-indexed factorial numbers (A000142), for k >= 2, then m-1 is a term, since the factorial-base representation of m is 1010...10, with the block "10" repeated k times, and the factorial-base representation of m-1 is the 1010...1001, with the block "10" repeated k-1 times and followed by "01" (these numbers are 25, 745, 41065, 3669865, 482671465, ...).

			25 is a term since the factorial-base representations of 25 and 26 are 1001 and 1010, respectively, and both have 2 odd digits and 2 even digits.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation..

Subsequence of A351895.
Cf. A000142, A007623.
Similar sequences: A337238, A373460.

With[{max = 7}, fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; s = Select[Range[1, max!], EvenQ[Length[(d = fctBaseDigits[#])]] && Count[d, _?EvenQ] == Length[d]/2 &]; ind = Position[Differences[s], 1] // Flatten; s[[ind]]]

iseq(n) = {my(p = 2, o = 0, e = 0); while(n > 0, if((n%p) %2  == 0, e++, o++); n \= p; p++); e == o;}
lista(kmax) = {my(q1 = 0, q2); for(k = 1, kmax, q2 = iseq(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A351896 Numbers k such that k and k+2 both have an equal number of odd and even digits in their factorial-base representations.

Original entry on oeis.org

71, 743, 791, 839, 862, 910, 983, 1031, 1079, 1102, 1150, 1223, 1271, 1319, 1342, 1390, 1583, 1631, 1823, 1871, 2063, 2111, 2183, 2231, 2279, 2302, 2350, 2423, 2471, 2519, 2542, 2590, 2663, 2711, 2759, 2782, 2830, 3023, 3071, 3263, 3311, 3503, 3551, 3623, 3671, 3719

Offset: 1

Amiram Eldar, Feb 24 2022

			71 is a term since the factorial-base representations of 71 and 73 are 2321 and 3001, respectively, and both have 2 odd digits and 2 even digits.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Subsequence of A351895.
Cf. A007623, A351893, A351894.
Similar sequence: A337238.

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; s = Select[Range[1, max!], EvenQ[Length[(d = fctBaseDigits[#])]] && Count[d, _?EvenQ] == Length[d]/2 &]; ind = Position[Differences[s], 2] // Flatten; s[[ind]]

A363243 Numbers with an equal number of odd and even digits in their primorial-base representation.

Original entry on oeis.org

2, 5, 31, 32, 35, 36, 40, 43, 44, 47, 48, 52, 55, 56, 59, 63, 67, 68, 71, 75, 79, 80, 83, 87, 91, 92, 95, 96, 100, 103, 104, 107, 108, 112, 115, 116, 119, 123, 127, 128, 131, 135, 139, 140, 143, 147, 151, 152, 155, 156, 160, 163, 164, 167, 168, 172, 175, 176, 179

Offset: 1

Amiram Eldar, May 23 2023

The sum of the first k odd-indexed primorial numbers (A002110) is a term, since its primorial-base representation is 1010...10, with the block "10" repeated k times (these numbers are 2, 32, 2342, 512852, 223605722, ...).

			5 is a term since its primorial-base representation, 21, has one odd digit, 1, and one even digit, 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base.

Cf. A002110, A049345.

Similar sequences: A031443 (binary), A227870 (decimal), A351895 (factorial base).

With[{max = 5}, bases = Prime@ Range[max, 1, -1]; nmax = Times @@ bases - 1; prmBaseDigits[n_] := IntegerDigits[n, MixedRadix[bases]]; Select[Range[nmax], EvenQ[Length[(d = prmBaseDigits[#])]] && Count[d, _?EvenQ] == Length[d]/2 &]]

is(n) = {my(p = 2, o = 0, e = 0); if(n < 1, return(0)); while(n > 0, if((n%p)%2 == 0, e++, o++); n \= p; p = nextprime(p+1)); e == o;}

cp's OEIS Frontend

A373505 Numbers k such that k and k+1 both have an equal number of odd and even digits in their factorial-base representations.

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A351896 Numbers k such that k and k+2 both have an equal number of odd and even digits in their factorial-base representations.

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A363243 Numbers with an equal number of odd and even digits in their primorial-base representation.

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Author

Keywords

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Examples

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Crossrefs

Programs

Mathematica

PARI